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+{"text":"\\section{Introduction}\n\nThe goal of this paper is the computation of correlation functions in the six-vertex model directly by Markov chain Monte-Carlo simulations. This model was introduced by Pauling~\\cite{Pauling} who proposed it to describe the crystal where the oxygen groups form a square lattice with a hydrogen atom between each pair of lattice sites.\nHe proposed the ice rule: each lattice site has two hydrogen atoms close to it and two further apart; see a recent historical review~\\cite{Harris}.\nAnother crystal with such a structure is the potassium dihydrogen phosphate.\nSlater was the first who suggested that the two dimensional case, known as the six-vertex model, is important \nto understand universal thermodynamic properties of these structures~\\cite{Slater}. \nThe states in this model are configurations of arrows on edges which satisfy the ice rules, see Fig.~\\ref{6v}.\nAn arrow indicates to which of two sites (the oxygen atoms) the hydrogen atom (which is approximately in the middle of an edge) is closer.\nEquivalently, the configurations of arrows can be regarded as configurations of lattice paths such that paths may meet at a vertex, \nturn or pass, as it is shown in Fig.~\\ref{6v}. Boltzmann weight of a configuration is the product of Boltzmann weights assigned to vertices. The weight of a vertex depends on the configurations of paths on adjacent edges,\nsee Fig. \\ref{6v}. The probability of state $\\sigma$ is\n\\[\nProb(\\sigma)=\\frac{1}{Z} \\, w(\\sigma),\n\\]\nwhere $w(\\sigma)=\\prod_v w_v(\\sigma)$ is the weight of  state $\\sigma$, $w_v(\\sigma)$ is the weight of the vertex $v$\nin the state $\\sigma$, and $Z=\\sum_\\sigma w(\\sigma)$ is the partition function. \n\nLocally, lattice paths of the six-vertex model on a planar lattice can be regarded as level curves of a step function\ndefined on faces~\\cite{R2010}. We assume it is increasing when we move to the right and up. This integer valued function is called \nthe {\\it height function} $\\chi(n,m)$. It is a random variable with values in integers $\\mathbb Z$ with the probability distribution given by Boltzmann weights described above.\nOn a planar simply connected lattice domain there is a bijection between configurations of paths with fixed positions on the boundary and height functions with corresponding boundary values\\footnote{We assume that the value of a height function is fixed at some reference point on the domain.}. \n\n\\begin{figure}[ht!]\n\\begin{center}\n\\includegraphics[width=1.0\\textwidth, angle=0.0]{figure01.pdf}\n\\end{center}\n\\caption{\\label{6v} Local configurations and weights of the six-vertex model. For the symmetric model, $\\tau=\\tau_{1}=\\tau_{2}$, where $\\tau=a,b,c$.}\n\\end{figure}\n\nThe first breakthrough in the study of the model came in works of Lieb, Yang, Sutherland and others\nwhere the Bethe ansatz was used for finding the spectrum of transfer matrices with periodic boundary conditions~\\cite{Lieb,Yang,Sutherland}.\nThen, came works of Baxter where the role of commuting transfer matrices became clear and the partition function of the eight-vertex model was obtained~\\cite{BaxterPaper}, see~\\cite{Baxter} for an overview of these developments.\nThen, many important algebraic structures came in the framework of the algebraic Bethe ansatz and the quantum inverse scattering method~\\cite{FT}, for an overview see~\\cite{BIK,BM,AGP}.\nIn the last decade, substantially better understanding of thermodynamic properties of the six-vertex model with domain wall boundary \nconditions (DW) on a square lattice has been achieved.\nThese boundary conditions correspond to paths coming through each edge on the top side of the square and leaving through edges on the right side only.\nThese particular boundary conditions are quite remarkable because the partition function in this case can be written as a determinant~\\cite{Izergin,BIK,PronkoTMF} as well as because of the relation to \nthe alternating sign matrices~\\cite{Ku}.\n\nIn the large volume limit (the thermodynamic limit, $N\\to \\infty$), properly normalized height function converges, as a random variable, \nto a deterministic function $h_{0}(x,y): D=[0,1]\\times [0,1]\\to \\mathbb R$ known as the limit shape\nheight function.\nSuch a behavior is known as the limit shape phenomenon, see~\\cite{CP,Ok} for an overview.\nThis phenomenon is an analogue of the central limit theorem in probability theory.\nIt predicts the following behavior of the height function as $N\\to \\infty$\\footnote{This means the convergence of $\\chi(n,m)\/N\\to h_0\\left(\\frac{n}{N}, \\frac{m}{N}\\right)$ and $\\chi(n,m)-Nh_0\\left(\\frac{n}{N}, \\frac{m}{N}\\right)\\to \\phi\\left(\\frac{n}{N}, \\frac{m}{N}\\right)$ in probability.}:\n\\begin{equation}\n\\label{eq1chi}\n\\chi \\left(n,m \\right) \\to N h_0\\left( \\frac{n}{N}, \\frac{m}{N} \\right)+\\phi \\left(\\frac{n}{N}, \\frac{m}{N} \\right).\n\\end{equation}\nHere $h_0(x,y)$  is the limit shape height function which can be computed using the variational principle.\nThe variational principle was developed and proved for dimer models in~\\cite{KOS,Ok}. It was adopted to the six-vertex model in~\\cite{ZJ,PR}.\nThe random variable $\\phi(x,y)$ is a free Gaussian quantum field in the Euclidean space time with the metric determined by the height function $h(x,y)$, see for example~\\cite{ZJ,GorinLectures}.\nFor generic values of parameters in the six-vertex model, mathematically, the variational principle is still a hypothesis~\\cite{PR,GBDJ}. It is proven in some special cases of stochastic weights in~\\cite{B-stoch} and\nit follows from~\\cite{CKP} for the free fermionic case $\\Delta=0$. \n\nAn important characteristic of the (symmetric) six-vertex model is the parameter~\\cite{BaxterPaper}\n\\begin{equation}\n\\label{delta}\n\\Delta=\\frac{a^2+b^2-c^2}{2ab}.\n\\end{equation}\nWhen $\\Delta=0$, the model can be mapped to a dimer model and the partition function and correlation functions\ncan be computed in terms of the determinant and the minors of the Kasteleyn matrix~\\cite{Ka}, respectively. \n\nIt has already been shown earlier~\\cite{R2005,AS} how to use configurations generated by a Markov chain Monte-Carlo simulations for calculating the limit shape of the height function of the six-vertex model with DW boundary conditions.\nIn this paper, we show how, based on the generated configurations, to compute numerically the two-point correlation function.\nWhen $\\Delta=0$ both the limit shape height function and the correlation functions are known from the exact solution because in this case the six-vertex can be mapped to a dimer model on a modified (decorated) square lattice, details can be found in~\\cite{PR,RS}. We demonstrate that in this case the usual averaging over time in Markov process gives an excellent agreement of numerical results with the exact ones.\n\nAfter that, we apply the same algorithm for other values of $\\Delta$.\nWhen $|\\Delta|\\leq 1$ the model is critical, i.e. the Gaussian field $\\phi(x,y)$ is a massless \nfield on the space time with the metric induced by the limit shape. The numerics confirms that \ncorrelation functions are conformal at short distances. Of course, we should not expect conformal invariance at all distances\nfor $\\Delta$ other than zero. \n\nWhen $\\Delta<-1$ an antiferroelectric diamond shape droplet forms\nin the middle of the limit shape. Because the antiferroelectric ground state is double degenerate,\nthe Markov process gets stuck in one of the ground states for exponentially long time. \nNumerical estimations for this case are given in the last section.\n\nThe paper is organized as follows. In section~\\ref{exact}, we outline a derivation of the\ntwo-point correlation function from the exact solution of the six-vertex model for $\\Delta=0$.\nIn section~\\ref{num}, we demonstrate the results of numerical Monte-Carlo simulations and comparisons with the exact solution.\nIn the appendices, we provide the technical details to specify the model to obtain the exact solution.\n \n{\\bf Acknowledgements.} We would like to thank A.~G.~Pronko for discussions and for sharing a draft of the manuscript~\\cite{Pronko},\nD. Keating and A. Sridhar for numerous discussions and the latest version of the Monte-Carlo code.\nWe also benefited from discussions with A.~A.~Nazarov.\nThe work of N.~Yu.~Reshetikhin was partly supported by the\nNSF grant DMS-1902226 and the RSF grant 18-11-00297.\nP.~A.~Belov is grateful to the Russian Science Foundation, grant no. 18-11-00297, for the financial support.\nThe calculations were carried out using the facilities of the SPbU Resource Center ``Computational Center of SPbU''.\n\n\\section{Correlation functions in the six-vertex model at the free fermionic point}\\label{exact}\n\n\\subsection{The free fermionic point of the six-vertex model and mapping to dimers}\\label{LSh}\nIn this paper, we focus on the symmetric six-vertex model with weights $a_{1}=a_{2}=a$, $b_{1}=b_{2}=b$, $c_{1}=c_{2}=c$.\nThe parameter $\\Delta$, Eq.~(\\ref{delta}), is an important characteristic of the model.\nIt determines the phases of the model on the $M\\times N$ torus when $M,\\, N \\to \\infty$.\nFor simplicity, in the following we assume that $M=N$.\nWhen $\\Delta>1$ the model develops a ferroelectric, totally ordered phase.\nFor $|\\Delta|<1$ it develops a disordered phase and for $\\Delta<-1$ it transitions to an antiferroelectric phase.\nWhen $\\Delta=\\pm 1$ the model undergoes phase transitions (in parameter $\\Delta$).\n\nWhen the weights of the six-vertex model satisfy the condition $\\Delta=0$\nthe six-vertex model can be mapped to the dimer model on a modified lattice, see for example\n\\cite{Wu,RS}.\nThe partition function of the dimer model is the sum of Pfaffians (the number of terms is determined by the topology of the lattice)~\\cite{Ka,FisherPR,McCoyWu}.\nEach Pfaffian can be regarded as the Gaussian Grassmann integral.\nBecause of this and because it implies that the multipoint correlators can be expressed as Pfaffians of the two-point correlation functions, the case of $\\Delta=0$ is also known as a free fermionic point of the six-vertex model.\n\nBecause the weights can be multiplied by an overall constant factor without changing the probability measure, we can put $c=1$.\nThen, we can parametrize weights $a$ and $b$ as\n$$\na=\\cos{(u)}, \\enskip b=\\sin{(u)}.\n$$\nThis is a particular case of Baxter's parametrization of \nweights of the six-vertex model \\cite{Baxter}. \nNote that the mapping $a\\mapsto b, \\ \\ b\\mapsto a$ is a symmetry of the \nprobability measure, see Appendix \\ref{Sym} for details. This is why we can assume,\nwithout loosing generality, that $b\/a\\leq 1$. \n\n\\subsection{The variational principle}\nHere we will recall the variational principle for deriving the limit shape.\nLet $\\sigma(s,t)$ be the free energy per site for the six-vertex model on a torus with $s$ and $t$ being fixed\ndensities of edges occupied vertical and horizontal paths respectively.\n\nThe limit shape height function $h_0(x,y)$ for the six-vertex model with DW boundary conditions is a real valued \nfunction on $\\mathcal D=[0,1]\\times [0,1]$ which minimizes the large deviation rate functional\n\\begin{equation}\\label{LDf}\nS[h]=-{\\int \\int}_{\\mathcal D} \\sigma(\\pa_xh,\\pa_yh) \\, dx \\, dy\n\\end{equation}\nin the space of functions with boundary conditions $h(0,y)=h(x,0)=0, \\ \\ h(1,y)=y, \\ \\ h(x,1)=x$ and the constraints\n\\[\n|h(x,y)-h(x',y)|\\leq |x-x'|,  \\ \\ |h(x,y)-h(x,y')|\\leq |y-y'|.\n\\]\nFor dimer models it follows from~\\cite{CKP}.\n\nThe critical value $S[h_{0}]$ is the minus free energy of the model.\nIf $Z_{N}$ is the partition function of the six-vertex model with DW boundary conditions, then\n$$\nS[h_{0}] = \\lim_{N\\to\\infty} \\frac{1}{N^{2}} \\ln Z_{N}.\n$$\n\nThe six-vertex model at the free fermionic point~($\\Delta=0$) can be mapped to the dimer model \non a decorated square lattice, see for example~\\cite{RS} and references therein.\nAs it was already stated earlier, the corresponding dimer model can be solved by the Pfaffian method.\nThis method gives the formula for the partition function of the model as a Pfaffian (or a determinant) of certain $N\\times N$\nmatrix, called the Kasteleyn matrix~\\cite{Ka}.\nIn this case $\\sigma(s,t)$ can be computed explicitly as the Legendre transform of the free energy $f(H,V)$\nof the six-vertex model on a torus (with $\\Delta=0$) in the presence of electric fields $H$ and $V$:\n$$\n\\sigma(s,t)=\\min_{H,V} \\left( Hs + Vt - f(H,V) \\right).\n$$\nThe function $f(H,V)$ is given by the double integral\n\\[\nf(H,V)=\\frac{1}{(2\\pi i)^2}\\int_{|z|=\\exp(H)}\\int_{|w|=\\exp(V)} \\ln|P(z,w)| \\frac{dz}{z}\\frac{dw}{w},\n\\]\nwhere\n\\begin{equation}\n\\label{SpectralCurve}\nP(z,w)=a(wz-1)+b(z+w)\n\\end{equation}\nis the spectral polynomial of the Kasteleyn matrix, see \\cite{Ka,McCoyWu,KOS}. \nNote that $f(H,V)$ is convex, $\\text{det}(\\partial_{i}\\partial_{j} f)>0$, and $\\sigma(s,t)$ is concave, $\\text{det}(\\partial_{i}\\partial_{j} \\sigma)<0$.\n\nEuler-Lagrange equations for the large deviation rate functional (\\ref{LDf}) can be written explicitly as follows (see \\cite{KOS,KO} for details).\nConsider complex valued \nfunctions $z(x,y)$ and $w(x,y)$ such that \n\\begin{equation}\\label{arg}\narg(z(x,y))=\\pi \\pa_x h(x,y), \\ \\ arg(w(x,y))=-\\pi \\pa_y h(x,y).\n\\end{equation}\nThen the Euler-Lagrange equations for $h(x,y)$ can be written as a system of \nequations for $z(x,y)$ and $w(x,y)$ as\n\\begin{equation}\\label{DEhf}\n\\pa_y \\log (z)+\\pa_x \\log(w)=0, \\ \\  P(z,w)=0.\n\\end{equation}\n\n\\begin{figure}[t!]\n\\begin{center}\n\\includegraphics[width=1.0\\textwidth, angle=0.0, scale=0.6]{figure02.pdf}\n\\end{center}\n\\caption{\\label{Ell-1} The $xy$ plane with the denoted domains. The ellipse is the boundary of the limit shape. The height function smooth\ninside the ellipse and linear outside. Here, we also show the lines $n_x=0$, $n_y=0$ and areas $A$, $B$, $C$, and $D$. }\n\\end{figure}\n\n\\subsection{The limit shape}\nHere we describe the limit shape height function $h_0(x,y)$ for DW boundary \nconditions. We use the result of \\cite{Pronko} where the density of the horizontal edges \noccupied by the paths is derived.\n\nDefine the function $D(x,y)$ as\n$$\nD(x,y) = \\alpha (1-\\alpha) \\left[ \\frac{(y-x)^{2}}{\\alpha}+\\frac{(1-x-y)^{2}}{1-\\alpha}-1 \\right].\n$$\nHere, the parameter $\\alpha$ is determined by values of Boltzmann weights of the model as\n$$\n\\alpha=\\frac{b}{a}.\n$$\nIn Baxter's parametrization $ \\alpha=\\tan{(u)}$.\nDefine the region $E= \\{ (x,y) | D(x,y) \\le 0 \\}$ as the interior of the ellipse  $\\partial E= \\{ (x,y) | D(x,y)= 0 \\}$\nwhich is inscribed  in the square $0\\leq x \\leq 1, 0\\leq y \\leq 1$ as it is shown in Fig. \\ref{Ell-1}. The \nellipse is the boundary of the limit shape, or the ``arctic curve''~\\cite{Pronko2011}.\n\nThe following expression was derived in \\cite{Pronko}: \n\\begin{equation}\\label{DyHF}\n\\partial_{y} h_0(x,y) = \\left\\{\n  \\begin{array}{cc}\n    \\frac{1}{\\pi} \\text{arccot} \\left( \\frac{-n_{y}}{\\sqrt{-D(x,y)}} \\right), &  (x,y) \\in E \\\\\n    0, & (x,y) \\in A \\cup B, \\enskip n_{y}<0 \\\\\n    1, & (x,y) \\in C \\cup D, \\enskip n_{y}>0 \\\\\n  \\end{array}\n\\right.\n\\end{equation}\nRegions $A,B,C,D$ are shown in Fig. \\ref{Ell-1}\nHere $n_{y}=(1-\\alpha)(y-x)+\\alpha (1-x-y)=x+(2\\alpha-1) y -\\alpha$.\n\nIntegrating this expression, we obtain the following formula for \nthe limit shape height function itself:\n\\begin{equation}\\label{HF}\nh_0(x,y) = \\left\\{\n  \\begin{array}{cc}\n    \\frac{1}{\\pi} \\left( y \\, \\text{arccot} \\left[ \\frac{-n_{y}}{\\sqrt{-D(x,y)}} \\right] -\\frac{1}{2} \\, \\text{arctan}\\left[\\frac{-x^2+(y-\\alpha)(\\alpha-1)+x(1+y-2y\\alpha)}{(1-x-\\alpha)\\sqrt{-D(x,y)}} \\right] + \\right. & \\\\\n \\left. +(\\frac{1}{2}-x) \\, \\text{arctan} \\left[ \\frac{-n_{x}}{\\sqrt{-D(x,y)}} \\right] \\right) +\\frac{x}{2} & (x,y) \\in E \\mbox{ }\\& \\mbox{ } x<1-\\alpha \\\\\n\\frac{1}{\\pi} \\left( y \\, \\text{arccot} \\left[ \\frac{-n_{y}}{\\sqrt{-D(x,y)}} \\right] -\\frac{1}{2} \\, \\text{arctan}\\left[\\frac{-x^2+(y-\\alpha)(\\alpha-1)+x(1+y-2y\\alpha)}{(1-x-\\alpha)\\sqrt{-D(x,y)}} \\right] + \\right. & \\\\\n \\left. +(\\frac{1}{2}-x) \\, \\text{arctan} \\left[ \\frac{-n_{x}}{\\sqrt{-D(x,y)}} \\right] \\right) +\\frac{x}{2}-\\frac{1}{2} & (x,y) \\in E \\mbox{ }\\& \\mbox{ } x \\ge 1-\\alpha \\\\\n    0 & (x,y) \\in A \\\\\n    x & (x,y) \\in B \\\\\n    x+y-1 & (x,y) \\in C \\\\\n    y & (x,y) \\in D\n  \\end{array}\n\\right.\n\\end{equation}\nHere $n_{x}=(1-\\alpha)(y-x)+\\alpha (x+y-1)=y+(2\\alpha-1) x -\\alpha$.\n\\begin{figure}[b!]\n\\begin{center}\n\\vspace{1mm}\\includegraphics[width=0.4\\textwidth, angle=0.0]{figure03.pdf}\n\\end{center}\n\\caption{\\label{f1} The density of horizontal edges occupied by paths, or $\\partial_{y} h_0(x,y)$, for $\\alpha=9\/25$.}\n\\end{figure}\n\nDifferentiating this expression in $x$, we obtain the density of edges occupied with horizontal paths:\n\\begin{equation}\n\\label{partialX}\n\\partial_{x} h_0(x,y) = \\left\\{\n  \\begin{array}{cc}\n    -\\frac{1}{\\pi} \\text{arctan} \\left( \\frac{-n_{x}}{\\sqrt{-D(x,y)}} \\right)+\\frac{1}{2}, &  (x,y) \\in E \\\\\n    0, & (x,y) \\in A \\cup D, \\enskip n_{x}<0 \\\\\n    1, & (x,y) \\in B \\cup C, \\enskip n_{x}>0 \\\\\n  \\end{array}\n\\right.\n\\end{equation}\n\nHere we use the branch of the function $\\text{arctan}$ which behaves as\n\\begin{equation}\n-\\frac{1}{\\pi} \\text{arctan} \\left( \\frac{-n_{x}}{\\sqrt{-D(x,y)}} \\right) \\to  \\left\\{\n  \\begin{array}{cc}\n    -\\frac{1}{2}, & n_{x}<0, \\enskip (x,y) \\in A \\cup D \\\\\n    \\frac{1}{2}, & n_{x}>0, \\enskip (x,y) \\in B \\cup C \\\\\n  \\end{array}\n\\right.\n\\end{equation}\nwhen $(x,y)$ approach to the boundary of $E$.\n\nAs an example, in Fig.~\\ref{f1} we show the partial derivative of the height function~(\\ref{HF}) for $\\alpha=9\/25$.\nInside the arctic curve, it is given by the nontrivial part of Eq.~(\\ref{DyHF}) and outside that one, it equals to zero or one. \nThe limit shape height function $h_{0}(x,y)$, Eq.~(\\ref{HF}), is shown in Fig.~\\ref{f2}.\n\\begin{figure}[t!]\n\\begin{center}\n\\vspace{1mm}\\includegraphics[width=0.4\\textwidth, angle=0.0]{figure04.pdf}\n\\end{center}\n\\caption{\\label{f2} The limit shape height function $h_{0}(x,y)$, Eq.~(\\ref{HF}), for $\\alpha=9\/25$.}\n\\end{figure}\n\n\\subsection{The function $z(x,y)$}\nAn important property of functions $z(x,y)$ and $w(x,y)$ is that $z$ maps \nthe inner part of the ellipse $D(x,y)=0$ (the arctic curve) to the upper half plane.\nIndeed, as we saw in the previous section, the $x$-derivative of the height function~(\\ref{partialX})\nis non-negative when $0\\leq x\\leq 1$ and, therefore, the imaginary part of the function $z(x,y)$\nis also non-negative.\n\nIn our case we already know the height function, so in order to find functions $z$ and $w$\nit is sufficient to solve the algebraic equation in (\\ref{DEhf}). \nMoreover, since we know the height function, we know the arguments of $z$ and $w$, therefore, we just have to solve\nthe equation $P(z,w)=0$ for absolute values of $z$ and $w$. This will give \nthe conformal mapping $z$ from the interior $E$ of our ellipse to the \nupper half of the complex plane.\n\nSolving the quadratic equation $P(z,w)=0$ for the absolute values of $z$ and $w$ and taking into account (\\ref{arg}),\nwe obtain:\n\\begin{eqnarray}\\label{zw}\n\\nonumber |z|=\\frac{1}{2 \\, a \\, b \\, \\sin[-\\pi \\pa_yh_0]}\\Bigg(a^2 \\sin[ \\pi (\\pa_xh_0-\\pa_yh_0) ]+b^2 \\sin[ \\pi (\\pa_xh_0+\\pa_yh_0)] \\\\\n\\hspace{1cm}\\mp \\sqrt{4 a^2 b^2 (\\sin[-\\pi \\pa_yh_0])^2+\\bigg(a^2 \\sin[\\pi (\\pa_xh_0-\\pa_yh_0)]+b^2 \\sin[\\pi (\\pa_xh_0+\\pa_yh_0)]\\bigg)^2}\\Bigg),\\label{z} \\\\\n\\nonumber |w|=\\frac{1}{2 \\, a \\, b \\, \\sin[\\pi \\pa_xh_{0}]}\\Bigg(a^2 \\sin[\\pi (\\pa_xh_0-\\pa_yh_0)]+b^2 \\sin[\\pi (\\pa_xh_0+\\pa_yh_0)] \\\\\n\\hspace{1cm}\\pm \\sqrt{4 a^2 b^2 (\\sin[-\\pi \\pa_yh_{0}])^2+\\bigg(a^2 \\sin[ \\pi (\\pa_xh_0-\\pa_yh_0)]+b^2 \\sin[ \\pi (\\pa_xh_0+\\pa_yh_0)])\\bigg)^2}\\Bigg) \\label{w}.\n\\end{eqnarray}\n\nWe almost constructed the mapping $z: E \\to H=\\{ z | Re(z) \\geq 0 \\},  \\pa E \\to \\mathbb R$. The last step is to determine the signs in (\\ref{z}).\nIn the Appendix~\\ref{A1}, we determine the signs and the mapping.\nIt maps the boundary of $E$ bijectively to the real line in the following way:\n\\begin{itemize}\n\\item $z: A\\cap \\pa E\\to (0, \\frac{a}{b})$, $A\\cap B\\mapsto 0$, $A\\cap D\\mapsto \\frac{a}{b}$\n\\item $z: B\\cap \\pa E\\to (-\\frac{b}{a}, 0)$, $B\\cap C\\mapsto -\\frac{b}{a}$, $B\\cap A\\mapsto 0$\n\\item $z: C\\cap \\pa E\\to (-\\infty, -\\frac{b}{a})$, $C\\cap D|_{C}\\mapsto -\\infty$, $C\\cap B\\mapsto -\\frac{b}{a}$\n\\item $z: D\\cap \\pa E\\to (\\frac{a}{b},\\infty)$, $D\\cap A\\mapsto \\frac{a}{b}$, $D\\cap C|_{D}\\mapsto \\infty$\n\\end{itemize}\n\n\\subsection{The two-point correlation function}\nIn the continuum limit, the fluctuations of the height function are described\nby the massless Euclidean quantum Bose field in the interior of the arctic curve with the metric determined\nby the second variation $S^{(2)}$ of the large deviation rate functional (\\ref{LDf}) computed at the limit shape height function.\nIt reads\n$$\nS^{(2)}[h_{0}]= \\frac{1}{2} \\iint_{\\mathcal D} \\left( \\partial_{1}^{2} \\sigma(\\vec{\\nabla} h_{0}) (\\partial_{x} \\phi)^{2}+\n2 \\partial_{1} \\partial_{2} \\sigma(\\vec{\\nabla} h_{0}) \\partial_{x} \\phi \\, \\partial_{y} \\phi +\n\\partial_{2}^{2} \\sigma(\\vec{\\nabla} h_{0}) (\\partial_{y} \\phi)^{2} \\right) \\, dx \\, dy.\n$$\n\nThe mapping $z$ brings the functional $S^{(2)}$ with the kernel defined on functions on the interior of $E$ to \nthe Dirichlet functional for the Laplace operator acting on functions on the upper half of the complex plane. \nThis defines the two-point correlation function for fluctuations of the height function on $E$\nas the Green's function for the Laplace operator on the upper half plane with Dirichlet boundary \nconditions on the real line:\n\\begin{equation}\n\\label{Corr2Pi}\n\\left \\langle \\phi(x_{1},y_{1}),\\phi(x_{2},y_{2}) \\right \\rangle = -\\frac{1}{2\\pi} \\ln \\Bigg| \\frac{z(x_{1},y_{1})-z(x_{2},y_{2})}{z(x_{1},y_{1})- \\overline{z(x_{2},y_{2})}} \\Bigg|.\n\\end{equation}\nHere, $\\phi$ is the fluctuation field from (\\ref{eq1chi}).\nThe formula (\\ref{Corr2Pi}) means that the two-point correlation function at the free fermionic point~($\\Delta=0$) has a logarithmic dependence on the distance between points $(x_{1},y_{1})$ and $(x_{2},y_{2})$, when this distance is small~\\cite{Kenyon}.\n\nFor free fermionic models, local correlation functions (multipoint correlation functions) $\\langle \\phi(\\vec{r}_{1}),\\ldots,\\phi(\\vec{r}_{n}) \\rangle$ of fluctuations of the height function are determined by the two-point correlation functions through the Wick's formula.\n\n\\begin{figure}[t!]\n\\begin{center}\n\\vspace{1mm}\\includegraphics[width=1.0\\textwidth, angle=0.0]{figure05.pdf}\n\\end{center}\n\\caption{\\label{thermalizationFig} The behavior of the normalized volume under the height function values as a function of a number of flips for three different lattice sizes and three values of $\\Delta$: $0$, $1\/2$, and $-7\/2$ of the six-vertex model.}\n\\end{figure}\n\n\\section{Numerical results}\\label{num}\n\n\\subsection{Computation of observables and the thermalization.}\n\nAs it was mentioned in the introduction, we use the Markov chain sampling algorithm to generate a sequence of random states of the six-vertex model~\\cite{R2005}.\nThis method is known as the Markov chain Monte-Carlo simulation.\nIt is based on the special choice of the transition probabilities to transfer from an arbitrary distribution to the desired one.\nIt is also known as the Metropolis algorithm~\\cite{Metropolis}.\nAn overview of these numerical methods and their applications to statistical mechanics can be found in~\\cite{LandauBinder2014}.\nSee~\\cite{Zvonarev,AS,Keesman,Korepin1,Korepin2} for related numerical simulations.\n\nThe idea of Markov sampling is to create a random process that will follow the most likely states in the model.\nThis is guaranteed by the choice of the matrix of transition probabilities which is symmetrizable (detailed balanced condition) by the diagonal matrix with entries given by Boltzmann weights of the system.\nThis condition (plus an assumption of nondegeneracy of the largest eigenvalue) \nalso guarantees the asymptotical convergence of the process to the Boltzmann distribution \nstarting from any distribution. Also, in this case the Boltzmann distribution is the Perron-Frobenius \neigenvector of the matrix of transition probabilities\\footnote{These are all standard facts about Markov processes, for details see\nfor example \\cite{LandauBinder2014,Seneta,Markov}.}.\n\nWhen a random process is constructed, the expectation values of observables with respect to the \nBoltzmann distribution can be computed by averaging along the random process. \nThis procedure is especially effective when the Boltzmann distribution is concentrated in a small neighborhood of\nthe most likely state (the limit shape). In probability theory this is known as large deviations, in non-equilibrium \nstatistical physics this is known as a hydrodynamic limit. \n\nIn dimer models it was proven rigorously~\\cite{CKP} \nthat there exists a most probable state and the probability for any other state to be ``macroscopically distant'' from\nit is exponentially suppressed:\n\\begin{equation}\\label{localization}\nProb(h)\\propto \\exp\\left[N^2(S[h_{0}]-S[h])\\right].\n\\end{equation}\nHere, $h_0$ is the height function corresponding to the limit shape~(\\ref{HF}). It minimizes the large deviation rate functional.\nThe minimal value $S[h_{0}]$ is exactly (minus) the free energy of the system. \n\nThe six-vertex model with $\\Delta=0$ is equivalent to a dimer model.\nTherefore, in this case we can use the probability distribution and results from the corresponding dimer model.\nFor other values of $\\Delta$ in the six-vertex model the analysis is more complicated, but we expect a similar \nstructure of the distribution, suggesting the formation of the limit shape $h_{0}$.\n\nThe localization (concentration) of random states near the limit shape makes the numerical computation of observables\neasy once the Markov process is thermalized i.e. when it moves along the states in a vicinity of the limit shape.\nThus, the main challenge for computing observables is to know when the process is thermalized.\nUnfortunately, it is very hard to have an effective criterium for thermalization.\nInstead, we use a simple empirical technique: we monitor the fluctuations of the normalized volume under the height\nfunction\n\\[\nvol(h)=\\frac{1}{N^3}\\sum_{(n,m)} h(n,m).\n\\]\nAs it is clear from Fig.~\\ref{thermalizationFig}, the normalized volume ``drifts'', when the process is not yet thermalized.\nThen it starts to fluctuate around the normalized volume under the limit shape $h_{0}$.\nThus, we can start measurements to compute observables using the Markov chain simulations.\nFor example, Fig.~\\ref{thermalizationFig} shows that for the lattice of size $90\\times 90$ and $\\Delta=1\/2$\nit is safe to start averaging after about $10^{7}$ flips~\\cite{R2005}.\n\\begin{figure}[t!]\n\\begin{center}\n\\begin{tabular}{cc}\n\\begin{minipage}{0.5\\linewidth}\n\\includegraphics[width=1.0\\textwidth, angle=0.0]{figure06.pdf}\n\\end{minipage}\n&\n\\begin{minipage}{0.5\\linewidth}\n\\includegraphics[width=1.0\\textwidth, angle=0.0]{figure07.pdf}\n\\end{minipage}\n\\end{tabular}\n\\end{center}\n\\caption{\\label{FigCompare4040} (\\textit{left plot}) The numerical height function for $\\Delta=0$ and for the lattice size of $40\\times 40$. The parameter $\\alpha=9\/25$. The height function is the result of averaging over measurements. (\\textit{right plot}) The difference between the theoretical limit shape height function and the one obtained from numerical simulation.}\n\\end{figure}\n\nOnce the thermalization is achieved, we compute an observable by time averaging:\n\\begin{equation}\\label{O}\n\\left \\langle O\\right \\rangle \\simeq \\frac{O(s_1)+\\dots +O(s_K)}{K}.\n\\end{equation}\nHere $s_i$  is a random state at time $T_i$ counting from the first measurement, $K$ is the total number of measurements.\nThe right side depends on random states $s_i$ and is a random variable, but as $K\\to \\infty$\nit converges to the Boltzmann expectation value.\nOf course, numerically $K\\to \\infty$ simply means large values.\nWe will use this to compute the limit shape $h_{0}$\nand correlation functions.\nIn particular, the two-point correlation function of points $(x_{i},y_{i})$ and $(x_{j},y_{j})$ is calculated as\n\\begin{equation}\\label{ncf}\n\\left \\langle\\phi(x_{i},y_{i}),\\phi(x_{j},y_{j})\\right \\rangle = \\left \\langle \\chi(x_{i},y_{i}) \\chi(x_{j},y_{j}) \\right \\rangle-\\left \\langle \\chi(x_{i},y_{i})\\right \\rangle \\left \\langle \\chi(x_{j},y_{j}) \\right \\rangle,\n\\end{equation}\nwhere \n\\begin{equation}\\label{nhf}\n\\left \\langle \\chi(x_{i},y_{i})\\right \\rangle = \\frac{1}{K}\\sum_{k=1}^K \\chi_k(x_{i},y_{i}),\n\\end{equation}\nthe height function $\\chi$ is from Eq.~(\\ref{eq1chi}),\nand indices $i,j=1,\\ldots,N$ numerate the lattice sites.\nHere $\\chi_{k}$ are random variables, but the sum represents a deterministic quantity as $K\\to\\infty$.\n\\begin{figure}[t!]\n\\begin{center}\n\\begin{tabular}{cc}\n\\begin{minipage}{0.5\\linewidth}\n\\includegraphics[width=1.0\\textwidth, angle=0.0]{figure08.pdf}\n\\end{minipage}\n&\n\\begin{minipage}{0.5\\linewidth}\n\\includegraphics[width=1.0\\textwidth, angle=0.0]{figure09.pdf}\n\\end{minipage} \\\\\n\\begin{minipage}{0.5\\linewidth}\n\\includegraphics[width=1.0\\textwidth, angle=0.0]{figure10.pdf}\n\\end{minipage}\n&\n\\begin{minipage}{0.5\\linewidth}\n\\includegraphics[width=1.0\\textwidth, angle=0.0]{figure11.pdf}\n\\end{minipage}\n\\end{tabular}\n\\end{center}\n\\caption{\\label{CorrCompareTheor406090} The plots of (a) the exact limit shape two-point correlation function inside the arctic curve for $\\Delta=0$, (b) the difference between the numerical correlation function on the square lattice of $60\\times 60$ and the exact one, the numerical ones on the square lattices of sizes (c) $40\\times 40$ and (d) $90\\times 90$. The parameter $\\alpha=9\/25$.}\n\\end{figure}\n\n\\subsection{Numerical computation of the limit shape and correlation functions at the free fermionic point.}\nWe start by comparison of the calculated height function with the exact one, the limit shape $h_{0}$ given by Eq.~(\\ref{HF}), for $\\Delta=0$ and $\\alpha=9\/25$.\nThe difference between the exact height function and the numerical one is shown in Fig.~\\ref{FigCompare4040}.\nIt should be noted that the numerical height function is smooth since it was averaged over a number of measurements, as described above.\nThe difference between $h_{0}$ and the numerical height function reveals the Airy asymptotic near the boundary of the limit shape.\nThe difference vanishes as the lattice size $N\\to \\infty$.\n\\begin{figure}[t!]\n\\begin{center}\n\\vspace{1mm}\\includegraphics[width=1.0\\textwidth, angle=0.0]{figure12.pdf}\n\\end{center}\n\\caption{\\label{DetailedCorrCompare6060} The values of the two-point correlation function along the slice $y=0.5$ for $\\Delta=0$. The curve shows the theoretical limit shape correlation function~(\\ref{Corr2Pi}). The points are the calculated values for different lattice sizes. The inset shows the values of numerically obtained coefficient against logarithm in Eq.~(\\ref{Corr2Pi}) with respect to the lattice size as well as a fit of these coefficients.}\n\\end{figure}\n\nThe results of computations of two-point correlation functions $\\left \\langle\\phi(x_{i},y_{i}),\\phi(x_{j},y_{j})\\right \\rangle$ are presented in Fig.~\\ref{CorrCompareTheor406090}.\nWe show plots of data~(\\ref{ncf}) for one point $(x_i,y_i)=(0.5,0.5)$ and another point $(x_j,y_j)$ running through the square $[0,1]\\times [0,1]$ of the lattice domain with step $1\/N$.\nThe plot (a) shows the theoretical limit shape correlation function, given by Eq.~(\\ref{Corr2Pi}).\nThe plots (c),(d) represent the computed values itself\nobtained in a ``single run'' of the Markov process described above for different linear sizes $N$ of the system.\nThe measurements are taken after thermalization after each several hundred thousand iterations of the process.\nThe total number of measurements is $10^{5}$.\nThe plot (b) shows the difference between the theoretical exact values and the numerical computation.\nAgain, the difference shows the Airy waves propagating from the boundary of the limit shape.\nAs in the case of the height function, one can see the Airy waves which decrease with increasing of $N$.\n\nThe agreement of theoretical values of the correlation function and the corresponding  numerical values can\nbe seen qualitatively by comparing  pictures in Fig.~\\ref{CorrCompareTheor406090}.\nWhen $\\Delta=0$ the six-vertex model maps to a dimer model and therefore in the limit $N\\to \\infty$\ncorrelation functions converge to conformally invariant correlation functions (\\ref{Corr2Pi}).\nOne can see the logarithmic behavior in the two-point correlation function\nin the vicinity of $(x_i,y_i)$, where $\\left \\langle\\phi(\\vec{r_{i}}),\\phi(\\vec{r}_{j})\\right \\rangle \\sim -1\/(2\\pi)\\, \\ln{|\\vec{r}_{i}-\\vec{r}_{j}|}$.\n\nBy plotting the results of numerics along the slice $y=0.5$ we can examine the logarithmic behavior carefully, see Fig.~\\ref{DetailedCorrCompare6060}.\nThere is a good agreement between the theoretical result and numerical data for different lattice sizes:\nthe calculated values converge to the theoretical prediction as the lattice size increases.\nThe numerical values of the coefficient against the logarithm in Eq.~(\\ref{Corr2Pi}) have been obtained from the fit to data.\nThey are shown in the inset of Fig.~\\ref{DetailedCorrCompare6060}.\nWe see that, as the lattice size increases, $N\\to \\infty$, the value of the coefficient against the logarithm tends to the exact one $-1\/(2\\pi)\\approx -0.159155$.\nFor example, a fit by logarithm for the lattice size $90\\times 90$ yields the coefficient $-0.167$.\nA fit of the values of the coefficient (red line in the inset), in turn, gives the approximate value for the infinite lattice to be $-0.1584\\pm0.0082$, which is close to the exact one.\n\n\\subsection{Numerical results for $\\Delta=1\/2$}\n\nThe agreement of theoretical and numerical results at the free fermionic point, $\\Delta=0$, suggests that the numerics should work equally well for other values of $\\Delta$, where the analytical results are still unknown.\nHere, we present numerical results for $\\Delta=1\/2$.\nWe choose this value of $\\Delta$ randomly, but note that it is also known as the combinatorial point where the model has many extra interesting features~\\cite{ZJdisser}.\n\nThe results of numerical computation of the two-point correlation function for $a\/c=b\/c=1$\nare shown in Fig.~\\ref{f1Delta05}.\nThree plots correspond to three lattice sizes: $40\\times 40$, $60\\times 60$, and $90\\times 90$.\nThe behavior of the correlation function at short distances, as expected, is very similar to that for the free fermionic point in Fig.~\\ref{CorrCompareTheor406090}.\n\n\n\\begin{figure}[t!]\n\\begin{center}\n\\begin{tabular}{ccc}\n\\begin{minipage}{0.33\\linewidth}\n\\includegraphics[width=1.0\\textwidth, angle=0.0]{figure13.pdf}\n\\end{minipage}\n&\n\\begin{minipage}{0.33\\linewidth}\n\\includegraphics[width=1.0\\textwidth, angle=0.0]{figure14.pdf}\n\\end{minipage}\n&\n\\begin{minipage}{0.33\\linewidth}\n\\includegraphics[width=1.0\\textwidth, angle=0.0]{figure15.pdf}\n\\end{minipage}\n\\end{tabular}\n\\end{center}\n\n\\caption{\\label{f1Delta05} The correlation function for the case $\\Delta=1\/2$, $a\/c=b\/c=1$. The shown data are for lattice sizes (a) $40\\times 40$, (b) $60\\times 60$, and (c) $90\\times 90$.}\n\\end{figure}\n\nThe numerical values of the two-point correlation function along the slice $y=0.5$ for $\\Delta=1\/2$ are shown in Fig.~\\ref{f2Delta05}.\nThe short distance asymptotic of the correlation function is again logarithmic. This is \nin agreement with the fact that the model is in the disordered phase. The difference with the \nfree fermionic case is that the global correlation function is not given by a conformal \nmapping anymore, but is given by an effective Gaussian field theory, see for example the discussion in~\\cite{GBDJ}.\nHowever, as in any disordered phase, at distances which are larger than the lattice step, but much smaller than the characteristic size of the lattice, the correlation functions are still given by an effective conformal field theory.\nIn the case of the six-vertex model, this is $c=1$ Gaussian CFT model with logarithmic correlators.\n\nThe fitted coefficient against logarithm in Eq.~(\\ref{Corr2Pi}) approaches the exact value $-1\/(2\\pi)$ as the lattice size $N \\to \\infty$ in this case as well.\nHowever, the numerical values of this coefficient when $\\Delta=1\/2$ are notably worse than the same values for $\\Delta=0$. For $\\Delta=1\/2$, the values for smaller lattices are systematically smaller than those for $\\Delta=0$.\nThey are naturally expected to converge to the exact value, but the rate of a convergence is less than for $\\Delta=0$.\nFor example, when the lattice size is $90\\times 90$, the fit by logarithm gives the value $-0.182$ for the coefficient against $\\log$.\nThe convergence is shown in the inset of Fig.~\\ref{f2Delta05} with the extrapolated value for the infinite lattice being $-0.1588\\pm0.0058$, which is still close to the expected $-1\/(2\\pi)$.\n\n\\begin{figure}[t!]\n\\begin{center}\n\\vspace{1mm}\\includegraphics[width=1.0\\textwidth, angle=0.0]{figure16.pdf}\n\\end{center}\n\\caption{\\label{f2Delta05}The values of the two-point correlation function along the slice $y=0.5$ for $\\Delta=1\/2$. The calculated values for different lattice sizes are shown. The inset shows the values of numerically obtained coefficient against logarithm in the logarithm-like fit with respect to the lattice size as well as a fit of these coefficients.}\n\\end{figure}\n\nAs in the  case $\\Delta=0$ one can see the Airy waves near the boundary of the limit shape. They disappear when $N$ is increasing.\n\n\\begin{figure}[t!]\n\\begin{center}\n\\vspace{1mm}\\includegraphics[width=0.750\\textwidth, angle=0.0]{figure17.pdf}\n\\end{center}\n\\caption{\\label{af}The values of the two-point correlation function for the antiferroelectric phase $\\Delta=-7\/2$. The lattice size is $N=90$. The sharp peak is given by the exponential fall which is fitted as $0.484+0.516 \\exp{\\left((-99\\pm 4) |\\vec{r}_{i}-\\vec{r}_{j}|\\right)}$.}\n\\end{figure}\n\n\\subsection{Numerical results for $\\Delta=-7\/2$}\n\nFor $\\Delta<-1$, the antiferroelectric phase of the six-vertex model is opened in the form of a diamond shape droplet.\nThis droplet has already been observed earlier~\\cite{R2005,Zvonarev,Cugliandolo,Korepin1}.\nThe Gaussian field $\\phi(x,y)$ in this region is massive which predicts the exponential decay of correlation functions at short distances.\nThe numerical observation of the exponential decay of correlation function in this phase is challenging. \nIn order to carry out such computations of correlation functions at distances deep in the antiferroelectric droplet, \nthe characteristic length of the droplet should be much larger than the correlation length. But for these values of $N$ and $\\Delta$ \nthe thermalization is expected exponentially long~\\cite{Ra}.\n\nWe carried out calculations of two-point correlation functions for $\\Delta=-7\/2$ and the lattice size $N=90$.\nThe result is given in Fig.~\\ref{af}, and the thermalization was presented in Fig.~\\ref{thermalizationFig}.\nThe numerics are in qualitative agreement with the theoretical prediction that the correlation function should exponentially decrease.\nOne can see a sharp peak over a relatively flat background.\nThe sharp peak is given by the exponential fall and can be fitted as $0.484+0.516 \\exp{\\left((-99\\pm 4) |\\vec{r}_{i}-\\vec{r}_{j}|\\right)}$.\n\n\nThe background ``pillow'' in Fig.~\\ref{af} is expected to be a result of ``mesoscopic'' effects. The lattice size $N=90$ is relatively small and \ncorrelation functions get affected by the Airy processes on the boundaries of the disordered region. \nThe parallel GPU computations on large lattices may resolve this issue~\\cite{AS}.\nMore careful analysis of comparative values of the linear size of the droplet and of the correlation length\nwill be given in a separate publication both numerically and from the exact solution.\n\n\\section{Conclusion}\nIn this paper, we numerically calculated the two-point correlation functions for the six-vertex model with the domain wall boundary conditions.\nThe disordered ($|\\Delta|<1$) phase has mainly been studied.\nParticular attention was paid to the free fermionic point ($\\Delta=0$), for which the correlation function has been also obtained analytically in the thermodynamic limit, $N\\to\\infty$.\nThe logarithm-like behavior of correlation functions at the small scales has been confirmed.\nFor antiferroelectric phase, the exponential decrease of the correlator has been observed.\nThe numerics for $N=90$ and $\\Delta=-7\/2$ show that it might be interesting to study correlation functions in the mesoscopic region where the size of the antiferroelectric droplet is comparable to the correlation length in the antiferroelectric phase.\nWe plan to continue studies of correlation functions and, in particular, their asymptotics in the limit $\\Delta \\to -1-0$\nwhen the relatively small characteristic size of the droplet requires computations on large lattices.\nFor such lattices, the implementation of Markov sampling on GPU may be of great practical significance.\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\nMost success in machine learning has been booked in the domain of supervised learning, when large quantities of labeled data are available.\nHowever it is commonly agreed that this is not a sustainable way forward, as we want algorithms that can learn new tasks and generalize to new domains even when there is no or very little labeled data available.\nOne succesful approach to unsupervised and semi-supervised learning are Generative Adversarial Networks (GANs) \\cite{goodfellow2014generative}, which are formulated as a min-max game between a generator $g_\\theta(z)$ which defines an implicit density $\\mathbb{Q}_\\theta$ over $\\X \\subset \\mathbb R^d$ from which we can sample, and a discriminator $D$ (or critic $f$) which measures a distance between the ``real'' $\\mathbb{P}_r$ and ``fake'' $\\mathbb{Q}_\\theta$ distribution.\nIn order to use GANs for semi-supervised learning \\cite{springenberg2015unsupervised,salimans2016improved} or unsupervised learning \\cite{chen2016infogan}, typically the discriminator network is in some sense shared with a learned classifier $p(y|x)$, with $y \\in \\{1 \\dots K\\}$ the discrete label set. \nThe underlying assumption is that features that are helpful towards telling real\/fake apart, are also helpful towards classification.\nSpecifically the idea of the discriminator having ``$K+1$'' output directions, with the $K+1$\\textsuperscript{th} ``fake'' direction competing with the K ``real'' directions \\cite{salimans2016improved} has lead to strong empirical results \\cite{dai2017good}.\n\nRecently the distance metric between distributions became a topic of interest \\cite{arjovsky2017towards,nowozin2016f,kaae2016amortised,mao2016least,arjovsky2017wasserstein,gulrajani2017improved,mroueh2017mcgan,mroueh2017fisher,li2017mmd,anon2017sobolev}.\nWe will focus on GANs with one class of distance metrics, Integral Probability Metrics or IPMs \\cite{muller1997integral,sriperumbudur2009integral,sriperumbudur2012empirical}.\nIPMs were first introduced in the GAN framework in Wasserstein GAN (WGAN) \\cite{arjovsky2017wasserstein,gulrajani2017improved},\nand also formed the basis for McGan \\cite{mroueh2017mcgan}, Fisher GAN \\cite{mroueh2017fisher} and Sobolev GAN \\cite{anon2017sobolev}.\nThe IPM distance between $\\mathbb{P}$ and $\\mathbb{Q}$ is defined as:\n\\[ \\sup_{f \\in \\mathcal{F}} \\mathbb{E}_{x\\sim \\mathbb{P}} f(x)- \\mathbb{E}_{x\\sim \\mathbb{Q}} f(x). \\]\nThe IPM definition looks for a witness function (or ``critic'') $f(x)$, which maximally discriminates between samples coming from the two distributions.\nThis naturally fits the GAN idea: we can parametrize the critic with a neural network which takes the place of the discriminator in the GAN framework.\nA crucial ingredient of the IPM metric is the function class $\\mathcal{F}$, which defines how the critic $f(x)$ is bounded,\nwhich in its turn \\emph{defines the metric} being measured between distributions.\n\nIn WGAN \\cite{arjovsky2017wasserstein,gulrajani2017improved} we approximate $\\mathcal{F}$ the class of Lipschitz functions.\nWGAN-GP \\cite{gulrajani2017improved} uses a pointwise gradient norm penalty.\nFisher GAN \\cite{mroueh2017fisher} introduces a tractable constraint on $\\mathbb{E}_{x \\sim \\mu} f^2(x)$ which is enforced on samples from $\\mu=\\frac{\\mathbb{P}+\\mathbb{Q}}{2}$.\nSobolev GAN \\cite{anon2017sobolev} introduces the tractable constraint on $\\mathbb{E}_{x \\sim \\mu } \\nor{\\nabla_x f(x)}^2$ on the same $\\mu$.\nIn this paper, we investigate which IPM formulations are amenable towards semi-supervised learning,\nand whether we can leverage the $K+1$ formulation of classical JSD-based GANs \\cite{salimans2016improved}.\n\n\\begin{figure}[t]\n\\centering\n\\includegraphics[width=0.75\\linewidth]{critic_ssl.pdf}\n\\caption{``K+1'' parametrization of the IPM critic for semi-supervised learning.}\n\\label{fig:sslcritic}\n\\end{figure}\n\nWe will investigate the ``$K+1$ parametrization\" of the critic as introduced in \\cite{mroueh2017fisher} (See Figure \\ref{fig:sslcritic}): \n\\[ \nf(x) =\\underbrace{ \\sum_{y=1}^K p(y|x)\\scalT{S_y}{\\Phi_{\\omega}(x)}}_{\\bm{f_{+}}: \\text{ ``real\" critic }} - \\underbrace{\\scalT{v}{\\Phi_{\\omega}(x)}}_{\\bm{f_-} :\\text{``fake\" critic}} \n\\]\n\\vskip -1.0em\nThe training objective for critic and classifier now becomes:\n\\[\n\n  \\max_{S, \\Phi_{\\omega}, f } \\L_{D} = \\frac{1}{N} \\left(\\sum_{x \\in \\text{unl} } f(x) - \\sum_{z \\sim p_z} f(g_\\theta(z)) \\right) + \\L^{\\text{C}} (f_+, f_-, g_\\theta) - \\lambda_{CE} \\sum_{(x,y) \\in \\text{lab} } CE (p(y|x),y )\n\\]\nWhere the constraint term $\\L^\\text{C}$ can contain the real and fake terms of the critic $f_+$ and $f_-$ separately.\nNote that $p(y|x)=\\rm{Softmax}(\\scalT{S}{\\Phi_{\\omega}(x)})_y$ appears both in the critic formulation and in the Cross-Entropy term.\nIntuitively this critic uses the $K$ class directions of the classifier $S_y$ to define the ``real'' direction, which competes with another $K+1$\\textsuperscript{th} direction $v$ that indicates fake samples.\n\nWe empirically investigate the merit of this $K+1$ formulation in IPM-based GANs in Section \\ref{sec:Kp1}.\nIn Section \\ref{sec:constraint} we investigate on which parts of the critic $f, f_+, f_-$ to apply the constraint terms.\nFinally, in Section \\ref{sec:normalization}, we investigate the influence of activation normalization in the critic $f$, such as batch normalization (BN) \\cite{ioffe2015batch} and layer normalization (LN) \\cite{ba2016layer}.\n\n\\section{Experiments}\nWe will provide experimental results on CIFAR-10 \\cite{cifar10} and SVHN \\cite{netzer2011reading} using the CNN architectures as in \\cite{mroueh2017fisher} for $g_\\theta$ and $f$,\nwhich is very close to the discriminator architecture used in \\cite{salimans2016improved,dumoulin2016adversarially}.\nUnless noted otherwise this CNN will have no normalization (LN or BN) in the critic, but $g_\\theta$ always includes BN.\nSimilar to the standard procedure in other GAN papers we use 4k labeled samples for CIFAR-10, 1k labeled samples for SVHN, and do hyperparameter and model selection on the standard CIFAR-10 and SVHN validation split.\nHyperparameter details are in Appendix \\ref{sec:hypers}.\nWe provide purely supervised baseline results of the critic architecture in Appendix \\ref{sec:supervised}.\n\n\n\\subsection{$K+1$ formulations.} \\label{sec:Kp1}\nWe show in Table \\ref{tab:Kp1} the results for plain vs $K+1$ critic formulation.\n``Plain $\\scal{v}{\\Phi_\\omega(x)}$'' indicates the plain critic $f=\\scal{v}{\\Phi_\\omega(x)}$ not interacting with the classifier.\nFor the $K+1$ formulations, the constraints act on the full critic, except for the combination where Fisher acts on $f$ but Sobolev is applied to $f_-$ (see next Section).\n\nThe third and sixth column of Table \\ref{tab:Kp1} present results for a modified $K+1$ formulation $f^H = f_+^H - f_-$, where \n\\[ \nf_+^{H}(x) \n= \\sum_{y=1}^K p(y|x) \\log( p(y|x) )\n= \\sum_{y=1}^K p(y|x)\\scalT{S_y}{\\Phi_{\\omega}(x)} - \\log Z_Y(x)\n= f_+(x) - \\log Z_Y(x)\n\\]\n\nIn this formulation, $f_+^H(x)$ is the negative entropy $-H[p(y|x)]$ of the classifier for a given $x$, which is what is being optimized as the sole objective in \\cite{springenberg2015unsupervised}.\nThis $f_+^H$ is not sensitive to the magnitude of, or an additive bias to, the $\\scalT{S_y}{\\Phi_\\omega(x)}$ because of the normalization constant $\\log Z_Y(x) = \\log( \\sum_{y'=1}^K \\exp \\scalT{S_{y'}}{\\Phi_\\omega(x)})$.\nThis could either stabilize the training and shift more focus on the $K+1$\\textsuperscript{th} direction $v$,\nor could limit the effectiveness of $f_+$ in contrasting between real and fake samples - which makes it interesting to investigate.\n\nWe see that in the two succesful IPMs (Fisher and Fisher+Sobolev), the $K+1$ formulations give a strong gain\nover the plain formulation.\nBest results are obtained for the (unmodified) $K+1$ formulation and Sobolev + Fisher constraint.\n\n\\begin{takeaway}\nThe $K+1$ formulation is superior over the plain $f(x)=\\scal{v}{\\Phi_\\omega(x)}$ formulation.\n\\end{takeaway}\n\n\\begin{table}[t]\n\\centering\n\\caption{Results for different critic formulations. Note that the $K+1$ formulation is better across the board (except for WGAN-GP because of the harmful gradient penalty on the full critic $f$, see Section \\ref{sec:constraint}).\nIn gray we indicate all settings which failed to improve over straight-up supervised training with the small set of labeled samples (Appendix \\ref{sec:supervised}).\n}\n\\label{tab:Kp1}\n\\resizebox{\\textwidth}{!}{\n\\begin{tabular}{| l | l |l | l | l | l | l |  }\n  \\toprule\n                 & \\multicolumn{3}{| c |}{CIFAR-10 (4k)}  & \\multicolumn{3}{| c |}{SVHN (1k)} \\\\\n                 &      Plain $\\scal{v}{\\Phi_\\omega(x)}$ &             $K+1$ &  $K+1, f_+^H$ &     Plain $\\scal{v}{\\Phi_\\omega(x)} $ &  $K+1$            &  $K+1, f_+^H$ \\\\\n\\midrule\nWGAN clip  &  \\cellcolor[gray]{0.9} $40.85$  & & & \\cellcolor[gray]{0.9} $28.65$ &  & \\\\\nWGAN-GP          &  \\cellcolor[gray]{0.9} $44.64 \\pm 0.61$ &  \\cellcolor[gray]{0.9} $48.92 \\pm 0.50$ &  \\cellcolor[gray]{0.9} $48.81 \\pm 0.41$ &                        $17.51 \\pm 0.23$ &  \\cellcolor[gray]{0.9} $31.59 \\pm 1.75$ &  \\cellcolor[gray]{0.9} $33.85 \\pm 1.46$ \\\\\nFisher           &                        $18.95 \\pm 0.32$ &                        $17.82 \\pm 0.43$ &                        $17.12 \\pm 0.11$ &                        $15.59 \\pm 0.67$ &                         $9.46 \\pm 0.32$ &                        $11.93 \\pm 0.26$ \\\\\nSobolev          &  \\cellcolor[gray]{0.9} $80.45 \\pm 0.59$ &  \\cellcolor[gray]{0.9} $45.35 \\pm 0.62$ &  \\cellcolor[gray]{0.9} $45.27 \\pm 0.33$ &  \\cellcolor[gray]{0.9} $80.46 \\pm 0.62$ &  \\cellcolor[gray]{0.9} $28.46 \\pm 1.04$ &  \\cellcolor[gray]{0.9} $26.00 \\pm 0.43$ \\\\\nFisher + Sobolev &  $21.44 \\pm 0.74$ &                        $\\bm{16.29 \\pm 0.42}$ &  $22.68 \\pm 0.31$ &                        $13.23 \\pm 0.53$ &                         $\\bm{8.88 \\pm 0.84}$ &                        $12.01 \\pm 0.19$ \\\\\n\\bottomrule\n\\end{tabular}\n}\n\\vskip -0.13in\n\\end{table}\n\n\\subsection{$K+1$: constraints on $f$, $f_+$ or $f_-$} \\label{sec:constraint}\n\n\\begin{table}[b]\n\\centering\n\\vskip -0.13in\n\\caption{ Four constraint combinations (GP, S, F, F+S), acting on either $f$, $f_-$, $f_+$.\n  Failed settings are color-marked:\n  blue indicates experiments that only include a penalty on $f_-$,\n  gray marks the experiments where a gradient constraint was acting on the full critic $f$.\n}\n\\label{tab:constraint}\n\\resizebox{0.40\\textwidth}{!}{\n\\begin{tabular}{| l | r | r |}\n\\toprule\n & CIFAR-10 (4k) & SVHN (1k) \\\\\n\\midrule\n\\rowcolor[gray]{0.9}\n$\\om_{GP}(f)$           &          $49.28$ &       $34.05$ \\\\\n\\rowcolor{myblue}\n$\\om_{GP}(f_{-})$        &          $73.78$ &       $70.61$ \\\\\n\\rowcolor[gray]{0.9}\n$\\om_S(f)$            &          $45.55$ &       $26.15$ \\\\\n\\rowcolor{myblue}\n$\\om_S(f_{-})$         &          $71.38$ &       $20.04$ \\\\\n$\\om_F(f)$            &          $17.49$ &        $9.16$ \\\\\n\\rowcolor{myblue}\n$\\om_F(f_{-})$         &          $72.08$ &       $38.33$ \\\\\n\\rowcolor[gray]{0.9}\n$\\om_F(f) , \\om_S(f)$       &          $45.55$ &       $29.30$ \\\\\n$\\om_F(f) , \\om_S(f_-)$   &          $\\bm{17.08}$ &        $8.27$ \\\\\n\\rowcolor[gray]{0.9}\n$\\om_F(f_-) , \\om_S(f)$   &          $47.15$ &       $26.31$ \\\\\n\\rowcolor{myblue}\n$\\om_F(f_-) , \\om_S(f_-)$ &          $68.62$ &       $37.50$ \\\\\n\\rowcolor[gray]{0.9}\n$\\om_F(f_+) , \\om_S(f)$   &          $47.79$ &       $27.99$ \\\\\n$\\om_F(f_+) , \\om_S(f_-)$ &          $17.12$ &        $\\bm{8.01}$ \\\\\n\\bottomrule\n\\end{tabular}\n}\n\\end{table}\n\nSimilar to the notation in \\cite{mroueh2017fisher,anon2017sobolev}\nwe write the Fisher and Sobolev constraint respectively as\n$\\om_{F}(f) = \\frac{1}{N} \\sum_{\\tilde{x} \\sim \\mu} f^2(\\tilde{x}) = 1$,\nand\n$\\om_{S}(f) = \\frac{1}{N} \\sum_{\\tilde{x} \\sim \\mu} \\nor{\\nabla_x f(\\tilde{x})}^2 = 1$.\nThe WGAN-GP constraint is\n$\\nor{\\nabla_x f(\\tilde{x})}=1$ for points interpolated between real and fake,\nthis is enforced through\n$\\om_{GP}(f) = \\sum_{\\tilde{x} \\sim \\mu_{GP}} (1 - \\nor{\\nabla_x f(\\tilde{x})})^2$.\nBoth $\\om_{F}$ and $\\om_{S}$ are enforced with augmented lagrange multipliers with hyperparameters $\\rho_F$ and $\\rho_S$,\nwhile $\\om_{GP}$ is enforced with a large fixed penalty weight $\\lambda_{GP}=10.0$.\n\nNote now that the constraints can be enforced on either $f_+$, $f_-$, or the full critic $f = f_+ - f_-$.\nTo ensure that the critic is bounded, at least some constraint has to be acting on $f_-$ directly or through $f$,\nwhile the boundedness of $f_+$ could in principle be ensured through the CE term on the small labeled set.\nNote there will be non-trivial interaction between $\\lambda_{GP}$, $\\rho_F$, $\\rho_{S}$ and $\\lambda_{CE}$.\n\nIn Table \\ref{tab:constraint} are results for the four different constraints and combinations acting on different parts of the critic.\nWe see that formulations with constraints only action on $f_-$ failed: clearly the CE term alone is not enough to constrain $f_+$.\nAnother important conclusion is that any combination where a gradient-norm constraint (Sobolev or WGAN-GP) is acting on the full critic $f$, the classifier is compromised:\nin these settings it is impossible for the network to fit even the small labeled training set (heavy underfitting), causing bad SSL performance.\n\n\\begin{takeaway}\nWe need some form of constraint acting on both $f_-$ and $f_+$; the CE term alone is not enough to control $f_+$.\n\\end{takeaway}\n\\begin{takeaway}\nConstraints including the gradient norm (WGAN-GP, Sobolev) should only act on $f_-$, otherwise the network underfits.\n\\end{takeaway}\n\n\n\\subsection{How to normalize the critic} \\label{sec:normalization}\n\\begin{table}[t]\n\\centering\n\\caption{How to normalize the critic. We see BN performs signficantly worse than other options for Fisher GAN (and is incompatible with GP\/Sobolev).\nThe layernorm formulation with singleton $\\mu,\\sigma^2$ statistics are superior to statistics per feature map.\nFisher GAN benefits from layernorm, while in Sobolev+Fisher no normalization is prefered - this gives the overall best result.\n}\n\\label{tab:normalization}\n\\resizebox{\\textwidth}{!}{\n\\begin{tabular}{lllll}\n\\toprule\n     &  CIFAR-10 (4k) &                   &     SVHN (1k) &                  \\\\\nIPM Definition &            Fisher &  Fisher + Sobolev &           Fisher & Fisher + Sobolev \\\\\n$f$ normalization         &                   &                   &                  &                  \\\\\n\\midrule\nBatch Normalization       &  $24.14 \\pm 0.26$ &                   & $10.76 \\pm 0.28$ &                  \\\\\nLN ($\\mu,\\sigma^2 \\in \\mathbb R^{1\\times 1\\times 1}$) ($\\bm{g}, \\bm{b} \\in \\mathbb R^{C\\times 1 \\times 1}$)\n&  $\\bm{16.45 \\pm 0.42}$ &  $16.79 \\pm 0.11$ &  $8.78 \\pm 0.68$ &  $9.02 \\pm 0.29$ \\\\\nLN ($\\mu,\\sigma^2 \\in \\mathbb R^{1\\times 1\\times 1}$) ($\\bm{g}, \\bm{b} \\in \\mathbb R^{1\\times H \\times W}$)\n&  $16.81 \\pm 0.36$ &  $17.24 \\pm 0.34$ &  $\\bm{8.55 \\pm 0.29}$ &  $8.70 \\pm 0.67$ \\\\\nLN ($\\mu,\\sigma^2 \\in \\mathbb R^{C\\times 1\\times 1}$) ($\\bm{g}, \\bm{b} \\in \\mathbb R^{C\\times 1 \\times 1}$)\n                &  $20.44 \\pm 0.24$ &  $20.09 \\pm 0.43$ &  $9.18 \\pm 0.12$ &  $9.19 \\pm 0.21$ \\\\\nLN ($\\mu,\\sigma^2 \\in \\mathbb R^{C\\times 1\\times 1}$) ($\\bm{g}, \\bm{b} \\in \\mathbb R^{1\\times H \\times W}$)\n                &  $19.93 \\pm 0.34$ &  $19.85 \\pm 0.29$ &  $9.69 \\pm 0.16$ &  $9.29 \\pm 0.16$ \\\\\n                No Normalization &  $17.73 \\pm 0.56$ &  $\\bm{16.33 \\pm 0.15}$ &  $9.06 \\pm 0.46$ &  $\\bm{8.31 \\pm 0.60}$ \\\\\n\\bottomrule\n\\end{tabular}\n}\n\\end{table}\n\nIn the original DCGAN \\cite{radford2015unsupervised} architecture, batch normalization (BN) \\cite{ioffe2015batch} was a crucial ingredient,\nboth in the generator and discriminator.\nEven though the original WGAN still relies on BN,\nboth WGAN-GP, Fisher GAN, and Sobolev GAN report strong results without any layerwise normalization.\nWhen the constraint involves a norm of the gradient, BN is problematic since it couples the different samples in the batch.\nHere we investigate as alternative to BN either layer normalization (LN) \\cite{ba2016layer} or having no normalization in the critic.\n\nOne important detail which usually glossed over, is how layernorm exactly is extended to the convolutional setting,\nwhere the activations at a given layer are $\\in \\mathbb R^{C \\times H \\times W}$.\nSpecifically we need to decide whether we will accumulate the statistics mean $\\mu$ and variance $\\sigma^2$ either into a singleton ($\\in \\mathbb R^{1\\times 1\\times 1}$), or separate per feature map ($\\in \\mathbb R^{C\\times 1\\times 1}$).\nSimilarly, we need to decide whether we will parametrize the scale $\\bm{g}$ and bias $\\bm{b}$ separate per feature map ($\\in \\mathbb R^{C\\times 1\\times 1}$), or separate per pixel ($\\in \\mathbb R^{1\\times H\\times W}$).\nIn the above, we used broadcasting notation, meaning that the singleton dimensions will be expanded to perform the elementwise operations.\nFor reference, in batch normalization for convolutional networks both mean $\\mu$, variance $\\sigma^2$, scale $\\bm{g}\/\\gamma$ and bias $\\bm{b}\/\\beta$ are collected separately per feature map, i.e. $\\in \\mathbb R^{C\\times 1\\times 1}$.\nImplementation-wise we follow \\cite{ren2016normalizing} in adding a small $\\epsilon$ \\emph{inside} the square root in $\\sqrt{\\sigma^2+\\epsilon}$.\nThe results in Table \\ref{tab:normalization} lead us to conclude:\n\n\\begin{takeaway}\nAvoid batchnorm, definitely when constraining the gradient norm in objective, but it also hurts for Fisher GAN!\n\\end{takeaway}\n\\begin{takeaway}\nThe layer normalization formulation with singleton stats ($\\mu,\\sigma^2 \\in \\mathbb R^{1\\times 1\\times 1}$) and parameters per feature map ($\\bm{g}, \\bm{b} \\in \\mathbb R^{C\\times 1 \\times 1}$) is superior.\nFisher GAN benefits from this LN, while for Sobolev+Fisher no normalization is better.\n\\end{takeaway}\n\n\\iffalse\n\\subsection{Class-conditional generator.}\nACGAN \\cite{odena2016conditional} defines a simple extension to GANs where a one-hot encoded class label is appended to the noise vector, which allows the generator to be conditioned on class $k$: $g_\\theta(z, y=k)$.\nAs described in \\cite{mroueh2017fisher}, we need to add a CE term to the objetive for the generator $\\L_G$ (with weight $\\lambda_{CE,G}=0.1$).\nIn Table \\ref{tab:conditional} we investigate whether a class-conditional generator helps with SSL.\n\n\\begin{table}[htb]\n\\centering\n\\caption{Class-conditional generator $g_\\theta(z, y=k)$ vs unconditional generator $g_\\theta(z)$.\nExperiments on CIFAR-10 with different number of labeled samples. All critics with $K+1$ formulation.\nIn almost all settings, class-conditioning slightly hurts rather than helps performance.\n}\n\\label{tab:conditional}\n\\resizebox{\\textwidth}{!}{\n\\begin{tabular}{lllll}\n\\toprule\nlabeledSamples &              1000 &              2000 &              4000 &              8000 \\\\\n\\midrule\nFisher BN Cond    &  $35.52$ &  $29.79$ &  $25.40$ &  $20.71$ \\\\\nFisher BN Uncond  &  $36.80$ &  $30.43$ &  $24.35$ &  $21.35$ \\\\\nFisher LN Cond   &  $20.99 \\pm 0.66$ &  $19.01 \\pm 0.21$ &  $17.41 \\pm 0.38$ &  $15.50 \\pm 0.41$ \\\\\nFisher LN Uncond &  $19.74 \\pm 0.21$ &  $17.87 \\pm 0.38$ &  $16.13 \\pm 0.53$ &  $14.81 \\pm 0.16$ \\\\\nFisher No Norm Cond   &  $22.58 \\pm 0.43$ &  $20.16 \\pm 0.27$ &  $18.16 \\pm 0.53$ &  $16.22 \\pm 0.18$ \\\\\nFisher No Norm Uncond &  $21.49 \\pm 0.18$ &  $19.20 \\pm 0.46$ &  $17.30 \\pm 0.30$ &  $15.57 \\pm 0.33$ \\\\\nSobolev+Fisher No Norm Cond   &  $21.40 \\pm 0.60$ &  $18.65 \\pm 0.59$ &  $16.96 \\pm 0.16$ &  $15.13 \\pm 0.22$ \\\\\nSobolev+Fisher No Norm Uncond &  $20.14 \\pm 0.21$ &  $17.38 \\pm 0.10$ &  $15.77 \\pm 0.19$ &  $14.20 \\pm 0.08$ \\\\\n\\bottomrule\n\\end{tabular}\n}\n\\end{table}\n\\fi\n\n\\section{Conclusion}\nWe empirically investigated how different types of IPM-based Generative Adversarial Networks \ncan be used for semi-supervised learning.\nA comparison with literature results is given in Appendix~\\ref{sec:literature}.\nOur main conclusions are (1) the $K+1$ formulation works, (2) batch normalization should be avoided, also in Fisher GAN,\nand (3) gradient penalty constraints should act on $f_-$ only, not on the full critic which includes the classifier $p(y|x)$.\n\n\\bibliographystyle{unsrt}\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\n\nWe are concerned with the following nonlinear Kirchhoff type equations\n\\begin{equation}\n\\left\\{\n\\begin{array}{ll}\n-\\left( a\\int_{\\mathbb{R}^{N}}|\\nabla u|^{2}dx+b\\right) \\Delta u+V(x)u=h(x,u)\n& \\text{ in }\\mathbb{R}^{N}, \\\\\nu\\in H^{1}(\\mathbb{R}^{N}), &\n\\end{array\n\\right.  \\label{1-0}\n\\end{equation\nwhere $N\\geq 1,a,b>0,$ $V\\in C(\\mathbb{R}^{N},\\mathbb{R})$ and $h\\in C\n\\mathbb{R}\\times \\mathbb{R}^{N},\\mathbb{R}).$\n\nKirchhoff type equations, of the form similar to Eq. $(\\ref{1-0}),$ are\nanalogous to the stationary case of equations that arise in the study of\nstring or membrane vibrations, namely,\n\\begin{equation}\nu_{tt}-\\left( a\\int_{\\Omega }|\\nabla u|^{2}dx+b\\right) \\Delta u=h(x,u)\\text{\nin }\\Omega ,  \\label{1-1}\n\\end{equation\nwhere $\\Omega $ is a bounded domain in $\\mathbb{R}^{N}.$ As an extension of\nthe classical D'Alembert's wave equation, Eq. $(\\ref{1-1})$ was first\npresented by Kirchhoff \\cite{K} in 1883 to describe the transversal\noscillations of a stretched string, particularly, taking into account the\nsubsequent change in string length caused by oscillations, where $u$ denotes\nthe displacement, $h$ is the external force and $b$ is the initial tension\nwhile $a$ is related to the intrinsic properties of the string, such as\nYoung's modulus. Equations of this type are often referred to as being\nnonlocal because of the presence of the integral.\n\nAfter the pioneering work of Pohozaev \\cite{P} and Lions \\cite{L}, the\nsolvability of the Kirchhoff type equation $(\\ref{1-1})$ has been\nwell-studied in general dimension by various authors, see for examples,\nD'Ancona-Shibata \\cite{DS}, D'Ancona-Spagnolo \\cite{DS1} and Nishihara \\cit\n{N}. More recently, the corresponding elliptic version like Eq. $(\\ref{1-0})\n$ has begun to receive much attention via variational methods. We refer the\nreader to \\cite{Az1,Az2,CKW,DPS,G,HL,HZ,LLS,LY,LLS1,N1,SZ,SW,SW1,TC,Y} and\nthe references therein.\n\nMost of researchers have of late years focused on the existence of positive\nsolutions, ground states, radial solutions and semiclassical states for Eq. \n(\\ref{1-0})$ in lower dimensions, i.e., $N=1,2,3.$ The typical way to deal\nwith such problem is to apply the mountain-pass theorem or the Nehari\nmanifold method. Owing to the fourth power of the nonlocal term, one usually\nassumes that the nonlinearity $h(x,u)$ is either $4$-superlinear at infinity\non $u$ in the sense tha\n\\begin{equation*}\n\\lim_{|u|\\rightarrow \\infty }\\frac{\\int_{0}^{u}h(x,s)ds}{u^{4}}=\\infty \\text{\nuniformly in }x\\in \\mathbb{R}^{N},\n\\end{equation*\nor satisfies the following (AR)-condition\n\\begin{equation}\n\\exists \\mu >4\\text{ such that }0<\\mu \\int_{0}^{u}h(x,s)ds\\leq h(x,u)u\\text{\nfor }u\\neq 0.  \\label{1-10}\n\\end{equation\nFor example, $h(x,u)=f(x)\\left\\vert u\\right\\vert ^{p-2}u$ with $4<p<2^{\\ast\n}(2^{\\ast }=\\infty $ for $N=1,2$ and $2^{\\ast }=2N\/(N-2)$ for $N\\geq 3).$ By\nso doing, one can easily verify the mountain-pass geometry and the\nboundedness of (PS) sequences for the energy functional. However, there have\na large number of functions $h(x,u)$ not satisfying the above assumptions,\nsuch as $h(x,u)=f(x)\\left\\vert u\\right\\vert ^{p-2}u(2<p<\\min \\left\\{\n4,2^{\\ast }\\right\\}).$ For that reason, some other approaches need to be\nintroduced in this case.\n\nIn studying the radial solutions for a class of autonomous Schr\\\"{o\ndinger-Poisson systems in $\\mathbb{R}^{3},$ Ruiz \\cite{R1} established a\nmanifold as follow\n\\begin{equation*}\n\\overline{\\mathbf{M}}=\\{u\\in H_{rad}^{1}(\\mathbb{R}^{3})\\backslash\n\\{0\\}:2\\left\\langle I^{\\prime }(u),u\\right\\rangle -P(u)=0\\},\n\\end{equation*\nwhere $I$ is the energy functional and $P(u)$ is the Pohozaev identity\ncorresponding to the system. It is usually called the Nehari-Pohozaev\nmanifold which is different from the Nehari manifold. By restricting the\nenergy functional to such manifold, the boundedness of (PS) sequences can be\nsolved effectively when the nonlinearity does not satisfy the (AR)-condition\nabove.\n\nInspired by Ruiz \\cite{R1}, Li-Ye \\cite{LY} applied the Nehari-Pohozaev\nmanifold (slightly different from $\\overline{\\mathbf{M}}$) to Kirchhoff type\nequations in $\\mathbb{R}^{3}$. By using the constraint minimization method,\ntogether with the monotonicity trick by Jeanjean \\cite{J}, they found one\nground state solution with positive energy of Eq. $(\\ref{1-0})$ when \nV(x)\\leq \\liminf_{|y|\\rightarrow \\infty }V(y)=V_{\\infty }<\\infty $ and \nh(x,u)=\\left\\vert u\\right\\vert ^{p-2}u(3<p<2^{\\ast })$. Later, using the\nsimilar approach to that in \\cite{LY}, Guo \\cite{G} and Tang-Chen \\cite{TC}\nalso obtained the existence of ground state solutions for Eq. $(\\ref{1-0})$\nwith a general nonlinearity $h(x,u)\\equiv h(u)$ in $\\mathbb{R}^{3}$,\nrespectively. In addition, Ye \\cite{Y} proved the existence of high energy\nsolutions for Eq. $(\\ref{1-0})$ with $h(x,u)\\equiv h(u)$ in $\\mathbb{R}^{3}$\nvia the Nehari-Pohozaev manifold and the linking theorem. It is worthy\nnoting that the nonlinearity $h(u)$ given by \\cite{G,TC,Y} can cover the\npower functions $\\left\\vert u\\right\\vert ^{p-2}u(2<p<4).$\n\nAzzollini \\cite{Az1,Az2} investigated a class of autonomous Kirchhoff type\nequations in higher dimensions $N\\geq 3,$ i.e., Eq. $(\\ref{1-0})$ with \nV(x)\\equiv 0$ and $h(x,u)\\equiv h(u)$ satisfying the Berestycki--Lions type\nconditions (see \\cite{BL}). With the aid of the radial ground state solution\n$\\overline{u}$ to the semilinear elliptic equation $-\\Delta u=h(u)$ in \n\\mathbb{R}^{N}(N\\geq 3)$, the following results were obtained:\\newline\n$(i)$ $N=3:$ one radial ground state solution exists for all $a,b>0;$\\newline\n$(ii)$ $N=4:$ one radial ground state solution exists if and only if \na<\\left( \\int_{\\mathbb{R}^{N}}|\\nabla \\overline{u}|^{2}dx\\right) ^{-1};\n\\newline\n$(iii)$ $N\\geq 5:$ one radial solution exists if and only if\n\\begin{equation*}\na\\leq \\left( \\frac{N-4}{N-2}\\right) ^{\\frac{N-2}{2}}\\frac{2}{(N-4)b^{\\frac\nN-4}{2}}\\int_{\\mathbb{R}^{N}}|\\nabla \\overline{u}|^{2}dx}.\n\\end{equation*}\n\nMotivated by these findings mentioned above, in the present paper we are\nlikewise interested in looking for positive solutions of Kirchhoff type\nequations. The problem we consider is thu\n\\begin{equation}\n\\left\\{\n\\begin{array}{ll}\n-\\left( a\\int_{\\mathbb{R}^{N}}|\\nabla u|^{2}dx+b\\right) \\Delta\nu+u=f(x)\\left\\vert u\\right\\vert ^{p-2}u & \\text{ in }\\mathbb{R}^{N}, \\\\\nu\\in H^{1}(\\mathbb{R}^{N}), &\n\\end{array\n\\right.  \\tag{$E_{a}$}\n\\end{equation\nwhere $N\\geq 1,a,b>0,2<p<\\min \\left\\{ 4,2^{\\ast }\\right\\} $ and the function\n$f(x)$ satisfies:\n\n\\begin{itemize}\n\\item[$(D1)$] $f\\in C(\\mathbb{R}^{N})\\cap L^{\\infty }(\\mathbb{R}^{N})$ with \nf_{\\min }=\\inf_{x\\in \\mathbb{R}^{N}}f\\left( x\\right) >0.$\n\\end{itemize}\n\nEq. $(E_{a})$ is variational, and its solutions correspond to critical\npoints of the energy functional $J_{a}:H^{1}(\\mathbb{R}^{N})\\rightarrow\n\\mathbb{R}$ given by\n\\begin{equation*}\nJ_{a}\\left( u\\right) =\\frac{a}{4}\\left( \\int_{\\mathbb{R}^{N}}|\\nabla\nu|^{2}dx\\right) ^{2}+\\frac{1}{2}\\int_{\\mathbb{R}^{N}}(b|\\nabla\nu|^{2}+u^{2})dx-\\frac{1}{p}\\int_{\\mathbb{R}^{N}}f(x)|u|^{p}dx.\n\\end{equation*\nFurthermore, one can see that $J_{a}$ is a $C^{1}$ functional with the\nderivative given b\n\\begin{equation*}\n\\left\\langle J_{a}^{\\prime }(u),\\varphi \\right\\rangle =\\left( a\\int_{\\mathbb\nR}^{N}}|\\nabla u|^{2}dx+b\\right) \\int_{\\mathbb{R}^{N}}\\nabla u\\nabla \\varphi\ndx+\\int_{\\mathbb{R}^{N}}(u\\varphi -f(x)|u|^{p-2}u\\varphi )dx\n\\end{equation*\nfor all $\\varphi \\in H^{1}(\\mathbb{R}^{N})$, where $J_{a}^{\\prime }$ denotes\nthe Fr\\'{e}chet derivative of $J_{a}.$\n\nDistinguishing from the existing literature, this paper is devoted to study\na series of questions as follows:\n\n\\begin{itemize}\n\\item[$(I)$] In spite of the amount of papers dealing with Eq. $(\\ref{1-0})\n, the geometric properties of the energy functional $J_{a}$ have not been\ndescribed in detail. One objective of this study is to shed some light on\nthe behavior of $J_{a}$. We will study whether $J_{a}$ is bounded below or\nnot, depending on the parameter $a$ and the dimension $N.$\n\n\\item[$(II)$] As we can see, the Nehari-Pohozaev manifold can help to find\npositive solutions with positive energy for Eq. $(\\ref{1-0})$ when the\nnonlinearity $h(x,u)$ does not satisfy the (AR)-condition (\\ref{1-10}) (see\n\\cite{G,LY,TC,Y}). However, to our knowledge, such approach is only valid\nfor the case of $N=3$. In our study, since the nonlinearity $f(x)\\left\\vert\nu\\right\\vert ^{p-2}u(2<p<\\min \\left\\{ 4,2^{\\ast }\\right\\} )$ does not\nsatisfy the (AR)-condition (\\ref{1-10}) as well, we would like to know\nwhether there exists an approach to study the existence of positive solution\nwith positive energy of Eq. $(E_{a})$ in any dimensions $N\\geq 1.$\n\n\\item[$(III)$] According to the geometry of the energy functional $J_{a},$\nwe think that $J_{a}$ should have two critical points in some dimensions $N,$\nwhere one is a global minimizer with negative energy and the other one is a\nlocal minimizer with positive energy. In view of this, another objective of\nthis study is to explore the existence of two positive solutions for Eq. \n(E_{a})$, which seems not to be involved in the literature.\n\\end{itemize}\n\nIn what follows, without loss of generality, we always assume $b=1.$ For any\n$u\\in H^{1}(\\mathbb{R}^{N})\\backslash \\left\\{ 0\\right\\} ,$ we defin\n\\begin{equation}\n\\overline{A}_{f}(u)=\\frac{\\left( \\int_{\\mathbb{R}^{N}}f(x)|u|^{p}dx\\right)\n^{2\/(p-2)}}{\\left\\Vert u\\right\\Vert _{D^{1,2}}^{4}\\left\\Vert u\\right\\Vert\n_{H^{1}}^{2(4-p)\/(p-2)}}\\text{ for }N\\geq 4  \\label{15-1}\n\\end{equation\nan\n\\begin{equation*}\n\\underline{A}_{f}(u)=\\frac{\\left( \\int_{\\mathbb{R}^{N}}f_{\\min\n}|u|^{p}dx\\right) ^{2\/(p-2)}}{\\left\\Vert u\\right\\Vert\n_{D^{1,2}}^{4}\\left\\Vert u\\right\\Vert _{H^{1}}^{2(4-p)\/(p-2)}}\\text{ for \nN=4,\n\\end{equation*\nwhere $\\left\\Vert u\\right\\Vert _{H^{1}}=\\left( \\int_{\\mathbb{R}^{N}}(|\\nabla\nu|^{2}+u^{2})dx\\right) ^{1\/2}$ and $\\left\\Vert u\\right\\Vert\n_{D^{1,2}}=\\left( \\int_{\\mathbb{R}^{N}}|\\nabla u|^{2}dx\\right) ^{1\/2}.$\nSince $\\overline{A}_{f}$\\ and $\\underline{A}_{f}$ are $0$-homogeneous, we\ncan denote the extremal values b\n\\begin{equation*}\n\\overline{\\mathbf{A}}_{f}=\\sup_{u\\in H^{1}(\\mathbb{R}^{N})\\backslash \\left\\{\n0\\right\\} }\\overline{A}_{f}(u)\\text{ and }\\underline{\\mathbf{A}\n_{f}=\\sup_{u\\in H^{1}(\\mathbb{R}^{N})\\backslash \\left\\{ 0\\right\\} \n\\underline{A}_{f}(u).\n\\end{equation*\nApplying the Gagliardo-Nirenberg and Young inequalities gives\n\\begin{eqnarray*}\n\\overline{A}_{f}(u) &\\leq &\\left( \\frac{f_{\\max }C_{p}^{p}\\left\\Vert\nu\\right\\Vert _{D^{1,2}}^{\\frac{N(p-2)}{2}}\\left\\Vert u\\right\\Vert\n_{L^{2}}^{p-\\frac{N(p-2)}{2}}}{\\left\\Vert u\\right\\Vert\n_{D^{1,2}}^{2(p-2)}\\left\\Vert u\\right\\Vert _{H^{1}}^{4-p}}\\right) ^{\\frac{2}\np-2}} \\\\\n&\\leq &\\left( f_{\\max }C_{p}^{p}\\right) ^{\\frac{2}{p-2}}\\left( \\frac\n\\left\\Vert u\\right\\Vert _{D^{1,2}}^{\\frac{\\left( N-4\\right) \\left(\np-2\\right) }{4-p}}\\left\\Vert u\\right\\Vert _{L^{2}}^{\\frac{2p-N\\left(\np-2\\right) }{4-p}}}{\\left\\Vert u\\right\\Vert _{D^{1,2}}^{2}+\\left\\Vert\nu\\right\\Vert _{L^{2}}^{2}}\\right) ^{\\frac{4-p}{p-2}} \\\\\n&\\leq &\\left( f_{\\max }C_{p}^{p}\\right) ^{\\frac{2}{p-2}}\\left( \\frac{\\max\n\\left\\{ (N-4)(p-2),(2p-N(p-2))\\right\\} }{2(4-p)}\\right) ^{\\frac{4-p}{p-2}}\n\\end{eqnarray*\nand\n\\begin{equation*}\n\\underline{A}_{f}\\left( u\\right) \\leq \\left( f_{\\min }C_{p}^{p}\\right) ^\n\\frac{2}{p-2}}\\left( \\frac{\\left\\Vert u\\right\\Vert _{L^{2}}^{2}}{\\left\\Vert\nu\\right\\Vert _{D^{1,2}}^{2}+\\left\\Vert u\\right\\Vert _{L^{2}}^{2}}\\right) ^\n\\frac{4-p}{p-2}}\\leq \\left( f_{\\min }C_{p}^{p}\\right) ^{\\frac{2}{p-2}},\n\\end{equation*\nwhere $f_{\\max }:=\\sup_{x\\in \\mathbb{R}^{N}}f\\left( x\\right) ,\\left\\Vert\nu\\right\\Vert _{L^{2}}=\\left( \\int_{\\mathbb{R}^{N}}u^{2}dx\\right) ^{1\/2}$and \nC_{p}>0$ is a sharp constant of Gagliardo-Nirenberg inequality. Thus, there\nexist two positive numbers $\\overline{C}_{0}(N,p,f)$ and $\\underline{C\n_{0}(p,f)$ such tha\n\\begin{equation*}\n0<\\overline{\\mathbf{A}}_{f}\\leq \\overline{C}_{0}(N,p,f)\\text{ for }N\\geq 4\n\\end{equation*\nand\n\\begin{equation*}\n0<\\underline{\\mathbf{A}}_{f}\\leq \\underline{C}_{0}(p,f)\\text{ for }N=4.\n\\end{equation*}\n\nLe\n\\begin{equation}\n\\overline{a}_{\\ast }=\\frac{2(p-2)}{4-p}\\left( \\frac{4-p}{p}\\right) ^{2\/(p-2)\n\\overline{\\mathbf{A}}_{f}\\text{ for }N\\geq 4  \\label{15-3}\n\\end{equation\nand\n\\begin{equation*}\n\\underline{a}_{\\ast }=\\frac{2(p-2)}{4-p}\\left( \\frac{4-p}{p}\\right)\n^{2\/(p-2)}\\underline{\\mathbf{A}}_{f}\\text{ for }N=4.\n\\end{equation*}\n\nWe now summarize the first part of our main results as follows.\n\n\\begin{theorem}\n\\label{t0-1}Suppose that $2<p<\\min \\left\\{ 4,2^{\\ast }\\right\\} $ and\ncondition $(D1)$ holds. Then the following statements are true.\\newline\n$(i)$ If $N=1,2,3,$ then $J_{a}$ is not bounded below on $H^{1}(\\mathbb{R\n^{N})$ for all $a>0;$\\newline\n$(ii)$ If $N=4,$ then for each $0<a<\\underline{a}_{\\ast },$ $J_{a}$ is not\nbounded below on $H^{1}(\\mathbb{R}^{N}),$ whereas for each $a>\\overline{a\n_{\\ast },$ $J_{a}$ is bounded below on $H^{1}(\\mathbb{R}^{N})$ and \n\\inf_{u\\in H^{1}(\\mathbb{R}^{N})\\backslash \\{0\\}}J_{a}(u)>0;$\\newline\n$(iii)$ If $N\\geq 5,$ then $J_{a}$ is bounded below on $H^{1}(\\mathbb{R\n^{N}) $ for all $a>0.$ More precisely, for each $0<a<\\overline{a}_{\\ast },$\nthere holds $-\\infty <\\inf_{u\\in H^{1}(\\mathbb{R}^{N})\\backslash\n\\{0\\}}J_{a}(u)<0,$ whereas for each $a>\\overline{a}_{\\ast },$ there holds \n\\inf_{u\\in H^{1}(\\mathbb{R}^{N})\\backslash \\{0\\}}J_{a}(u)>0.$\n\\end{theorem}\n\nFor brevity, we sum up the main result of Theorem \\ref{t0-1} with the table\nbelow\n\\begin{equation*}\n\\begin{tabular}{|c|c|c|c|c|}\n\\hline\n& $a>0$ & $0<a<\\underline{a}_{\\ast }$ & $0<a<\\overline{a}_{\\ast }$ & $a\n\\overline{a}_{\\ast }$ \\\\ \\hline\n$N=1,2,3$ & $\\inf J_{a}(u)=-\\infty $ & - & - & - \\\\ \\hline\n$N=4$ & - & $\\inf J_{a}(u)=-\\infty $ & - & $\\inf J_{a}(u)>0$ \\\\ \\hline\n$N\\geq 5$ & $\\inf J_{a}(u)>-\\infty $ & - & $\\inf J_{a}(u)<0$ & $\\inf\nJ_{a}(u)>0$ \\\\ \\hline\n\\end{tabular\n\\end{equation*}\n\nNote that $\\frac{p^{2\/\\left( p-2\\right) }}{2^{p\/\\left( p-2\\right) }}>1,$\nsince $2<p<2^{\\ast }$ for $N\\geq 4.$ From Theorem \\ref{t0-1} $(ii)-(iii)$,\none can see that $\\inf_{u\\in H^{1}(\\mathbb{R}^{N})\\backslash \\{0\\}}J_{a}(u)>0\n$ for $a>\\overline{a}_{\\ast }$, if $N\\geq 4$. However, we obtain the\nfollowing nonexistence result.\n\n\\begin{theorem}\n\\label{t0-2}Suppose that $N\\geq 4$ and condition $(D1)$ holds. Then for each\n$a>\\frac{p^{2\/\\left( p-2\\right) }}{2^{p\/\\left( p-2\\right) }}\\overline{a\n_{\\ast },$ Eq. $(E_{a})$ does not admit any nontrivial solutions.\n\\end{theorem}\n\nNext, we need the following assumption on $f.$\n\n\\begin{itemize}\n\\item[$\\left( D2\\right) $] $\\lim_{\\left\\vert x\\right\\vert \\rightarrow \\infty\n}f\\left( x\\right) =f_{\\infty }>0\\ $uniformly on$\\ \\mathbb{R}^{N}.$\n\\end{itemize}\n\n\\begin{theorem}\n\\label{t0-3}$(i)$ Suppose that $N\\geq 5$ and $f(x)\\equiv f_{\\infty }>0.$\nThen for each $0<a<\\overline{a}_{\\ast },$ Eq. $(E_{a})$ has a positive\nground state solution $v_{a}^{+}\\in H^{1}(\\mathbb{R}^{N})$ satisfying\n\\begin{equation*}\nJ_{a}^{\\infty }(v_{a}^{+})=\\inf_{u\\in H^{1}(\\mathbb{R}^{N})\\backslash\n\\{0\\}}J_{a}^{\\infty }(u)<0,\n\\end{equation*\nwhere $J_{a}^{\\infty }=J_{a}$ with $f(x)\\equiv f_{\\infty }.$\\newline\n$(ii)$ Suppose that $N\\geq 5$ and conditions $(D1)-(D2)$ hold. In addition,\nwe assume that\n\n\\begin{itemize}\n\\item[$(D3)$] $\\int_{\\mathbb{R}^{N}}(f(x)-f_{\\infty })(v_{a}^{+})^{p}dx>0,$\nwhere $v_{a}^{+}$ is the positive solution as described in part $(i)$.\n\\end{itemize}\n\nThen for each $0<a<\\overline{a}_{\\ast },$ Eq. $(E_{a})$ has a positive\nground state solution $u_{a}^{+}\\in H^{1}(\\mathbb{R}^{N})$ satisfying\n\\begin{equation*}\nJ_{a}(u_{a}^{+})=\\inf_{u\\in H^{1}(\\mathbb{R}^{N})\\backslash \\{0\\}}J_{a}(u)<0.\n\\end{equation*}\n\\end{theorem}\n\nIn order to obtain the existence of positive solution with positive energy\nfor Eq. $(E_{a}),$ it is necessary to introduce the filtration of the Nehari\nmanifold. That is,\n\\begin{equation*}\n\\mathbf{M}_{a}(c)=\\{u\\in \\mathbf{M}_{a}:J_{a}(u)<c\\}\\text{ for some }c>0,\n\\end{equation*\nwhere $\\mathbf{M}_{a}=\\{u\\in H^{1}(\\mathbb{R}^{N})\\backslash\n\\{0\\}:\\left\\langle J_{a}^{\\prime }(u),u\\right\\rangle =0\\}$ is the Nehari\nmanifold. We will show that $\\mathbf{M}_{a}(c)$ can be divided into two\nparts\n\\begin{equation*}\n\\mathbf{M}_{a}^{(1)}(c)=\\{u\\in \\mathbf{M}_{a}(c):\\Vert u\\Vert\n_{H^{1}}<C_{1}\\}\\ \\text{and}\\ \\mathbf{M}_{a}^{(2)}(c)=\\{u\\in \\mathbf{M\n_{a}(c):\\Vert u\\Vert _{H^{1}}>C_{2}\\},\n\\end{equation*\nin which each local minimizer of the functional $J_{a}$ is a critical point\nof $J_{a}$ in $H^{1}(\\mathbb{R}^{N})$. Our approach is to minimize the\nenergy functional $J_{a}$ on $\\mathbf{M}_{a}^{(1)}(c)$, where $J_{a}$ is\nbounded below and the minimizing sequence is bounded. In fact, such approach\nhas been applied in the study of Schrodinger-Poisson systems in $\\mathbb{R\n^{3}$ by us (see \\cite{SWF1,SWF2}).\n\nWe assume that $f$ satisfies the following condition:\n\n\\begin{itemize}\n\\item[$\\left( D4\\right) $] $f_{\\max }=\\sup_{x\\in \\mathbb{R}^{N}}f\\left(\nx\\right) <\\frac{f_{\\infty }}{D(p)^{(p-2)\/2}},$ wher\n\\begin{equation*}\nD(p)=\\left\\{\n\\begin{array}{ll}\n\\left( \\frac{4-p}{2}\\right) ^{1\/(p-2)}, & \\text{ if }2<p\\leq 3, \\\\\n\\frac{1}{2}, & \\text{ if }3<p<\\min \\{4,2^{\\ast }\\}\n\\end{array\n\\right.\n\\end{equation*}\n\\end{itemize}\n\n\\begin{remark}\n\\label{r-1} By a direct calculation, we obtain that for $2<p<4,$\n\\begin{equation*}\n\\frac{1}{2}\\leq D(p)<\\frac{1}{\\sqrt{e}}\\text{ and }D(p)\\left( \\frac{2}{4-p\n\\right) ^{2\/(p-2)}>1.\n\\end{equation*}\n\\end{remark}\n\nLet\n\\begin{equation}\n\\Lambda _{0}=\\left[ 1-D(p)\\left( \\frac{f_{\\max }}{f_{\\infty }}\\right)\n^{2\/(p-2)}\\right] \\left( \\frac{f_{\\infty }}{S_{p}^{p}}\\right) ^{2\/(p-2)},\n\\label{1-7}\n\\end{equation\nwhere $S_{p}$ is the best Sobolev constant for the embedding of $H^{1}\n\\mathbb{R}^{N})$ in $L^{p}(\\mathbb{R}^{N}).$ In particular, if $f(x)\\equiv\nf_{\\infty },$ then equality (\\ref{1-7}) becomes\n\\begin{equation*}\n\\Lambda _{0}=(1-D(p))\\left( \\frac{f_{\\infty }}{S_{p}^{p}}\\right) ^{2\/(p-2)}.\n\\end{equation*\nSe\n\\begin{equation*}\n\\Lambda =\\left\\{\n\\begin{array}{ll}\n\\frac{4-p}{2}\\left( \\frac{f_{\\infty }(4-p)}{2pS_{p}^{p}}\\right) ^{2\/(p-2)} &\n\\text{ if }N=1,2,3, \\\\\n\\min \\left\\{ \\frac{p-2}{2(4-p)}\\left( \\frac{4-p}{p}\\right) ^{2\/(p-2)}\\Lambda\n_{0},\\overline{a}_{\\ast }\\right\\} & \\text{ if }N\\geq 4\n\\end{array\n\\right.\n\\end{equation*\nLet $w_{0}$ be the unique positive solution of the following Schr\\\"{o}dinger\nequatio\n\\begin{equation}\n\\begin{array}{ll}\n-\\Delta u+u=f_{\\infty }|u|^{p-2}u & \\text{ in }\\mathbb{R}^{N}\n\\end{array}\n\\tag*{$\\left( E_{0}^{\\infty }\\right) $}\n\\end{equation\nFrom \\cite{K}, we see that\n\\begin{equation}\n\\left\\Vert w_{0}\\right\\Vert _{H^{1}}^{2}=\\int_{\\mathbb{R}^{N}}f_{\\infty\n}|w_{0}|^{p}dx=\\left( \\frac{S_{p}^{p}}{f_{\\infty }}\\right) ^{2\/(p-2)},\n\\label{1-8}\n\\end{equation\nand\n\\begin{equation*}\nJ_{0}^{\\infty }(w_{0})=\\frac{p-2}{2p}\\left( \\frac{S_{p}^{p}}{f_{\\infty }\n\\right) ^{2\/(p-2)},\n\\end{equation*\nwhere $J_{0}^{\\infty }$ is the energy functional of equation $(E_{0}^{\\infty\n})$ in $H^{1}(\\mathbb{R}^{N})$ in the form\n\\begin{equation*}\nJ_{0}^{\\infty }(u)=\\frac{1}{2}\\left\\Vert u\\right\\Vert _{H^{1}}^{2}-\\frac{1}{\n}\\int_{\\mathbb{R}^{N}}f_{\\infty }|u|^{p}dx.\n\\end{equation*}\n\nWe now summarize the second part of our main results as follows.\n\n\\begin{theorem}\n\\label{t1}Assume that $f(x)\\equiv f_{\\infty }>0.$ Then the following\nstatements are true.\\newline\n$\\left( i\\right) $ If $N\\geq 1,$ then for each $0<a<\\Lambda ,$ Eq. $(E_{a})$\nhas at least a positive solution $v_{a}^{-}\\in H^{1}(\\mathbb{R}^{N})$\nsatisfying\n\\begin{equation*}\n\\left\\Vert v_{a}^{-}\\right\\Vert _{H^{1}}<\\left( \\frac{2S_{p}^{p}}{f_{\\infty\n}(4-p)}\\right) ^{1\/(p-2)}\\text{ and }J_{a}^{\\infty }(v_{a}^{-})>\\frac{p-2}{2\n}\\left( \\frac{S_{p}^{p}}{f_{\\infty }}\\right) ^{2\/(p-2)}>0;\n\\end{equation*\n\\newline\n$\\left( ii\\right) $ If $1\\leq N\\leq 4,$ then for each $0<a<\\Lambda ,$ Eq. \n(E_{a})$ has a unique positive solution $v_{a}^{-}\\in H^{1}(\\mathbb{R}^{N}),$\nwhich is radially symmetric;\\newline\n$\\left( iii\\right) $ If $N\\geq 5,$ then for each $0<a<\\Lambda ,$ Eq. \n(E_{a}) $ has at least two positive solutions $v_{a}^{-},v_{a}^{+}\\in H^{1}\n\\mathbb{R}^{N})$ satisfyin\n\\begin{equation*}\n\\left\\Vert v_{a}^{-}\\right\\Vert _{H^{1}}<\\left( \\frac{2S_{p}^{p}}{f_{\\infty\n}(4-p)}\\right) ^{1\/(p-2)}<\\sqrt{2}\\left( \\frac{2S_{p}^{p}}{f_{\\infty }(4-p)\n\\right) ^{1\/(p-2)}<\\left\\Vert v_{a}^{+}\\right\\Vert _{H^{1}}\n\\end{equation*\nand $J_{a}^{\\infty }(v_{a}^{+})<0<J_{a}^{\\infty }(v_{a}^{-}).$ In\nparticular, $v_{a}^{+}$ is a ground state solution of Eq. $(E_{a}).$\n\\end{theorem}\n\nWith the aid of Theorem \\ref{t1}, we obtain the following results in the\nnonautnomous case.\n\n\\begin{theorem}\n\\label{t2}Suppose that $N\\geq 1$ and conditions $(D1)-(D2),(D4)$ hold. In\naddition, we assume that\n\n\\begin{itemize}\n\\item[$(D5)$] $\\int_{\\mathbb{R}^{N}}(f(x)-f_{\\infty })(v_{a}^{-})^{p}dx>0,$\nwhere $v_{a}^{-}$ is the positive solution as described in Theorem \\ref{t1}.\n\\end{itemize}\n\nThen for each $0<a<\\Lambda ,$ Eq. $(E_{a})$ has at least a positive solution\n$u_{a}^{-}\\in H^{1}(\\mathbb{R}^{N})$ satisfying\n\\begin{equation*}\n\\left\\Vert u_{a}^{-}\\right\\Vert _{H^{1}}<\\left( \\frac{2S_{p}^{p}}{f_{\\max\n}\\left( 4-p\\right) }\\right) ^{1\/(p-2)}\\text{ and }J_{a}(u_{a}^{-})>\\frac{p-\n}{4p}\\left( \\frac{S_{p}^{p}}{f_{\\max }}\\right) ^{2\/(p-2)}.\n\\end{equation*}\n\\end{theorem}\n\n\\begin{theorem}\n\\label{t3}Suppose that $N\\geq 5$ and conditions $(D1)-(D5)$ hold. Then for\neach $0<a<\\Lambda ,$ Eq. $(E_{a})$ has at least two positive solutions \nu_{a}^{-},u_{a}^{+}\\in H^{1}(\\mathbb{R}^{N})$ satisfying\n\\begin{equation*}\n\\left\\Vert u_{a}^{-}\\right\\Vert _{H^{1}}<\\left( \\frac{2S_{p}^{p}}{f_{\\max\n}(4-p)}\\right) ^{1\/(p-2)}<\\sqrt{2}\\left( \\frac{2S_{p}^{p}}{f_{\\max }(4-p)\n\\right) ^{1\/(p-2)}<\\left\\Vert u_{a}^{+}\\right\\Vert _{H^{1}}\n\\end{equation*\nand\n\\begin{equation*}\nJ_{a}(u_{a}^{+})<0<\\frac{p-2}{4p}\\left( \\frac{S_{p}^{p}}{f_{\\max }}\\right)\n^{2\/(p-2)}<J_{a}(u_{a}^{-}).\n\\end{equation*\nIn particular, $u_{a}^{+}$ is a ground state solution of Eq. $(E_{a}).$\n\\end{theorem}\n\n\\begin{theorem}\n\\label{t4} Let $u_{a}^{-}$ be the positive solution of Eq. $(E_{a})$ as\ndescribed in Theorem \\ref{t1} or \\ref{t2}. Then we have the following\nconclusions:\\newline\n$\\left( i\\right) $ If $N=1$ and $f(x)$ is weakly differentiable satisfying\n\\begin{equation*}\n(p-1)(p-2)f\\left( x\\right) +2\\langle \\nabla f(x),x\\rangle \\geq 0,\n\\end{equation*\nthen $u_{a}^{-}$ is a positive ground state solution of Eq. $(E_{a}).\n\\newline\n$\\left( ii\\right) $ If $N=2$ and $f(x)$ is weakly differentiable satisfyin\n\\begin{equation*}\n(p-2)f(x)+\\langle \\nabla f(x),x\\rangle \\geq 0,\n\\end{equation*\nthen $u_{a}^{-}$ is a positive ground state solution of Eq. $(E_{a}).$\n\\end{theorem}\n\nThe following table sums up the above main results:\n\n\\begin{equation*}\n\\begin{tabular}{|c|c|c|}\n\\hline\n& $a$ small & $a$ large \\\\ \\hline\n$N=1,2,3$ & One solution & - \\\\ \\hline\n$N=4$ & One solution & No solution \\\\ \\hline\n$N\\geq 5$ & Two solutions & No solution \\\\ \\hline\n\\end{tabular\n\\end{equation*}\n\nIn the above table, \"Two solutions\" (respectively \"One solution\") means that\nthere exist at least two (respectively one) positive solutions. On the other\nhand, \"No solution\" means that there are no nontrivial solutions.\n\nRecently, Azzollini \\cite{Az1,Az2} has proved that Eq. $(E_{a})$ with \nf(x)\\equiv f_{\\infty }$ admits a ground state solution with positive energy\nfor all $a>0$ when $N=3$ and for $a>0$ sufficiently small when $N=4.$ In the\nfollowing, we shall further describe some characteristics of such solution\ndepending on $a$ and $f_{\\infty }$, which are not concerned in \\cite{Az1,Az2\n.\n\nDefine the fibering map $h_{a,u}:t\\rightarrow J_{a}\\left( tu\\right) $ a\n\\begin{equation*}\nh_{a,u}\\left( t\\right) =\\frac{t^{2}}{2}\\left\\Vert u\\right\\Vert _{H^{1}}^{2}\n\\frac{at^{4}}{4}\\left( \\int_{\\mathbb{R}^{N}}\\left\\vert \\nabla u\\right\\vert\n^{2}dx\\right) ^{2}-\\frac{t^{p}}{p}\\int_{\\mathbb{R}^{N}}f(x)\\left\\vert\nu\\right\\vert ^{p}dx\\text{ for }t>0.\n\\end{equation*\nAbout its theory and application, we refer the reader to \\cite{BZ,DP}. Note\nthat for $u\\in H^{1}(\\mathbb{R}^{N})\\backslash \\left\\{ 0\\right\\} $ and $t>0,$\n$h_{a,u}^{\\prime }\\left( t\\right) =0$ holds if and only if $tu\\in \\mathbf{M\n_{a}$. In particular, $h_{a,u}^{\\prime }\\left( 1\\right) =0$ holds if and\nonly if $u\\in \\mathbf{M}_{a}.$ It is natural to split $\\mathbf{M}_{a}$ into\nthree parts corresponding to the local minima, local maxima and points of\ninflection. As a consequence, following \\cite{T}, we can define\n\\begin{eqnarray*}\n\\mathbf{M}_{a}^{+} &=&\\{u\\in \\mathbf{M}_{a}:h_{a,u}^{\\prime \\prime }\\left(\n1\\right) >0\\}, \\\\\n\\mathbf{M}_{a}^{0} &=&\\{u\\in \\mathbf{M}_{a}:h_{a,u}^{\\prime \\prime }\\left(\n1\\right) =0\\}, \\\\\n\\mathbf{M}_{a}^{-} &=&\\{u\\in \\mathbf{M}_{a}:h_{a,u}^{\\prime \\prime }\\left(\n1\\right) <0\\}.\n\\end{eqnarray*}\n\nFor $2<p<4,$ we se\n\\begin{eqnarray}\nA_{0} &=&\\frac{3\\left( p-1\\right) (-p^{2}+2p+12)}{p^{2}(p-2)}\\left( \\frac\nS_{p}^{p}}{f_{\\infty }}\\right) ^{2\/(p-2)}>0,  \\label{1-3} \\\\\n\\overline{A}_{0} &=&\\frac{p^{2}}{16}\\left( \\frac{f_{\\infty }}{S_{p}^{p}\n\\right) ^{2\/(p-2)}>0,  \\label{1-4} \\\\\nA_{0}^{\\ast } &=&\\frac{p-2}{2}\\left( \\frac{4-p}{p}\\right) ^{\\left(\n4-p\\right) \/(p-2)}(f_{\\infty }C_{p}^{p})^{2\/(p-2)}>0.  \\label{1-5}\n\\end{eqnarray}\n\nIt is clearly that $\\overline{A}_{0}>\\Lambda .$ We now state the last part\nof our main results as follows.\n\n\\begin{theorem}\n\\label{t5}Let $u_{0}$ be a nontrivial solution of Eq. $(E_{a})$ with \nf(x)\\equiv f_{\\infty }$. Then the following statements are true.\\newline\n$(i)$ When $N=3,$ for each $a>0$ with $\\sqrt{a^{2}+4}+\\frac{2}{a}\\geq A_{0},$\nthere holds $u_{0}\\in \\mathbf{M}_{a}^{-}.$ In particular, $v_{a}^{-}$ is a\nground state solution as in Theorem \\ref{t1} $\\left( i\\right) $ for $N=3.\n\\newline\n$(ii)$ When $N=4,$ for each $0<a\\leq \\overline{A}_{0},$ there holds \nu_{0}\\in \\mathbf{M}_{a}^{-},$ whereas for each $a>A_{0}^{\\ast },$ there\nholds $u_{0}\\in \\mathbf{M}_{a}^{+}.$ In particular, $v_{a}^{-}$ is a ground\nstate solution as in Theorem \\ref{t1} $\\left( i\\right) $ for $N=4.$\n\\end{theorem}\n\n\\begin{remark}\n$(i)$ Note that $\\inf_{a>0}\\left( \\sqrt{a^{2}+4}+\\frac{2}{a}\\right) >0.$\nThen whe\n\\begin{equation*}\nf_{\\infty }\\geq \\left[ \\frac{3\\left( p-1\\right) (-p^{2}+2p+12)}\np^{2}(p-2)\\inf_{a>0}\\left( \\sqrt{a^{2}+4}+\\frac{2}{a}\\right) }\\right]\n^{\\left( p-2\\right) \/2}S_{p}^{p},\n\\end{equation*\nthere holds $\\sqrt{a^{2}+4}+\\frac{2}{a}\\geq A_{0}$ for all $a>0.$ This shows\nthat $u_{0}\\in \\mathbf{M}_{a}^{-}$ for all $a>0.$\\newline\n$(ii)$ If there is a number $a_{0}>0$ such that $\\sqrt{a_{0}^{2}+4}+\\frac{2}\na_{0}}<A_{0},$ then there exist positive numbers $a_{1},a_{2}$ such that \n\\sqrt{a^{2}+4}+\\frac{2}{a}\\geq A_{0}$ for all $a\\in (0,a_{1}]\\cup \\lbrack\na_{2},\\infty )$ and $\\sqrt{a^{2}+4}+\\frac{2}{a}<A_{0}$ for all $a\\in\n(a_{1},a_{2}).$\n\\end{remark}\n\nThe structure of this paper is as follows. After briefly introducing some\ntechnical lemmas in Section 2, we prove Theorems \\ref{t0-1}--\\ref{t0-3} in\nSection 3, and demonstrate proofs of Theorems \\ref{t1} and \\ref{t2} in\nSections 4 and 5, respectively. Section 6 is dedicated to the proof of\nTheorems \\ref{t4} and \\ref{t5}.\n\n\\section{Preliminaries}\n\nFirstly, we consider the boundedness below for energy functional $J_{a}$ on \nH^{1}(\\mathbb{R}^{N})$ for $N\\geq 4.$ For $u\\in H^{1}(\\mathbb{R\n^{N})\\backslash \\{0\\},$ we define\n\\begin{equation}\nT_{f}(u)=\\left( \\frac{\\left\\Vert u\\right\\Vert _{H^{1}}^{2}}{\\int_{\\mathbb{R\n^{N}}f(x)|u|^{p}dx}\\right) ^{1\/(p-2)}.  \\label{2-0}\n\\end{equation}\n\n\\begin{lemma}\n\\label{g4}Suppose that $N\\geq 4$ and condition $(D1)$ holds. Then the\nfollowing results are true.\\newline\n$\\left( i\\right) $ For each $0<a<\\overline{a}_{\\ast }$ and $u\\in H^{1}\n\\mathbb{R}^{N})\\backslash \\{0\\},$ there exists a constant $\\widehat{t\n_{a}^{(0)}>\\left( \\frac{p}{4-p}\\right) ^{1\/(p-2)}T_{f}(u)$ such tha\n\\begin{equation}\n\\inf_{t\\geq 0}J_{a}(tu)=\\inf_{\\left( \\frac{p}{4-p}\\right)\n^{1\/(p-2)}T_{f}(u)<t<\\widehat{t}_{a}^{(0)}}J_{a}(tu)<0.  \\label{eqq17}\n\\end{equation\n$\\left( ii\\right) $ For each $a\\geq \\overline{a}_{\\ast }$ and $u\\in H^{1}\n\\mathbb{R}^{N})\\backslash \\left\\{ 0\\right\\} ,$ there holds $J_{a}\\left(\ntu\\right) \\geq 0$ for all $t>0.$\n\\end{lemma}\n\n\\begin{proof}\n$\\left( i\\right) $ For $u\\in H^{1}(\\mathbb{R}^{N})\\backslash \\{0\\}$ and $t>0\n, it has\n\\begin{eqnarray*}\nh_{a,u}(t) &=&J_{a}(tu)=\\frac{t^{2}}{2}\\left\\Vert u\\right\\Vert _{H^{1}}^{2}\n\\frac{at^{4}}{4}\\left( \\int_{\\mathbb{R}^{N}}|\\nabla u|^{2}dx\\right) ^{2}\n\\frac{t^{p}}{p}\\int_{\\mathbb{R}^{N}}f(x)|u|^{p}dx \\\\\n&=&t^{4}\\left[ g(t)+\\frac{a}{4}\\left( \\int_{\\mathbb{R}^{N}}|\\nabla\nu|^{2}dx\\right) ^{2}\\right] ,\n\\end{eqnarray*\nwhere\n\\begin{equation*}\ng(t)=\\frac{t^{-2}}{2}\\left\\Vert u\\right\\Vert _{H^{1}}^{2}-\\frac{t^{p-4}}{p\n\\int_{\\mathbb{R}^{N}}f(x)|u|^{p}dx.\n\\end{equation*\nClearly, $J_{a}(tu)=0$ if and only if\n\\begin{equation*}\ng(t)+\\frac{a}{4}\\left( \\int_{\\mathbb{R}^{N}}|\\nabla u|^{2}dx\\right) ^{2}=0.\n\\end{equation*\nIt is easy to see that\n\\begin{equation*}\ng(\\hat{t}_{a})=0,\\ \\lim_{t\\rightarrow 0^{+}}g(t)=\\infty \\ \\text{and \n\\lim_{t\\rightarrow \\infty }g(t)=0,\n\\end{equation*\nwhere $\\hat{t}_{a}=\\left( \\frac{p}{2}\\right) ^{1\/(p-2)}T_{f}(u).$ By\ncalculating the derivative of $g(t)$, we obtain\n\\begin{equation*}\ng^{\\prime }(t)=t^{-3}\\left[ -\\left\\Vert u\\right\\Vert _{H^{1}}^{2}+\\frac\n(4-p)t^{p-2}}{p}\\int_{\\mathbb{R}^{N}}f(x)|u|^{p}dx\\right] ,\n\\end{equation*\nwhich implies that $g(t)$ is decreasing when $0<t<\\left( \\frac{p}{4-p\n\\right) ^{1\/(p-2)}T_{f}(u)$ and is increasing when $t>\\left( \\frac{p}{4-p\n\\right) ^{1\/(p-2)}T_{f}(u).$ This indicates tha\n\\begin{eqnarray}\n\\inf_{t>0}g(t) &=&g\\left( \\left( \\frac{p}{4-p}\\right)\n^{1\/(p-2)}T_{f}(u)\\right)  \\notag \\\\\n&=&-\\frac{p-2}{2(4-p)}\\left( \\frac{p\\left\\Vert u\\right\\Vert _{H^{1}}^{2}}\n(4-p)\\int_{\\mathbb{R}^{N}}f(x)|u|^{p}dx}\\right) ^{-2\/(p-2)}\\Vert u\\Vert\n_{H^{1}}^{2}.  \\label{3-5}\n\\end{eqnarray\nNote that\n\\begin{equation*}\n0<a<\\frac{2(p-2)(4-p)^{(4-p)\/(p-2)}}{p^{2\/(p-2)}}\\overline{\\mathbf{A}}_{f}.\n\\end{equation*\nThen there exists $u\\in H^{1}(\\mathbb{R}^{N})\\backslash \\{0\\}$ such tha\n\\begin{equation*}\n\\frac{2(p-2)(4-p)^{(4-p)\/(p-2)}\\left( \\int_{\\mathbb{R}^{N}}f(x)|u|^{p}d\n\\right) ^{2\/(p-2)}}{p^{2\/(p-2)}\\left\\Vert u\\right\\Vert\n_{H^{1}}^{2(4-p)\/(p-2)}\\left\\Vert u\\right\\Vert _{D^{1,2}}^{4}}>a.\n\\end{equation*\nUsing the above inequality, together with $(\\ref{3-5}),$ leads t\n\\begin{equation*}\n\\inf_{t>0}g(t)<-\\frac{a}{4}\\left\\Vert u\\right\\Vert _{D^{1,2}}^{4}.\n\\end{equation*\nThis implies that there exist two numbers $\\widehat{t}_{a}^{(0)}$ and \n\\widehat{t}_{a}^{(1)}$ satisfying\n\\begin{equation*}\n0<\\widehat{t}_{a}^{(1)}<\\left( \\frac{p}{4-p}\\right) ^{1\/(p-2)}T_{f}(u)\n\\widehat{t}_{a}^{(0)}\n\\end{equation*\nsuch that\n\\begin{equation*}\ng\\left( \\widehat{t}_{a}^{(j)}\\right) +\\frac{a}{4}\\left\\Vert u\\right\\Vert\n_{D^{1,2}}^{4}=0\\text{ for }j=0,1.\n\\end{equation*\nThat is,\n\\begin{equation*}\nJ_{a}\\left( \\widehat{t}_{a}^{(j)}u\\right) =0\\text{ for }j=0,1.\n\\end{equation*\nThus,\n\\begin{eqnarray*}\nJ_{a}\\left( \\left( \\frac{p}{4-p}\\right) ^{1\/(p-2)}T_{f}(u)u\\right)&=&\\left[\n\\left( \\frac{p}{4-p}\\right) ^{1\/(p-2)}T_{f}(u)\\right]^{4}\\left[ g\\left(\n\\left( \\frac{p}{4-p}\\right) ^{1\/(p-2)}T_{f}(u)\\right) +\\frac{a}{4}\\left\\Vert\nu\\right\\Vert _{D^{1,2}}^{4}\\right] \\\\\n&<&0,\n\\end{eqnarray*\nand s\n\\begin{equation*}\n\\inf_{t\\geq 0}J_{a}(tu)<0.\n\\end{equation*\nNote tha\n\\begin{equation*}\nh_{a,u}^{\\prime }(t)=4t^{3}\\left( g(t)+\\frac{a}{4}\\left\\Vert u\\right\\Vert\n_{D^{1,2}}^{4}\\right) +t^{4}g^{\\prime }(t).\n\\end{equation*\nThen we have\n\\begin{equation*}\nh_{a,u}^{\\prime }(t)<0\\text{ for all }t\\in \\left( \\widehat{t\n_{a}^{(1)},\\left( \\frac{p}{4-p}\\right) ^{1\/(p-2)}T_{f}(u)\\right]\n\\end{equation*\nan\n\\begin{equation*}\nh_{a,u}^{\\prime }\\left( \\widehat{t}_{a}^{(0)}\\right) >0.\n\\end{equation*\nConsequently, we arrive at inequality (\\ref{eqq17}).\\newline\n$\\left( ii\\right) $ For each $u\\in H^{1}(\\mathbb{R}^{N})\\backslash \\{0\\},$\nwe can find a unique $t_{0}:=t_{0}(u)>0$ such that $h_{a,u}(t_{0})=0$ and \nh_{a,u}^{\\prime }(t_{0})=0.$ In fact, we only need to solve the system with\nrespect to the variables $t,a\n\\begin{equation*}\n\\left\\{\n\\begin{array}{c}\nh_{a,u}(t)=t^{2}\\left( \\frac{1}{2}\\left\\Vert u\\right\\Vert _{H^{1}}^{2}+\\frac\nat^{2}}{4}\\left\\Vert u\\right\\Vert _{D^{1,2}}^{4}-\\frac{t^{p-2}}{p}\\int_\n\\mathbb{R}^{N}}f(x)|u|^{p}dx\\right) =0, \\\\\nh_{a,u}^{\\prime }(t)=t\\left( \\left\\Vert u\\right\\Vert\n_{H^{1}}^{2}+at^{2}\\left\\Vert u\\right\\Vert _{D^{1,2}}^{4}-t^{p-2}\\int_\n\\mathbb{R}^{N}}f(x)|u|^{p}dx\\right) =0\n\\end{array\n\\right.\n\\end{equation*\nA direct calculation shows tha\n\\begin{equation*}\nt_{0}(u)=\\left( \\frac{2(p-2)\\int_{\\mathbb{R}^{N}}f(x)|u|^{p}dx}{ap\\left\\Vert\nu\\right\\Vert _{D^{1,2}}^{4}}\\right) ^{1\/(4-p)},\n\\end{equation*\nand accordingly,\n\\begin{eqnarray*}\na_{0}(u) &=&\\frac{2(p-2)(4-p)^{(4-p)\/(p-2)}\\left( \\int_{\\mathbb{R\n^{N}}f(x)|u|^{p}dx\\right) ^{2\/(p-2)}}{p^{2\/(p-2)}\\left\\Vert u\\right\\Vert\n_{D^{1,2}}^{4}\\left\\Vert u\\right\\Vert _{H^{1}}^{2(4-p)\/(p-2)}} \\\\\n&=&\\frac{2(p-2)(4-p)^{(4-p)\/(p-2)}}{p^{2\/(p-2)}}\\overline{A}_{f}(u).\n\\end{eqnarray*\nSince\n\\begin{equation*}\n\\overline{a}_{\\ast }=\\frac{2(p-2)(4-p)^{(4-p)\/(p-2)}}{p^{2\/(p-2)}}\\sup_{u\\in\nH^{1}(\\mathbb{R}^{N})\\backslash \\{0\\}}\\overline{A}_{f}(u)=\\frac\n2(p-2)(4-p)^{(4-p)\/(p-2)}}{p^{2\/(p-2)}}\\overline{\\mathbf{A}}_{f},\n\\end{equation*\nwe have for each $a\\geq \\overline{a}_{\\ast }$ and $u\\in H^{1}(\\mathbb{R\n^{N})\\backslash \\{0\\},$\n\\begin{equation*}\nh_{a,u}(t)=J_{a}(tu)\\geq 0\\text{ for all }t>0.\n\\end{equation*\nThis completes the proof.\n\\end{proof}\n\n\\begin{corollary}\n\\label{g17}Suppose that $N=4$ and condition $(D1)$ holds. Then for all \na\\geq \\overline{a}_{\\ast }$, $J_{a}$ is bounded below on $H^{1}(\\mathbb{R\n^{N})$ and $\\inf_{u\\in H^{1}(\\mathbb{R}^{N})\\backslash \\{0\\}}J_{a}(u)\\geq 0.$\n\\end{corollary}\n\n\\begin{lemma}\n\\label{g11}Suppose that $N\\geq 5$ and condition $(D1)$ holds. Then for all \na>0,$ $J_{a}$ is bounded below on $H^{1}(\\mathbb{R}^{N})$ and there exist\nnumbers $\\widehat{r},\\widehat{R}_{a}>0$ such tha\n\\begin{equation*}\nJ_{a}(u)>0\\text{ for all }u\\in H^{1}(\\mathbb{R}^{N})\\text{ with \n0<\\left\\Vert u\\right\\Vert _{H^{1}}<\\widehat{r}\\text{ or }\\left\\Vert\nu\\right\\Vert _{H^{1}}\\geq \\widehat{R}_{a}.\n\\end{equation*\nFurthermore, for each $0<a<\\overline{a}_{\\ast },$ there hold\n\\begin{equation*}\n-\\infty <\\inf_{\\widehat{r}<\\left\\Vert u\\right\\Vert _{H^{1}}<\\widehat{R\n_{a}}J_{a}(u)=\\inf_{u\\in H^{1}(\\mathbb{R}^{N})\\backslash \\{0\\}}J_{a}(u)<0.\n\\end{equation*}\n\\end{lemma}\n\n\\begin{proof}\nApplying the Gagliardo-Nirenberg and Young inequalities leads to\n\\begin{eqnarray*}\nJ_{a}\\left( u\\right) &\\geq &\\frac{a}{4}\\left\\Vert u\\right\\Vert\n_{D^{1,2}}^{4}+\\frac{1}{2}\\left\\Vert u\\right\\Vert _{H^{1}}^{2}-\\frac{f_{\\max\n}C_{p}^{p}}{p}\\left\\Vert u\\right\\Vert _{D^{1,2}}^{\\alpha p}\\left\\Vert\nu\\right\\Vert _{L^{2}}^{(1-\\alpha )p} \\\\\n&\\geq &\\frac{a}{4}\\left\\Vert u\\right\\Vert _{D^{1,2}}^{4}+\\frac{1}{2\n\\left\\Vert u\\right\\Vert _{H^{1}}^{2}-\\frac{2^{\\ast }}{\\alpha p^{2}}\\left(\nf_{\\max }C_{p}^{p}\\beta ^{-\\frac{(1-\\alpha )p}{2}}\\right) ^{\\frac{2^{\\ast }}\n\\alpha p}}\\left\\Vert u\\right\\Vert _{D^{1,2}}^{2^{\\ast }}-\\frac{\\beta }\n(1-\\alpha )p^{2}}\\left\\Vert u\\right\\Vert _{L^{2}}^{2} \\\\\n&\\geq &\\left\\Vert u\\right\\Vert _{D^{1,2}}^{2^{\\ast }}\\left[ \\frac{a}{4\n\\left\\Vert u\\right\\Vert _{D^{1,2}}^{4-2^{\\ast }}-\\frac{2^{\\ast }}{\\alpha\np^{2}}\\left( f_{\\max }C_{p}^{p}\\beta ^{-\\frac{(1-\\alpha )p}{2}}\\right) ^\n\\frac{2^{\\ast }}{\\alpha p}}\\right] \\\\\n&\\geq &-D_{0}\\text{ for some }D_{0}>0,\n\\end{eqnarray*\nwhere $\\alpha =\\frac{2^{\\ast }(p-2)}{p(2^{\\ast }-2)}$ and $0<\\beta <\\frac\np(2^{\\ast }-p)}{2^{\\ast }-2}.$ This implies that $J_{a}(u)$ is bounded below\non $H^{1}(\\mathbb{R}^{N})$ for all $a>0.$ Moreover, for each $a>0,$ there\nexist\n\\begin{equation*}\nR_{a}:=\\left[ \\frac{42^{\\ast }}{a\\alpha p^{2}}\\left( f_{\\max }C_{p}^{p}\\beta\n^{-\\frac{(1-\\alpha )p}{2}}\\right) ^{\\frac{2^{\\ast }}{\\alpha p}}\\right]\n^{1\/(4-2^{\\ast })}>0\n\\end{equation*\nsuch tha\n\\begin{equation*}\nJ_{a}(u)>0\\text{ for all }u\\in H^{1}(\\mathbb{R}^{N})\\text{ with }\\left\\Vert\nu\\right\\Vert _{D^{1,2}}>R_{a}.\n\\end{equation*}\n\nLet\n\\begin{equation*}\n\\widehat{R}_{a}=\\left[ R_{a}+\\left( \\frac{1}{2}-\\frac{1}{(1-\\alpha )p\n\\right) ^{-1}\\frac{2^{\\ast }}{\\alpha p}R_{a}^{2^{\\ast }}\\right] ^{1\/2}.\n\\end{equation*\nWe now prove that\n\\begin{equation*}\nJ_{a}(u)>0\\text{ for all }u\\in H^{1}(\\mathbb{R}^{N})\\text{ with }\\left\\Vert\nu\\right\\Vert _{H^{1}}>\\widehat{R}_{a}.\n\\end{equation*\nLet $u\\in H^{1}(\\mathbb{R}^{N})$ with $\\left\\Vert u\\right\\Vert _{H^{1}}\\geq\n\\widehat{R}_{a}.$ If $\\left\\Vert u\\right\\Vert _{D^{1,2}}>R_{a},$ then the\nresult is done clearly. If $\\left\\Vert u\\right\\Vert _{D^{1,2}}<R_{a},$ then\nit is enough to show that $J_{a}(u)\\geq 0$ when\n\\begin{equation*}\n\\int_{\\mathbb{R}^{N}}u^{2}dx>\\left( \\frac{1}{2}-\\frac{\\beta }{\\left(\n1-\\alpha \\right) p^{2}}\\right) ^{-1}\\frac{2^{\\ast }}{\\alpha p}\\left( f_{\\max\n}C_{p}^{p}\\beta ^{-\\frac{(1-\\alpha )p}{2}}\\right) ^{\\frac{2^{\\ast }}{\\alpha \n}}R_{a}^{2^{\\ast }}.\n\\end{equation*\nIndeed, note tha\n\\begin{equation}\n\\frac{f_{\\max }}{p}\\int_{\\mathbb{R}^{N}}|u|^{p}dx\\leq \\frac{2^{\\ast }}\n\\alpha p^{2}}\\left( f_{\\max }C_{p}^{p}\\beta ^{-\\frac{(1-\\alpha )p}{2\n}\\right) ^{\\frac{2^{\\ast }}{\\alpha p}}R_{a}^{2^{\\ast }}+\\frac{\\beta }\n(1-\\alpha )p^{2}}\\int_{\\mathbb{R}^{N}}u^{2}dx.  \\notag\n\\end{equation\nThen we hav\n\\begin{eqnarray*}\nJ_{a}(u) &\\geq &\\frac{a}{4}\\left\\Vert u\\right\\Vert _{D^{1,2}}^{4}+\\frac{1}{2\n\\left\\Vert u\\right\\Vert _{H^{1}}^{2}-\\frac{f_{\\max }}{p}\\int_{\\mathbb{R\n^{N}}|u|^{p}dx \\\\\n&>&\\left( \\frac{1}{2}-\\frac{\\beta }{\\left( 1-\\alpha \\right) p^{2}}\\right)\n\\int_{\\mathbb{R}^{N}}u^{2}dx-\\frac{2^{\\ast }}{\\alpha p^{2}}\\left( f_{\\max\n}C_{p}^{p}\\beta ^{-\\frac{(1-\\alpha )p}{2}}\\right) ^{\\frac{2^{\\ast }}{\\alpha \n}}R_{a}^{2^{\\ast }} \\\\\n&>&0.\n\\end{eqnarray*\nThus, we obtain that there exists a positive number $\\widehat{R}_{a}>R_{a}$\nsuch tha\n\\begin{equation*}\nJ_{a}(u)>0\\text{ for all }u\\in H^{1}(\\mathbb{R}^{N})\\text{ with }\\left\\Vert\nu\\right\\Vert _{H^{1}}\\geq \\widehat{R}_{a}.\n\\end{equation*}\n\nMoreover, using the Sobolev inequality give\n\\begin{equation*}\nJ_{a}(u)\\geq \\left\\Vert u\\right\\Vert _{H^{1}}^{2}\\left( \\frac{1}{2}-\\frac\nf_{\\max }}{pS_{p}^{p}}\\left\\Vert u\\right\\Vert _{H^{1}}^{p-2}\\right) ,\n\\end{equation*\nwhich implies that there exists a numbe\n\\begin{equation*}\n\\widehat{r}:=\\left( \\frac{pS_{p}^{p}}{2f_{\\max }}\\right) ^{1\/(p-2)}>0\n\\end{equation*\nsuch tha\n\\begin{equation*}\nJ_{a}(u)>0\\text{ for all }u\\in H^{1}(\\mathbb{R}^{N})\\text{ with \n0<\\left\\Vert u\\right\\Vert _{H^{1}}<\\widehat{r}.\n\\end{equation*\nHence, we hav\n\\begin{equation*}\nJ_{a}(u)>0\\text{ for all }u\\in H^{1}(\\mathbb{R}^{N})\\text{ with \n0<\\left\\Vert u\\right\\Vert _{H^{1}}<\\widehat{r}\\text{ or }\\left\\Vert\nu\\right\\Vert _{H^{1}}>\\widehat{R}_{a}.\n\\end{equation*}\n\nIt follows from Lemma \\ref{g4}$(i)$ that for each $0<a<\\overline{a}_{\\ast },\n\\begin{equation*}\n-\\infty <\\inf_{\\widehat{r}\\leq \\left\\Vert u\\right\\Vert _{H^{1}}\\leq \\widehat\nR}_{a}}J_{a}(u)=\\inf_{u\\in H^{1}(\\mathbb{R}^{N})\\backslash \\{0\\}}J_{a}(u)<0.\n\\end{equation*\nConsequently, this completes the proof.\n\\end{proof}\n\nAs pointed out in the section of Introduction, the Nehari manifold $\\mathbf{\n}_{a}$ given b\n\\begin{equation*}\n\\mathbf{M}_{a}=\\{u\\in H^{1}(\\mathbb{R}^{N})\\backslash \\{0\\}:\\left\\langle\nJ_{a}^{\\prime }(u),u\\right\\rangle =0\\},\n\\end{equation*\ncan be decomposed into three parts, i.e.,\n\\begin{equation*}\n\\mathbf{M}_{a}=\\mathbf{M}_{a}^{-}\\cup \\mathbf{M}_{a}^{0}\\cup \\mathbf{M\n_{a}^{+}.\n\\end{equation*\nThen we have the following result.\n\n\\begin{lemma}\n\\label{g2}Assume that $u_{0}$ is a local minimizer for $J_{a}$ on $\\mathbf{M\n_{a}$ and $u_{0}\\notin \\mathbf{M}_{a}^{0}.$ Then $J_{a}^{\\prime }(u_{0})=0$\nin $H^{-1}(\\mathbb{R}^{N}).$\n\\end{lemma}\n\n\\begin{proof}\nThe proof of Lemma \\ref{g2} is essentially same as that in Brown-Zhang \\cite\nTheorem 2.3]{BZ}, so we omit it here.\n\\end{proof}\n\nNote that $u\\in \\mathbf{M}_{a}$ if and only if $\\left\\Vert u\\right\\Vert\n_{H^{1}}^{2}+a\\left( \\int_{\\mathbb{R}^{N}}|\\nabla u|^{2}dx\\right) ^{2}-\\int_\n\\mathbb{R}^{N}}f(x)|u|^{p}dx=0.$ By the Sobolev inequality one ha\n\\begin{equation*}\n\\left\\Vert u\\right\\Vert _{H^{1}}^{2}\\leq \\left\\Vert u\\right\\Vert\n_{H^{1}}^{2}+a\\left( \\int_{\\mathbb{R}^{N}}|\\nabla u|^{2}dx\\right) ^{2}\\leq\nS_{p}^{-p}f_{\\max }\\left\\Vert u\\right\\Vert _{H^{1}}^{p}\\text{ for all }u\\in\n\\mathbf{M}_{a},\n\\end{equation*\nwhich leads to\n\\begin{equation}\n\\int_{\\mathbb{R}^{N}}f(x)|u|^{p}dx\\geq \\left\\Vert u\\right\\Vert\n_{H^{1}}^{2}\\geq \\left( \\frac{S_{p}^{p}}{f_{\\max }}\\right) ^{2\/(p-2)}>0\\text{\nfor all }u\\in \\mathbf{M}_{a}.  \\label{2}\n\\end{equation\nMoreover, for all $u\\in \\mathbf{M}_{a},$ we have\n\\begin{eqnarray}\nh_{a,u}^{\\prime \\prime }(1) &=&\\left\\Vert u\\right\\Vert _{H^{1}}^{2}+3a\\left(\n\\int_{\\mathbb{R}^{N}}|\\nabla u|^{2}dx\\right) ^{2}-(p-1)\\int_{\\mathbb{R\n^{N}}f(x)|u|^{p}dx  \\notag \\\\\n&=&-(p-2)\\left\\Vert u\\right\\Vert _{H^{1}}^{2}+a(4-p)\\left( \\int_{\\mathbb{R\n^{N}}|\\nabla u|^{2}dx\\right) ^{2}  \\notag \\\\\n&=&-2\\left\\Vert u\\right\\Vert _{H^{1}}^{2}+(4-p)\\int_{\\mathbb{R\n^{N}}f(x)|u|^{p}dx.  \\label{2-2}\n\\end{eqnarray\nThus, using $\\left( \\ref{2}\\right) $ and $(\\ref{2-2})$ gives\n\\begin{equation*}\nJ_{a}(u)=\\frac{1}{4}\\left\\Vert u\\right\\Vert _{H^{1}}^{2}-\\frac{(4-p)}{4p\n\\int_{\\mathbb{R}^{N}}f(x)|u|^{p}dx>\\frac{p-2}{4p}\\left( \\frac{S_{p}^{p}}\nf_{\\max }}\\right) ^{2\/(p-2)}\\text{ for all }u\\in \\mathbf{M}_{a}^{-}\n\\end{equation*\nWe need the following conclusion.\n\n\\begin{lemma}\n\\label{g1}Suppose that $N\\geq 1$ and $2<p<\\min \\{4,2^{\\ast }\\}.$ Then $J_{a}$\nis coercive and bounded below on $\\mathbf{M}_{a}^{-}.$ Furthermore, there\nholds\n\\begin{equation*}\nJ_{a}(u)>\\frac{p-2}{4p}\\left( \\frac{S_{p}^{p}}{f_{\\max }}\\right) ^{2\/(p-2)\n\\text{ for all }u\\in \\mathbf{M}_{a}^{-}.\n\\end{equation*}\n\\end{lemma}\n\n\\begin{lemma}\n\\label{g6}Suppose that $N=1,2,3$ and condition $(D1)$ holds$.$ Then for each\n$a>0$ and $u\\in H^{1}(\\mathbb{R}^{N})\\backslash \\{0\\}$ satisfying\n\\begin{equation*}\n\\int_{\\mathbb{R}^{N}}f(x)|u|^{p}dx>\\frac{p}{4-p}\\left( \\frac{2a(4-p)}{p-2\n\\right) ^{\\left( p-2\\right) \/2}\\left\\Vert u\\right\\Vert _{H^{1}}^{p},\n\\end{equation*\nthen there exist two numbers $t_{a}^{+}$ and $t_{a}^{-}$ satisfying\n\\begin{equation*}\nT_{f}(u)<t_{a}^{-}<\\sqrt{D(p)}\\left( \\frac{2}{4-p}\\right)\n^{1\/(p-2)}T_{f}(u)<\\left( \\frac{2}{4-p}\\right) ^{1\/(p-2)}T_{f}(u)<t_{a}^{+}\n\\end{equation*\nsuch that $t_{a}^{\\pm }u\\in \\mathbf{M}_{a}^{\\pm }$ and $J_{a}(t_{a}^{-}u)\n\\sup_{0\\leq t\\leq t_{a}^{+}}J_{a}(tu),$ and\n\\begin{equation*}\nJ_{a}(t_{a}^{+}u)=\\inf_{t\\geq t_{a}^{-}}J_{a}(tu)=\\inf_{t\\geq 0}J_{a}(tu)<0,\n\\end{equation*\nwhere $T_{f}(u)$ is defined as $(\\ref{2-0}).$\n\\end{lemma}\n\n\\begin{proof}\nLet\n\\begin{equation*}\nm(t)=t^{-2}\\left\\Vert u\\right\\Vert _{H^{1}}^{2}-t^{p-4}\\int_{\\mathbb{R\n^{N}}f(x)|u|^{p}dx\\text{ for }t>0.\n\\end{equation*\nClearly, $tu\\in \\mathbf{M}_{a}$ if and only if $m(t)+a\\left( \\int_{\\mathbb{R\n^{N}}|\\nabla u|^{2}dx\\right) ^{2}=0.$ A straightforward evaluation gives\n\\begin{equation*}\nm(T_{f}(u))=0,\\ \\lim_{t\\rightarrow 0^{+}}m(t)=\\infty \\text{ and \n\\lim_{t\\rightarrow \\infty }m(t)=0.\n\\end{equation*\nSince $2<p<4$ and\n\\begin{equation*}\nm^{\\prime }\\left( t\\right) =t^{-3}\\left( -2\\left\\Vert u\\right\\Vert\n_{H^{1}}^{2}+(4-p)t^{p-2}\\int_{\\mathbb{R}^{N}}f(x)|u|^{p}dx\\right) ,\n\\end{equation*\nwe find that $m(t)$ is decreasing when $0<t<\\left( \\frac{2}{4-p}\\right)\n^{1\/(p-2)}T_{f}(u)$ and is increasing when $t>\\left( \\frac{2}{4-p}\\right)\n^{1\/(p-2)}T_{f}(u).$ This indicates that\n\\begin{equation}\n\\inf_{t>0}m(t)=m\\left( \\left( \\frac{2}{4-p}\\right) ^{1\/(p-2)}T_{f}(u)\\right)\n.  \\label{2-4}\n\\end{equation\nFor each $a>0$ and $u\\in H^{1}(\\mathbb{R}^{N})\\backslash \\left\\{ 0\\right\\} $\nsatisfying\n\\begin{equation*}\n\\int_{\\mathbb{R}^{N}}f(x)|u|^{p}dx>\\frac{p}{4-p}\\left( \\frac{2a(4-p)}{p-2\n\\right) ^{(p-2)\/2}\\left\\Vert u\\right\\Vert _{H^{1}}^{p},\n\\end{equation*\nwe can conclude that\n\\begin{equation*}\nm\\left( \\left( \\frac{2}{4-p}\\right) ^{1\/(p-2)}T_{f}(u)\\right) <-2a\\left(\n\\frac{p}{2}\\right) ^{2\/(p-2)}\\left\\Vert u\\right\\Vert\n_{H^{1}}^{4}<-a\\left\\Vert u\\right\\Vert _{D^{1,2}}^{4},\n\\end{equation*\nwhere we have used the fact of $\\left( \\frac{p}{2}\\right) ^{2\/(p-2)}>1$.\nMoreover, by Remark $\\ref{r-1}$ we have\n\\begin{equation}\nT_{f}(u)<\\sqrt{D(p)}\\left( \\frac{2}{4-p}\\right) ^{1\/(p-2)}T_{f}(u)<\\left(\n\\frac{2}{4-p}\\right) ^{1\/(p-2)}T_{f}(u),  \\label{2-3}\n\\end{equation\nand a direct calculation shows that\n\\begin{equation}\n\\frac{\\left( \\frac{2}{4-p}\\right) D(p)^{(p-2)\/2}-1}{D(p)\\left( \\frac{2}{4-p\n\\right) ^{2\/(p-2)}}>\\frac{p-2}{2(4-p)}\\left( \\frac{4-p}{p}\\right) ^{2\/(p-2)}.\n\\label{2-1}\n\\end{equation\nIt follows from $(\\ref{2-4})-(\\ref{2-1})$ tha\n\\begin{equation*}\nm\\left( \\sqrt{D(p)}\\left( \\frac{2}{4-p}\\right) ^{1\/(p-2)}T_{f}(u)\\right)\n<-a\\left\\Vert u\\right\\Vert _{D^{1,2}}^{4}.\n\\end{equation*\nThus, there exist two numbers $t_{a}^{+},t_{a}^{-}>0$ which satisfy\n\\begin{equation*}\nT_{f}(u)<t_{a}^{-}<\\sqrt{D(p)}\\left( \\frac{2}{4-p}\\right)\n^{1\/(p-2)}T_{f}(u)<\\left( \\frac{2}{4-p}\\right) ^{1\/(p-2)}T_{f}(u)<t_{a}^{+}\n\\end{equation*\nsuch that\n\\begin{equation*}\nm(t_{a}^{\\pm })+a\\left\\Vert u\\right\\Vert _{D^{1,2}}^{4}=0,\n\\end{equation*\nleading to $t_{a}^{\\pm }u\\in \\mathbf{M}_{a}.$ By calculating the second\norder derivatives, we find\n\\begin{eqnarray*}\nh_{a,t_{a}^{-}u}^{\\prime \\prime }(1) &=&-2\\left\\Vert t_{a}^{-}u\\right\\Vert\n_{H^{1}}^{2}+(4-p)\\int_{\\mathbb{R}^{N}}f(x)|t_{a}^{-}u|^{p}dx \\\\\n&=&(t_{a}^{-})^{5}m^{\\prime }(t_{a}^{-})<0,\n\\end{eqnarray*\nand\n\\begin{eqnarray*}\nh_{a,t_{a}^{+}u}^{\\prime \\prime }(1) &=&-2\\left\\Vert t_{a}^{+}u\\right\\Vert\n_{H^{1}}^{2}+(4-p)\\int_{\\mathbb{R}^{N}}f(x)|t_{a}^{+}u|^{p}dx \\\\\n&=&(t_{a}^{-})^{5}m^{\\prime }(t_{a}^{+})>0.\n\\end{eqnarray*\nThese imply that $t_{a}^{\\pm }u\\in \\mathbf{M}_{a}^{\\pm }.$ Note that\n\\begin{equation*}\nh_{a,u}^{\\prime }(t)=t^{3}\\left( m(t)+a\\left( \\int_{\\mathbb{R}^{N}}|\\nabla\nu|^{2}dx\\right) ^{2}\\right) .\n\\end{equation*\nThen one can see that $h_{a,u}^{\\prime }(t)>0$ for all $t\\in\n(0,t_{a}^{-})\\cup (t_{a}^{+},\\infty )$ and $h_{a,u}^{\\prime }(t)<0$ for all \nt\\in (t_{a}^{-},t_{a}^{+})$. It leads to\n\\begin{equation*}\nJ_{a}(t_{a}^{-}u)=\\sup_{0\\leq t\\leq t_{a}^{+}}J_{a}(tu)\\text{ and \nJ_{a}(t_{a}^{+}u)=\\inf_{t\\geq t_{a}^{-}}J_{a}(tu),\n\\end{equation*\nand so $J_{a}\\left( t_{a}^{+}u\\right) <J_{a}(t_{a}^{-}u).$ Similar to the\nargument of Lemma \\ref{g4}$(i),$ we have\n\\begin{equation*}\nJ_{a}(t_{a}^{+}u)=\\inf_{t\\geq 0}J_{a}(tu)<0.\n\\end{equation*\nThis completes the proof.\n\\end{proof}\n\n\\begin{lemma}\n\\label{g15}Suppose that $N\\geq 4$ and condition $(D1)$ holds. Then for each \n0<a<\\overline{a}_{\\ast },$ there exist two numbers $t_{a}^{+}$ and \nt_{a}^{-} $ satisfying\n\\begin{equation*}\nT_{f}(u)<t_{a}^{-}<\\sqrt{D(p)}\\left( \\frac{2}{4-p}\\right)\n^{1\/(p-2)}T_{f}(u)<\\left( \\frac{2}{4-p}\\right) ^{1\/(p-2)}T_{f}(u)<t_{a}^{+}\n\\end{equation*\nsuch that $t_{a}^{\\pm }u\\in \\mathbf{M}_{a}^{\\pm }$ and $J_{a}(t_{a}^{-}u)\n\\sup_{0\\leq t\\leq t_{a}^{+}}J_{a}(tu)$ and $J_{a}(t_{a}^{+}u)=\\inf_{t\\geq\nt_{a}^{-}}J_{a}(tu)=\\inf_{t\\geq 0}J_{a}(tu)<0.$\n\\end{lemma}\n\n\\begin{proof}\nThe proof is analogous to those of Lemmas \\ref{g4} and \\ref{g6}, so we omit\nit here.\n\\end{proof}\n\nNow, we follow a part of idea in \\cite{SWF1}, for any $u\\in \\mathbf{M}_{a}$\nwith $J_{a}(u)<\\frac{D(p)(p-2)}{2p}\\left( \\frac{2S_{p}^{p}}{f_{\\infty }(4-p)\n\\right) ^{2\/(p-2)},$ deduce that\n\\begin{eqnarray*}\n\\frac{D\\left( p\\right) (p-2)}{2p}\\left( \\frac{2S_{p}^{p}}{f_{\\infty }\\left(\n4-p\\right) }\\right) ^{2\/(p-2)} &>&J_{a}(u) \\\\\n&\\geq &\\frac{p-2}{2p}\\left\\Vert u\\right\\Vert _{H^{1}}^{2}-\\frac{a(4-p)}{4p\n\\left\\Vert u\\right\\Vert _{H^{1}}^{4},\n\\end{eqnarray*\nwhich implies that for $0<a<\\frac{p-2}{2(4-p)}\\left( \\frac{4-p}{p}\\right)\n^{2\/(p-2)}\\Lambda _{0},$ there exist two positive numbers $D_{1}$ and $D_{2}$\nsatisfyin\n\\begin{equation*}\n\\sqrt{D(p)}\\left( \\frac{2S_{p}^{p}}{f_{\\infty }(4-p)}\\right)\n^{1\/(p-2)}<D_{1}<\\left( \\frac{2S_{p}^{p}}{f_{\\max }(4-p)}\\right) ^{1\/(p-2)}\n\\sqrt{2}\\left( \\frac{2S_{p}^{p}}{f_{\\infty }(4-p)}\\right) ^{1\/(p-2)}<D_{2}\n\\end{equation*\nsuch tha\n\\begin{equation*}\n\\left\\Vert u\\right\\Vert _{H^{1}}<D_{1}\\text{ or }\\left\\Vert u\\right\\Vert\n_{H^{1}}>D_{2}.\n\\end{equation*\nThus, we obtain that\n\\begin{eqnarray}\n&&\\mathbf{M}_{a}\\left( \\frac{D(p)(p-2)}{2p}\\left( \\frac{2S_{p}^{p}}\nf_{\\infty }(4-p)}\\right) ^{\\frac{2}{p-2}}\\right)  \\notag \\\\\n&=&\\left\\{ u\\in \\mathbf{M}_{a}:J_{a}(u)<\\frac{D(p)(p-2)}{2p}\\left( \\frac\n2S_{p}^{p}}{f_{\\infty }(4-p)}\\right) ^{\\frac{2}{p-2}}\\right\\}  \\notag \\\\\n&=&\\mathbf{M}_{a}^{(1)}\\cup \\mathbf{M}_{a}^{(2)},  \\label{4-4}\n\\end{eqnarray\nwhere\n\\begin{equation*}\n\\mathbf{M}_{a}^{(1)}:=\\left\\{ u\\in \\mathbf{M}_{a}\\left( \\frac{D(p)(p-2)}{2p\n\\left( \\frac{2S_{p}^{p}}{f_{\\infty }(4-p)}\\right) ^{2\/(p-2)}\\right)\n:\\left\\Vert u\\right\\Vert _{H^{1}}<D_{1}\\right\\}\n\\end{equation*\nand\n\\begin{equation*}\n\\mathbf{M}_{a}^{(2)}:=\\left\\{ u\\in \\mathbf{M}_{a}\\left( \\frac{D(p)(p-2)}{2p\n\\left( \\frac{2S_{p}^{p}}{f_{\\infty }(4-p)}\\right) ^{2\/(p-2)}\\right)\n:\\left\\Vert u\\right\\Vert _{H^{1}}>D_{2}\\right\\} .\n\\end{equation*\nWe further have\n\\begin{equation}\n\\left\\Vert u\\right\\Vert _{H^{1}}<D_{1}<\\left( \\frac{2S_{p}^{p}}{f_{\\max\n}(4-p)}\\right) ^{\\frac{1}{p-2}}\\text{ for all }u\\in \\mathbf{M}_{a}^{(1)}\n\\label{4-1}\n\\end{equation\nan\n\\begin{equation}\n\\left\\Vert u\\right\\Vert _{H^{1}}>D_{2}>\\sqrt{2}\\left( \\frac{2S_{p}^{p}}\nf_{\\infty }(4-p)}\\right) ^{\\frac{1}{p-2}}\\text{ for all }u\\in \\mathbf{M\n_{a}^{(2)}.  \\label{4-2}\n\\end{equation\nUsing the Sobolev inequality, $(\\ref{2-2})$ and $(\\ref{4-1})$ gives\n\\begin{equation*}\nh_{a,u}^{\\prime \\prime }(1)\\leq -2\\left\\Vert u\\right\\Vert\n_{H^{1}}^{2}+(4-p)S_{p}^{-p}f_{\\max }\\left\\Vert u\\right\\Vert _{H^{1}}^{p}<\n\\text{ for all }u\\in \\mathbf{M}_{a}^{(1)}.\n\\end{equation*\nBy $(\\ref{4-2}),$ we derive that\n\\begin{eqnarray*}\n\\frac{1}{4}\\left\\Vert u\\right\\Vert _{H^{1}}^{2}-\\frac{(4-p)}{4p}\\int_\n\\mathbb{R}^{N}}f(x)|u|^{p}dx &=&J_{a}(u)<\\frac{D(p)(p-2)}{2p}\\left( \\frac\n2S_{p}^{p}}{f_{\\infty }(4-p)}\\right) ^{2\/(p-2)} \\\\\n&<&\\frac{p-2}{2p}\\left( \\frac{2S_{p}^{p}}{f_{\\infty }(4-p)}\\right) ^{2\/(p-2)}\n\\\\\n&<&\\frac{p-2}{4p}\\left\\Vert u\\right\\Vert _{H^{1}}^{2}\\text{ for all }u\\in\n\\mathbf{M}_{a}^{(2)},\n\\end{eqnarray*\nwhich implies that\n\\begin{equation*}\n2\\left\\Vert u\\right\\Vert _{H^{1}}^{2}<(4-p)\\int_{\\mathbb{R}^{N}}f(x)|u|^{p}d\n\\text{ for all }u\\in \\mathbf{M}_{a}^{(2)}.\n\\end{equation*\nUsing the above inequality, together with $(\\ref{2-2})$, yields\n\\begin{equation*}\nh_{a,u}^{\\prime \\prime }\\left( 1\\right) =-2\\left\\Vert u\\right\\Vert\n_{H^{1}}^{2}+\\left( 4-p\\right) \\int_{\\mathbb{R}^{N}}f(x)\\left\\vert\nu\\right\\vert ^{p}dx>0\\text{ for all }u\\in \\mathbf{M}_{a}^{(2)}.\n\\end{equation*\nHence, we have the following result.\n\n\\begin{lemma}\n\\label{g7}For $N\\geq 1$ and $0<a<\\frac{p-2}{2(4-p)}\\left( \\frac{4-p}{p\n\\right) ^{2\/(p-2)}\\Lambda _{0},$ we have $\\mathbf{M}_{a}^{(1)}\\subset\n\\mathbf{M}_{a}^{-}$ and $\\mathbf{M}_{a}^{(2)}\\subset \\mathbf{M}_{a}^{+}$\nboth are $C^{1}$ sub-manifolds. Furthermore, each local minimizer of the\nfunctional $J_{a}$ in the sub-manifolds $\\mathbf{M}_{a}^{(1)}$ and $\\mathbf{\n}_{a}^{(2)}$ is a critical point of $J_{a}$ in $H^{1}(\\mathbb{R}^{N}).$\n\\end{lemma}\n\nAt the end of this section, similar to \\cite[Lemma 3.4]{LY}, we introduce a\nglobal compactness result, which is applicable to Kirchhoff type equations.\n\n\\begin{proposition}\n\\label{l0}Suppose that $N\\geq 1$ and conditions $(D1)-(D2)$ hold. Let \n\\{u_{n}\\}$ be a bounded $(PS)_{\\beta }$--sequence in $H^{1}(\\mathbb{R}^{N})$\nfor $J_{a}.$ There exist $u_{0}\\in H^{1}(\\mathbb{R}^{N})$ and $A\\in\n\\mathbb{R}\n$ such that $I_{a}^{\\prime }(u)=0,$ wher\n\\begin{equation*}\nI_{a}(u)=\\frac{(1+aA^{2})}{2}\\int_{\\mathbb{R}^{N}}|\\nabla u|^{2}dx+\\frac{1}{\n}\\int_{\\mathbb{R}^{N}}u^{2}dx-\\frac{1}{p}\\int_{\\mathbb{R}^{N}}f(x)|u|^{p}dx,\n\\end{equation*\nand either\\newline\n$(i)$ $u_{n}\\rightarrow u_{0}$ in $H^{1}(\\mathbb{R}^{N}),$or\\newline\n$(ii)$ there exists a number $m\\in \\mathbb{N}$ and $\\{x_{n}^{i}\\}_{n=1}^\n\\infty }\\subset \\mathbb{R}^{N}$ with $|x_{n}^{i}|\\rightarrow \\infty $ as \nn\\rightarrow \\infty $ for each $1\\leq i\\leq m,$ nontrivial solutions \nw^{1},w^{2},...,w^{m}\\in H^{1}(\\mathbb{R}^{N})$ of the following equatio\n\\begin{equation*}\n-(aA^{2}+1)\\Delta u+u=f_{\\infty }|u|^{p-2}u\n\\end{equation*\nsuch tha\n\\begin{equation*}\n\\beta +\\frac{aA^{4}}{4}=I_{a}(u_{0})+\\underset{i=1}{\\overset{m}{\\sum }\nI_{a}^{\\infty }(w^{i}),\n\\end{equation*\nan\n\\begin{equation*}\nu_{n}=u_{0}+\\underset{i=1}{\\overset{m}{\\sum }}w^{i}(\\cdot -x_{n}^{i})+o(1\n\\text{ strongly in }H^{1}(\\mathbb{R}^{N}),\n\\end{equation*\nan\n\\begin{equation*}\nA^{2}=\\int_{\\mathbb{R}^{N}}|\\nabla u_{0}|^{2}dx+\\underset{i=1}{\\overset{m}\n\\sum }}\\int_{\\mathbb{R}^{N}}|\\nabla w^{i}|^{2}dx.\n\\end{equation*}\n\\end{proposition}\n\n\\section{Proofs of Theorems \\protect\\ref{t0-1}, \\protect\\ref{t0-2} and\n\\protect\\ref{t0-3}}\n\n\\textbf{At the beginning of this section, we prove Theorem \\ref{t0-1}: }$(i)$\nFor $0<k<\\frac{4-N}{4-p}$ and $u\\in H^{1}(\\mathbb{R}^{N})\\backslash \\{0\\},$\nlet $v_{t}(x)=t^{k}u(t^{-1}x).$ Then we have\\newline\n$(i-A)$ $\\int_{\\mathbb{R}^{N}}|\\nabla v_{t}(x)|^{2}dx=t^{2k-2}\\int_{\\mathbb{\n}^{N}}|\\nabla u(t^{-1}x)|^{2}dx=t^{2k-2+N}\\int_{\\mathbb{R}^{N}}|\\nabla\nu(y)|^{2}dy;$\\newline\n$(i-B)$ $\\int_{\\mathbb{R}^{N}}|v_{t}(x)|^{2}dx=t^{2k}\\int_{\\mathbb{R\n^{N}}|u(tx)|^{2}dx=t^{2k+N}\\int_{\\mathbb{R}^{N}}|u(y)|^{2}dy;$\\newline\n$(i-C)$ $\\int_{\\mathbb{R}^{N}}|v_{t}(x)|^{p}dx=t^{pk}\\int_{\\mathbb{R\n^{N}}|u(tx)|^{p}dx=t^{pk+N}\\int_{\\mathbb{R}^{N}}|u(y)|^{p}dy.$\\newline\nUsing the above conclusions give\n\\begin{eqnarray*}\nJ_{a}(v_{t}(x)) &=&\\frac{a}{4}\\left( \\int_{\\mathbb{R}^{N}}|\\nabla\nv_{t}(x)|^{2}dx\\right) ^{2}+\\frac{1}{2}\\left( \\int_{\\mathbb{R}^{N}}|\\nabla\nv_{t}(x)|^{2}dx+\\int_{\\mathbb{R}^{N}}|v_{t}(x)|^{2}dx\\right) \\\\\n&&-\\frac{1}{p}\\int_{\\mathbb{R}^{N}}f(x)|v_{t}(x)|^{p}dx \\\\\n&\\leq &\\frac{a}{4}t^{4k-4+2N}\\left\\Vert u\\right\\Vert _{D^{1,2}}^{4}+\\frac{1}\n2}t^{2k-2+N}\\left\\Vert u\\right\\Vert _{D^{1,2}}^{2}+\\frac{1}{2}t^{2k+N}\\int_\n\\mathbb{R}^{N}}|u|^{2}dx \\\\\n&&-\\frac{f_{\\min }}{p}t^{pk+N}\\int_{\\mathbb{R}^{N}}|u|^{p}dx\\rightarrow\n-\\infty \\text{ as }t\\rightarrow \\infty ,\n\\end{eqnarray*\nsince $pk+N>4k-4+2N.$ This implies that $J_{a}$ is not bounded below on \nH^{1}(\\mathbb{R}^{N})\\ $for $N=1,2,3.$\\newline\n$(ii)$ It follows from Corollary \\ref{g17} that for each $a>\\overline{a\n_{\\ast },$ the energy functional $J_{a}$ is bounded below on $H^{1}(\\mathbb{\n}^{4})$ and $\\inf_{u\\in H^{1}(\\mathbb{R}^{4})\\backslash \\{0\\}}J_{a}(u)>0.$\nNext, we claim that for each $0<a<\\underline{a}_{\\ast },$ $J_{a}$ is not\nbounded below on $H^{1}(\\mathbb{R}^{4}),$ i.e., $\\inf_{u\\in H^{1}(\\mathbb{R\n^{4})\\backslash \\{0\\}}J_{a}(u)=-\\infty .$ Le\n\\begin{equation*}\nI(u)=\\frac{a}{4}\\left\\Vert u\\right\\Vert _{D^{1,2}}^{4}+\\frac{1}{2}\\int_\n\\mathbb{R}^{4}}u^{2}dx-\\frac{f_{\\min }}{p}\\int_{\\mathbb{R}^{4}}|u|^{p}d\n\\text{ for }u\\in H^{1}(\\mathbb{R}^{N}).\n\\end{equation*\nThen for $s>0,$ we hav\n\\begin{equation*}\nI(su)=s^{4}\\left( \\frac{a}{4}\\left\\Vert u\\right\\Vert _{D^{1,2}}^{4}\n\\overline{g}(s)\\right) ,\n\\end{equation*\nwhere\n\\begin{equation*}\n\\overline{g}(s)=\\frac{s^{-2}}{2}\\int_{\\mathbb{R}^{4}}u^{2}dx-\\frac{f_{\\min\n}s^{p-4}}{p}\\int_{\\mathbb{R}^{4}}|u|^{p}dx.\n\\end{equation*\nClearly, $I(su)=0$ if and only if $\\overline{g}(s)+\\frac{a}{4}\\left\\Vert\nu\\right\\Vert _{D^{1,2}}^{4}=0.$ It is not difficult to observe that \n\\overline{g}\\left( s_{a}\\right) =0,\\ \\lim_{s\\rightarrow 0^{+}}\\overline{g\n(s)=\\infty \\ $and$\\ \\lim_{s\\rightarrow \\infty }\\overline{g}(s)=0,$ where\n\\begin{equation*}\ns_{a}=\\left( \\frac{p\\int_{\\mathbb{R}^{4}}u^{2}dx}{2f_{\\min }\\int_{\\mathbb{R\n^{4}}\\left\\vert u\\right\\vert ^{p}dx}\\right) ^{1\/(p-2)}>0.\n\\end{equation*\nConsidering the derivative of $\\overline{g}(s)$, we find\n\\begin{equation*}\n\\overline{g}^{\\prime }(s)=s^{-3}\\left[ \\frac{s^{p-2}f_{\\min }(4-p)}{p}\\int_\n\\mathbb{R}^{4}}|u|^{p}dx-\\int_{\\mathbb{R}^{4}}u^{2}dx\\right] ,\n\\end{equation*\nwhich implies that $\\overline{g}(s)$ is decreasing when $0<t<\\left( \\frac\np\\int_{\\mathbb{R}^{4}}u^{2}dx}{(4-p)f_{\\min }\\int_{\\mathbb{R}^{4}}|u|^{p}dx\n\\right) ^{1\/(p-2)}$ and is increasing when $t>\\left( \\frac{p\\int_{\\mathbb{R\n^{4}}u^{2}dx}{(4-p)f_{\\min }\\int_{\\mathbb{R}^{4}}|u|^{p}dx}\\right)\n^{1\/(p-2)},$ and s\n\\begin{eqnarray}\n\\inf_{s>0}\\overline{g}(t) &=&\\overline{g}\\left( \\left( \\frac{p\\int_{\\mathbb{\n}^{4}}u^{2}dx}{(4-p)f_{\\min }\\int_{\\mathbb{R}^{4}}|u|^{p}dx}\\right)\n^{1\/(p-2)}\\right)  \\notag \\\\\n&=&-\\frac{p-2}{2}\\left( \\frac{f_{\\min }\\int_{\\mathbb{R}^{4}}|u|^{p}dx}{p\n\\right) ^{2\/(p-2)}\\left( \\frac{4-p}{\\int_{\\mathbb{R}^{4}}u^{2}dx}\\right)\n^{\\left( 4-p\\right) \/(p-2)}<0.  \\label{10-0}\n\\end{eqnarray\nSince $0<a<\\underline{a}_{\\ast },$ there exists $u\\in H^{1}(\\mathbb{R\n^{4})\\backslash \\{0\\}$ such tha\n\\begin{equation*}\na<2(p-2)\\left( \\frac{4-p}{\\int_{\\mathbb{R}^{4}}u^{2}dx}\\right) ^{\\left(\n4-p\\right) \/(p-2)}\\left( \\frac{f_{\\min }\\int_{\\mathbb{R}^{4}}|u|^{p}dx}{p\n\\right) ^{2\/(p-2)}\\left\\Vert u\\right\\Vert _{D^{1,2}}^{-4}\\leq \\underline{a\n_{\\ast }.\n\\end{equation*\nUsing the above inequality, together with $(\\ref{10-0}),$ leads to \n\\inf_{s>0}\\overline{g}(s)<-\\frac{a}{4}\\left\\Vert u\\right\\Vert\n_{D^{1,2}}^{4}. $ Se\n\\begin{equation*}\ns_{0}(u)=\\left( \\frac{p\\int_{\\mathbb{R}^{4}}u^{2}dx}{(4-p)f_{\\min }\\int_\n\\mathbb{R}^{4}}|u|^{p}dx}\\right) ^{1\/(p-2)}.\n\\end{equation*\nThen we obtain\n\\begin{equation}\nI(s_{0}(u)u)=s_{0}^{4}(u)\\left[ \\overline{g}(s_{0}(u))+\\frac{a}{4}\\left\\Vert\nu\\right\\Vert _{D^{1,2}}^{4}\\right] <0.  \\label{10-11}\n\\end{equation\nLet $u_{0}=s_{0}\\left( u\\right) u$ and $v_{t}(x)=u_{0}(t^{-1}x).$ Then we\nhave\\newline\n$(ii-A)$ $\\int_{\\mathbb{R}^{4}}|\\nabla v_{t}(x)|^{2}dx=t^{2}\\int_{\\mathbb{R\n^{4}}|\\nabla u_{0}(x)|^{2}dx;$\\newline\n$(ii-B)$ $\\int_{\\mathbb{R}^{4}}|v_{t}(x)|^{2}dx=t^{4}\\int_{\\mathbb{R\n^{4}}|u_{0}(x)|^{2}dx;$\\newline\n$(ii-C)$ $\\int_{\\mathbb{R}^{4}}|v_{t}(x)|^{p}dx=t^{4}\\int_{\\mathbb{R\n^{4}}|u_{0}(x)|^{p}dx.$\\newline\nCombining the above conclusions with $(\\ref{10-11})$ give\n\\begin{eqnarray*}\nJ_{a}(v_{t}(x)) &\\leq &t^{4}\\left( \\frac{a}{4}\\left\\Vert u_{0}\\right\\Vert\n_{D^{1,2}}^{4}+\\frac{1}{2}\\int_{\\mathbb{R}^{N}}u_{0}^{2}dx-\\frac{f_{\\min }}{\n}\\int_{\\mathbb{R}^{4}}|u_{0}|^{p}dx\\right) +\\frac{t^{2}}{2}\\left\\Vert\nu_{0}\\right\\Vert _{D^{1,2}}^{2} \\\\\n&=&t^{4}I(u_{0})+\\frac{t^{2}}{2}\\left\\Vert u_{0}\\right\\Vert _{D^{1,2}}^{2} \\\\\n&\\rightarrow &-\\infty \\text{ as }t\\rightarrow \\infty ,\n\\end{eqnarray*\nwhich implies that for each $0<a<\\underline{a}_{\\ast },$ $J_{a}$ is not\nbounded below on $H^{1}(\\mathbb{R}^{4}),$ i.e., $\\inf_{u\\in H^{1}(\\mathbb{R\n^{4})\\backslash \\{0\\}}J_{a}(u)=-\\infty .$\\newline\n$\\left( iii\\right) $ By Lemmas \\ref{g4} and \\ref{g11}, we can arrive at the\nconclusion.\n\n\\textbf{Next, we are ready to prove Theorem \\ref{t0-2}: }For $u\\in H^{1}\n\\mathbb{R}^{N})\\backslash \\left\\{ 0\\right\\} ,$ we know that $tu\\in \\mathbf{M\n_{a}^{0}$ if and only if $h_{a,tu}^{\\prime }\\left( 1\\right)\n=h_{a,tu}^{\\prime \\prime }\\left( 1\\right) =0,$ i.e., the following system of\nequations is satisfied\n\\begin{equation}\n\\left\\{\n\\begin{array}{c}\nt\\left\\Vert u\\right\\Vert _{H^{1}}^{2}+at^{3}\\left( \\int_{\\mathbb{R\n^{N}}\\left\\vert \\nabla u\\right\\vert ^{2}dx\\right) ^{2}-t^{p-1}\\int_{\\mathbb{\n}^{N}}f(x)\\left\\vert u\\right\\vert ^{p}dx=0, \\\\\n\\left\\Vert u\\right\\Vert _{H^{1}}^{2}+3at^{2}\\left( \\int_{\\mathbb{R\n^{N}}\\left\\vert \\nabla u\\right\\vert ^{2}dx\\right) ^{2}-(p-1)t^{p-2}\\int_\n\\mathbb{R}^{N}}f(x)\\left\\vert u\\right\\vert ^{p}dx=0\n\\end{array\n\\right.  \\label{15-2}\n\\end{equation\nBy solving the system $(\\ref{15-2})$ with respect to the variables $t$ and \na,$ we hav\n\\begin{equation*}\nt(u)=\\left( \\frac{2\\left\\Vert u\\right\\Vert _{H^{1}}^{2}}{(4-p)\\int_{\\mathbb{\n}^{N}}f(x)\\left\\vert u\\right\\vert ^{p}dx}\\right) ^{1\/(p-2)}\n\\end{equation*\nan\n\\begin{equation*}\na(u)=\\frac{p-2}{4-p}\\left( \\frac{4-p}{2}\\right) ^{2\/(p-2)}\\overline{A\n_{f}(u),\n\\end{equation*\nwhere $\\overline{A}_{f}(u)$ is as $(\\ref{15-1}).$ We conclude that $a(u)$ is\nthe unique parameter $a>0$ for which the fibering map $h_{a,u}$ has a\ncritical point with second derivative zero at $t(u)$. Moreover, if $a>a(u)$,\nthen $h_{a,u}$ is increasing on $(0,\\infty )$ and has no critical point.\nNote that $\\sup_{u\\in H^{1}(\\mathbb{R}^{N})\\backslash \\left\\{ 0\\right\\}\n}a(u)=\\frac{p^{2\/\\left( p-2\\right) }}{2^{p\/\\left( p-2\\right) }}\\overline{a\n_{\\ast }$ by (\\ref{15-3}). Hence, the energy functional $J_{a}$ has no any\nnontrivial critical points for $a>\\frac{p^{2\/\\left( p-2\\right) }}\n2^{p\/\\left( p-2\\right) }}\\overline{a}_{\\ast }.$ Consequently, we complete\nthe proof.\n\nTo prove that Theorem \\ref{t0-3}\\textbf{, }we need the following result.\n\n\\begin{lemma}\n\\label{g12}Suppose that $N\\geq 5$ and condition $(D1)$ holds. Let $0<a\n\\overline{a}_{\\ast }$ Then every minimizing sequence for $J_{a}$ in $H^{1}\n\\mathbb{R}^{N})$ is bounded.\n\\end{lemma}\n\n\\begin{proof}\nUsing Lemma \\ref{g11} give\n\\begin{equation*}\n-\\infty <\\inf_{u\\in H^{1}(\\mathbb{R}^{N})\\backslash \\{0\\}}J_{a}(u)<0.\n\\end{equation*\nLet $\\left\\{ u_{n}\\right\\} $ be a minimizing sequence for $J_{a}$ in $H^{1}\n\\mathbb{R}^{N}).$ Then by Lemma \\ref{g11} and the fact of $\\inf_{u\\in H^{1}\n\\mathbb{R}^{N})\\backslash \\{0\\}}J_{a}(u)<0$, there exists a number $\\widehat\nR}_{a}>0$ such that\n\\begin{equation*}\n\\left\\Vert u_{n}\\right\\Vert _{H^{1}}\\leq \\widehat{R}_{a}\\text{ for }n\\text{\nlarge enough.}\n\\end{equation*\nConsequently, we complete the proof.\n\\end{proof}\n\n\\textbf{At the end of this section, we begin to prove Theorem \\ref{t0-3}: }\n(i)$ By Lemma \\ref{g12} and the Ekeland variational principle, for each $0<a\n\\overline{a}_{\\ast }$ there exists a bounded minimizing sequence \n\\{u_{n}\\}\\subset H^{1}(\\mathbb{R}^{N})$ such that\n\\begin{equation*}\nJ_{a}^{\\infty }(u_{n})=\\inf_{u\\in H^{1}(\\mathbb{R}^{N})\\backslash\n\\{0\\}}J_{a}^{\\infty }(u)+o(1)\\text{ and }(J_{a}^{\\infty })^{\\prime\n}(u_{n})=o(1)\\text{ in }H^{-1}(\\mathbb{R}^{N}).\n\\end{equation*\nSimilar to the argument of Theorem \\ref{g10} in Appendix, we can prove that\nthe compactness for the sequence $\\{u_{n}\\}$ holds. Then for each $\\theta >0$\nthere exist a number $R=R(\\theta )>0$ and a sequence $\\{z_{n}\\}\\subset\n\\mathbb{R}^{N}$ such tha\n\\begin{equation}\n\\int_{\\lbrack B_{R}(z_{n})]^{c}}(|\\nabla\nu_{n}(x)|^{2}+u_{n}^{2}(x))dx<\\theta \\text{ uniformly for }n\\geq 1.\n\\label{10-2}\n\\end{equation\nDefine a new sequence of functions $v_{n}:=u_{n}(\\cdot +z_{n})\\in H^{1}\n\\mathbb{R}^{N}).$ Clearly, $\\left\\langle (J_{a}^{\\infty })^{\\prime\n}(v_{n}),v_{n}\\right\\rangle =o(1)$ and $J_{a}^{\\infty }(v_{n})=\\inf_{u\\in\nH^{1}(\\mathbb{R}^{N})\\backslash \\{0\\}}J_{a}^{\\infty }(u)+o(1).$ By virtue of\n$(\\ref{10-2})$, for each $\\theta >0$ there exists a number $R=R(\\theta )>0$\nsuch tha\n\\begin{equation}\n\\int_{\\lbrack B_{R}(0)]^{c}}(|\\nabla v_{n}(x)|^{2}+v_{n}^{2}(x))dx<\\theta\n\\text{ uniformly for }n\\geq 1.  \\label{10-3}\n\\end{equation\nSince $\\{v_{n}\\}$ is bounded in $H^{1}(\\mathbb{R}^{N}),$ one can assume that\nthere exist a subsequence $\\{v_{n}\\}$ and $v_{a}^{+}\\in H^{1}(\\mathbb{R\n^{N}) $ such tha\n\\begin{eqnarray}\nv_{n} &\\rightharpoonup &v_{a}^{+}\\text{ weakly in }H^{1}(\\mathbb{R}^{N}),\n\\label{10-4} \\\\\nv_{n} &\\rightarrow &v_{a}^{+}\\text{ strongly in }L_{loc}^{r}(\\mathbb{R}^{N}\n\\text{ for }2\\leq r<2^{\\ast },  \\label{10-5} \\\\\nv_{n} &\\rightarrow &v_{a}^{+}\\text{ a.e. in }\\mathbb{R}^{N}.  \\notag\n\\end{eqnarray\nBy $(\\ref{10-3})-(\\ref{10-5})$ and Fatou's Lemma, for any $\\theta >0$ and\nsufficiently large $n$, there exists a number $R>0$ such tha\n\\begin{eqnarray*}\n&&\\int_{\\mathbb{R}^{3}}|v_{n}-v_{a}^{+}|^{p}dx \\\\\n&\\leq\n&\\int_{B_{R}(0)}|v_{n}-v_{a}^{+}|^{p}dx\n\\int_{[B_{R}(0)]^{c}}|v_{n}-v_{a}^{+}|^{p}dx \\\\\n&\\leq &\\theta +S_{p}^{-p}\\left[ \\int_{[B_{R}(0)]^{c}}(|\\nabla\nv_{n}|^{2}+v_{n}^{2})dx+\\int_{[B_{R}(0)]^{c}}(|\\nabla\nv_{a}^{+}|^{2}+(v_{a}^{+})^{2})dx\\right] ^{\\frac{p}{2}} \\\\\n&\\leq &\\theta +S_{p}^{-p}(2\\theta )^{\\frac{p}{2}},\n\\end{eqnarray*\nwhich implies that for every $p\\in (2,2^{\\ast }),\n\\begin{equation}\nv_{n}\\rightarrow v_{a}^{+}\\text{ strongly in }L^{p}(\\mathbb{R}^{N}).\n\\label{10-6}\n\\end{equation\nSince $\\left\\langle (J_{a}^{\\infty })^{\\prime }(v_{n}),v_{n}\\right\\rangle\n=o\\left( 1\\right) $ and $\\widehat{r}<\\left\\Vert v_{n}\\right\\Vert _{H^{1}}\n\\widehat{R}_{a},$ using $(\\ref{10-6})$ gives\n\\begin{equation*}\n\\int_{\\mathbb{R}^{N}}f_{\\infty }|v_{a}^{+}|^{p}dx\\geq \\left( \\frac{S_{p}^{p\n}{f_{\\infty }}\\right) ^{2\/(p-2)}>0,\n\\end{equation*\nwhich indicates that $v_{a}^{+}\\not\\equiv 0.$\n\nNext, we show that $v_{n}\\rightarrow v_{a}^{+}$ strongly in $H^{1}(\\mathbb{R\n^{N}).$ Suppose on the contrary. Then we have\n\\begin{equation}\n\\left\\Vert v_{a}^{+}\\right\\Vert _{H^{1}}<\\liminf_{n\\rightarrow \\infty }\\Vert\nv_{n}\\Vert _{H^{1}}.  \\label{10-7}\n\\end{equation\nSimilar to the argument of Lemma \\ref{g6}, there exists a unique $t_{a}>0$\nsuch that\n\\begin{equation}\n(h_{a,v_{a}^{+}}^{\\infty })^{\\prime }(t_{a})=0,  \\label{10-8}\n\\end{equation\nwhere $h_{a,u}^{\\infty }(t)=h_{a,u}\\left( t\\right) $ with $f(x)\\equiv\nf_{\\infty }$. Since $\\left\\langle (J_{a}^{\\infty })^{\\prime\n}(v_{n}),v_{n}\\right\\rangle =o(1)$, it follows from $(\\ref{10-6})-(\\ref{10-7\n)$ that\n\\begin{equation}\n(h_{a,v_{a}^{+}}^{\\infty })^{\\prime }(1)<0.  \\label{10-9}\n\\end{equation\nCombining $(\\ref{10-8})-(\\ref{10-9})$ with the profile of \nh_{a,v_{a}^{+}}^{\\infty }(t)$ gives $t_{a}<1$. By $\\left( \\ref{10-6}\\right)\n-\\left( \\ref{10-7}\\right) $ again, we see $(h_{a,v_{n}}^{\\infty })^{\\prime\n}(t_{a})>0$ for sufficiently large $n$. Note that\n\\begin{equation*}\n\\left( h_{a,v_{n}}^{\\infty }\\right) ^{\\prime }(1)=o(1),\n\\end{equation*\nbecause of $\\left\\langle (J_{a}^{\\infty })^{\\prime\n}(v_{n}),v_{n}\\right\\rangle =o(1)$. Similar to the proof of Lemma \\ref{g6},\nwe obtain\n\\begin{equation*}\n\\left( h_{a,v_{n}}^{\\infty }\\right) ^{\\prime }(t)=t^{3}\\left( m^{\\infty\n}(t)+a\\left( \\int_{\\mathbb{R}^{N}}|\\nabla v_{n}|^{2}dx\\right) ^{2}\\right)\n\\text{ for }t>0,\n\\end{equation*\nwhere\n\\begin{equation*}\nm^{\\infty }(t):=t^{-2}\\left\\Vert v_{n}\\right\\Vert _{H^{1}}^{2}-t^{p-4}\\int_\n\\mathbb{R}^{N}}f_{\\infty }|v_{n}|^{p}dx.\n\\end{equation*\nOne can see that $m^{\\infty }(t)$ is decreasing for\n\\begin{equation*}\n0<t<\\left( \\frac{2\\left\\Vert v_{n}\\right\\Vert _{H^{1}}^{2}}{(4-p)\\int_\n\\mathbb{R}^{N}}f_{\\infty }|v_{n}|^{p}dx}\\right) ^{1\/(p-2)},\n\\end{equation*\nand\n\\begin{equation*}\n\\left( \\frac{2\\left\\Vert v_{n}\\right\\Vert _{H^{1}}^{2}}{(4-p)\\int_{\\mathbb{R\n^{N}}f_{\\infty }|v_{n}|^{p}dx}\\right) ^{1\/(p-2)}>1.\n\\end{equation*\nThis indicates that $\\left( h_{a,v_{n}}^{\\infty }\\right) ^{\\prime }(t)>0$\nfor $0<t<1$, which implies that $h_{a,v_{n}}^{\\infty }$ is increasing on \n(t_{a},1)$ for sufficiently large $n$. Thus, $h_{a,v_{n}}^{\\infty\n}(t_{a})<h_{a,v_{n}}^{\\infty }(1)$ holds for sufficiently large $n$. This\nimplies that\n\\begin{equation*}\nJ_{a}^{\\infty }(t_{a}v_{n})<J_{a}^{\\infty }(v_{n})\\text{ for sufficiently\nlarge }n,\n\\end{equation*\nand so, we have\n\\begin{equation*}\nJ_{a}^{\\infty }(t_{a}v_{a}^{+})<\\liminf_{n\\rightarrow \\infty }J_{a}^{\\infty\n}(t_{a}v_{n})\\leq \\liminf_{n\\rightarrow \\infty }J_{a}^{\\infty\n}(v_{n})=\\inf_{u\\in H^{1}(\\mathbb{R}^{N})\\backslash \\{0\\}}J_{a}^{\\infty }(u),\n\\end{equation*\nwhich is a contradiction. Thus, we obtain that $v_{n}\\rightarrow v_{a}^{+}$\nstrongly in $H^{1}(\\mathbb{R}^{N})$ and\n\\begin{equation*}\nJ_{a}^{\\infty }(v_{n})\\rightarrow J_{a}^{\\infty }(v_{a}^{+})=\\inf_{u\\in\nH^{1}(\\mathbb{R}^{N})\\backslash \\{0\\}}J_{a}^{\\infty }(u)\\text{ as \nn\\rightarrow \\infty .\n\\end{equation*\nHence, $v_{a}^{+}$ is a minimizer for $J_{a}^{\\infty }$ on $H^{1}(\\mathbb{R\n^{N}).$ Since\n\\begin{equation*}\n\\left\\vert v_{a}^{+}\\right\\vert \\in H^{1}(\\mathbb{R}^{N})\\backslash \\left\\{\n0\\right\\} \\ \\text{and}\\ J_{a}^{\\infty }(\\left\\vert v_{a}^{+}\\right\\vert\n)=J_{a}^{\\infty }(v_{a}^{+})=\\inf_{u\\in H^{1}(\\mathbb{R}^{N})\\backslash\n\\{0\\}}J_{a}^{\\infty }(u),\n\\end{equation*\none can see that $v_{a}^{+}$ is a positive solution of Eq. $(E_{a})$.\n\nBefore proving Theorem \\ref{t0-3} $(ii)$, we need the following compactness\nlemma which is regarded as a corollary of Proposition \\ref{g10}.\n\n\\begin{lemma}\n\\label{lem1}Suppose that $N\\geq 5$ and conditions $(D1)-(D2)$ hold. Let \n\\{u_{n}\\}$ be a $(PS)_{\\beta }$--sequence in $H^{1}(\\mathbb{R}^{N})$ for \nJ_{a}$ with $\\beta <\\inf_{u\\in H^{1}(\\mathbb{R}^{N})\\backslash\n\\{0\\}}J_{a}^{\\infty }(u)<0.$ Then there exist a subsequence $\\{u_{n}\\}$ and\na nonzero $u_{0}$ in $H^{1}(\\mathbb{R}^{N})$ such that $u_{n}\\rightarrow\nu_{0}$ strongly in $H^{1}(\\mathbb{R}^{N})$ and $J_{a}(u_{0})=\\beta .$\nFurthermore, $u_{0}$ is a nonzero solution of Eq. $(E_{a}).$\n\\end{lemma}\n\n\\textbf{The proof of Theorem \\ref{t0-3} }$(ii):$ By condition $(D3),$ we hav\n\\begin{equation*}\n\\inf_{u\\in H^{1}(\\mathbb{R}^{N})\\backslash \\{0\\}}J_{a}(u)\\leq\nJ_{a}(v_{a}^{+})<J_{a}^{\\infty }(v_{a}^{+})=\\inf_{u\\in H^{1}(\\mathbb{R\n^{N})\\backslash \\{0\\}}J_{a}^{\\infty }(u).\n\\end{equation*\nMoreover, by Lemmas \\ref{g11}, \\ref{g12}$(i)$ and the Ekeland variational\nprinciple, there exists a bounded minimizing sequence $\\{u_{n}\\}\\subset\nH^{1}(\\mathbb{R}^{N})$ with $\\widehat{r}<\\left\\Vert u_{n}\\right\\Vert\n_{H^{1}}<\\widehat{R}_{a}$ satisfying\n\\begin{equation*}\nJ_{a}(u_{n})=\\inf_{u\\in H^{1}(\\mathbb{R}^{N})\\backslash \\{0\\}}J_{a}(u)+o(1\n\\text{ and }J_{a}^{\\prime }(u_{n})=o\\left( 1\\right) \\text{ in }H^{-1}\n\\mathbb{R}^{N}).\n\\end{equation*\nBy virtue of Lemma \\ref{lem1}, we know that for each $0<a<\\overline{a}_{\\ast\n},$ Eq. $(E_{a})$ has a nontrivial solution $u_{a}^{+}$ such that \nJ_{a}(u_{a}^{+})=\\inf_{u\\in H^{1}(\\mathbb{R}^{N})\\backslash\n\\{0\\}}J_{a}(u)<0. $ Since $J_{a}(|u_{a}^{+}|)=J_{a}(u_{a}^{+})=\\inf_{u\\in\nH^{1}(\\mathbb{R}^{N})\\backslash \\{0\\}}J_{a}(u)$, we may assume that \nu_{a}^{+}$ is a positive solution of Eq. $(E_{a}).$ Consequently, we\ncomplete the proof.\n\n\\section{Proofs of Theorem \\protect\\ref{t1}}\n\nIn this section, we consider the following limit proble\n\\begin{equation}\n\\left\\{\n\\begin{array}{ll}\n-\\left( a\\int_{\\mathbb{R}^{N}}|\\nabla u|^{2}dx+1\\right) \\Delta u+u=f_{\\infty\n}|u|^{p-2}u & \\text{ in }\\mathbb{R}^{N}, \\\\\nu\\in H^{1}(\\mathbb{R}^{N}). &\n\\end{array\n\\right.  \\tag{$E_{a}^{\\infty }$}\n\\end{equation\nIt is clear that solutions of Eq. $(E_{a}^{\\infty })$ are critical points of\nthe energy functional $J_{a}^{\\infty }$ defined by\n\\begin{equation*}\nJ_{a}^{\\infty }(u)=\\frac{1}{2}\\left\\Vert u\\right\\Vert _{H^{1}}^{2}+\\frac{a}{\n}\\left( \\int_{\\mathbb{R}^{N}}|\\nabla u|^{2}dx\\right) ^{2}-\\frac{1}{p}\\int_\n\\mathbb{R}^{N}}f_{\\infty }|u|^{p}dx.\n\\end{equation*\nMoreover, $u\\in \\mathbf{M}_{a}^{\\infty }$ if and only if $\\left\\Vert\nu\\right\\Vert _{H^{1}}^{2}+a\\left( \\int_{\\mathbb{R}^{N}}\\left\\vert \\nabla\nu\\right\\vert ^{2}dx\\right) ^{2}-\\int_{\\mathbb{R}^{N}}f_{\\infty }|u|^{p}dx=0,$\nwhere $\\mathbf{M}_{a}^{\\infty }=\\mathbf{M}_{a}$ with $f(x)\\equiv f_{\\infty\n}. $ We denote by\n\\begin{equation*}\n\\mathbf{M}_{a}^{\\infty ,(j)}=\\mathbf{M}_{a}^{(j)}\\text{ with }f(x)\\equiv\nf_{\\infty }\\text{ for }j=1,2.\n\\end{equation*\nSince $w_{0}$ is the unique positive solution of Eq. $(E_{0}^{\\infty }),$ it\nfollows from $\\left( \\ref{1-8}\\right) $ that\n\\begin{equation*}\n\\int_{\\mathbb{R}^{N}}f_{\\infty }|w_{0}|^{p}dx=f_{\\infty\n}S_{p}^{-p}\\left\\Vert w_{0}\\right\\Vert _{H^{1}}^{p}>\\frac{p}{4-p}\\left(\n\\frac{2a(4-p)}{p-2}\\right) ^{(p-2)\/2}\\left\\Vert w_{0}\\right\\Vert\n_{H^{1}}^{p},\n\\end{equation*\nfor all $0<a<\\frac{p-2}{2(4-p)}\\left( \\frac{4-p}{p}\\right) ^{2\/(p-2)}\\Lambda\n_{0}.$ Thus, by Lemma \\ref{g6}, there exist two numbers $t_{a}^{\\infty\n,-},t_{a}^{\\infty ,+}>0$ satisfyin\n\\begin{equation*}\n1<t_{a}^{\\infty ,-}<\\sqrt{D(p)}\\left( \\frac{2}{4-p}\\right)\n^{1\/(p-2)}<t_{a}^{\\infty ,+},\n\\end{equation*\nsuch that $t_{a}^{\\infty ,\\pm }w_{0}\\in \\mathbf{M}_{a}^{\\infty ,\\pm },$\nwhere $\\mathbf{M}_{a}^{\\infty ,\\pm }=\\mathbf{M}_{a}^{\\pm }$ with $f(x)\\equiv\nf_{\\infty }$. Furthermore, there holds\n\\begin{equation*}\nJ_{a}^{\\infty }(t_{a}^{\\infty ,-}w_{0})=\\sup_{0\\leq t\\leq t_{a}^{\\infty\n,+}}J_{a}^{\\infty }(tw_{0})\n\\end{equation*\nand\n\\begin{equation}\nJ_{a}^{\\infty }(t_{a}^{\\infty ,+}w_{0})=\\inf_{t\\geq t_{a}^{\\infty\n,-}}J_{a}^{\\infty }(tw_{0})=\\inf_{t\\geq 0}J_{a}^{\\infty }(tw_{0})<0.\n\\label{28}\n\\end{equation\nA direct calculation shows tha\n\\begin{equation}\nJ_{a}^{\\infty }(t_{a}^{\\infty ,-}w_{0})<D(p)\\frac{p-2}{2p}\\left( \\frac\n2S_{p}^{p}}{f_{\\infty }(4-p)}\\right) ^{2\/(p-2)}.  \\label{27}\n\\end{equation\nIt follows from $(\\ref{28})-(\\ref{27})$ that $t_{a}^{\\infty ,-}w_{0}\\in\n\\mathbf{M}_{a}^{\\infty ,(1)}$ and $t_{a}^{\\infty ,+}w_{0}\\in \\mathbf{M\n_{a}^{\\infty ,(2)}.$ Namely, $\\mathbf{M}_{a}^{\\infty ,\\left( j\\right)\n}(j=1,2)$ are nonempty.\n\nDefine\n\\begin{equation*}\n\\alpha _{a}^{\\infty ,-}=\\inf_{u\\in \\mathbf{M}_{a}^{\\infty\n,(1)}}J_{a}^{\\infty }(u)=\\inf_{u\\in \\mathbf{M}_{a}^{\\infty ,-}}J_{a}^{\\infty\n}(u)\\text{ for }N\\geq 1.\n\\end{equation*\nThen we have\n\\begin{equation*}\n\\frac{p-2}{4p}\\left( \\frac{S_{p}^{p}}{f_{\\infty }}\\right) ^{2\/(p-2)}\\leq\n\\alpha _{a}^{\\infty ,-}<\\frac{D(p)(p-2)}{2p}\\left( \\frac{2S_{p}^{p}}\nf_{\\infty }(4-p)}\\right) ^{2\/(p-2)}\n\\end{equation*\nby Lemma \\ref{g1} and $(\\ref{27}).$\n\n\\textbf{We are now ready to prove Theorem \\ref{t1}: }$\\left( i\\right) $ By\nLemma \\ref{g7} and the Ekeland variational principle, there exists a\nsequence $\\{u_{n}\\}\\subset \\mathbf{M}_{a}^{\\infty ,(1)}$ satisfies\n\\begin{equation*}\nJ_{a}^{\\infty }(u_{n})=\\alpha _{a}^{\\infty ,-}+o(1)\\text{ and \n(J_{a}^{\\infty })^{\\prime }(u_{n})=o(1)\\text{ in }H^{-1}(\\mathbb{R}^{N}).\n\\end{equation*\nApplying Theorem \\ref{g10} in Appendix, we obtain that for $0<a<\\Lambda ,$\ncompactness holds for the sequence $\\{u_{n}\\}.$ Then for each $\\theta >0$\nthere exist a positive constant $R=R\\left( \\theta \\right) $ and a sequence \n\\{z_{n}\\}\\subset \\mathbb{R}^{N}$ such tha\n\\begin{equation}\n\\int_{\\left[ B_{R}(z_{n})\\right] ^{c}}(|\\nabla\nu_{n}(x)|^{2}+u_{n}^{2}(x))dx<\\theta \\text{ uniformly for }n\\geq 1.\n\\label{18-4}\n\\end{equation\nDefine a new sequence of functions $v_{n}:=u_{n}(\\cdot +z_{n})\\in H^{1}\n\\mathbb{R}^{N}).$ Then we have $\\{v_{n}\\}\\subset \\mathbf{M}_{a}^{\\infty\n,(1)} $ and $J_{a}^{\\infty }(v_{n})=\\alpha _{a}^{\\infty ,-}+o(1).$ By virtue\nof $(\\ref{18-4})$, for each $\\theta >0$ there exists a constant $R=R(\\theta\n)>0$ such tha\n\\begin{equation*}\n\\int_{\\left[ B_{R}(z_{n})\\right] ^{c}}(|\\nabla\nv_{n}(x)|^{2}+v_{n}^{2}(x))dx<\\theta \\text{ uniformly for }n\\geq 1.\n\\end{equation*\nSince $\\{v_{n}\\}$ is bounded in $H^{1}(\\mathbb{R}^{N}),$ one can assume that\nthere exist a subsequence $\\{v_{n}\\}$ and $v_{a}^{-}\\in H^{1}(\\mathbb{R\n^{N}) $ such tha\n\\begin{eqnarray*}\nv_{n} &\\rightharpoonup &v_{a}^{-}\\text{ weakly in }H^{1}(\\mathbb{R}^{N}); \\\\\nv_{n} &\\rightarrow &v_{a}^{-}\\text{ strongly in }L_{loc}^{r}(\\mathbb{R}^{N}\n\\text{ for }2\\leq r<2^{\\ast }; \\\\\nv_{n} &\\rightarrow &v_{a}^{-}\\text{ a.e. in }\\mathbb{R}^{N}.\n\\end{eqnarray*\nIn the following, by adapting the argument of Theorem \\ref{t0-3} $(i),$ we\nobtain\n\\begin{equation*}\nv_{n}\\rightarrow v_{a}^{-}\\text{ strongly in }H^{1}(\\mathbb{R}^{N})\n\\end{equation*\nand\n\\begin{equation*}\nJ_{a}^{\\infty }(v_{n})\\rightarrow J_{a}^{\\infty }(v_{a}^{-})=\\alpha\n_{a}^{\\infty ,-}\\text{ as }n\\rightarrow \\infty .\n\\end{equation*\nThus, $v_{a}^{-}$ is a minimizer for $J_{a}^{\\infty }$ on $\\mathbf{M\n_{a}^{\\infty ,-}$ for each $0<a<\\Lambda .$ By $(\\ref{27})$ one has\n\\begin{equation*}\nJ_{a}^{\\infty }(v_{a}^{-})=\\alpha _{a}^{\\infty ,-}\\leq J_{a}^{\\infty\n}(t_{a}^{\\infty ,-}w_{0})<\\frac{p-2}{2p}D(p)\\left( \\frac{2S_{p}^{p}}\nf_{\\infty }(4-p)}\\right) ^{\\frac{2}{p-2}},\n\\end{equation*\nwhich implies that $v_{a}^{-}\\in \\mathbf{M}_{a}^{\\infty ,(1)}$. Since \n|v_{a}^{-}|\\in \\mathbf{M}_{a}^{\\infty ,-}\\ $and$\\ J_{a}^{\\infty\n}(|v_{a}^{-}|)=J_{a}^{\\infty }(v_{a}^{-})=\\alpha _{a}^{\\infty ,-},$ one can\nsee that $v_{a}^{-}$ is a positive solution of Eq. $(E_{a})$ according to\nLemma \\ref{g2}. Note that for $2<p<\\min \\{4,2^{\\ast }\\},$ there holds\n\\begin{equation*}\n(4-p)\\int_{\\mathbb{R}^{N}}f_{\\infty }|v_{a}^{-}|^{p}dx<2\\left\\Vert\nv_{a}^{-}\\right\\Vert _{H^{1}}^{2}\\text{ and }T_{f_{\\infty\n}}(v_{a}^{-})v_{a}^{-}\\in \\mathbf{M}_{0}^{\\infty },\n\\end{equation*\nwhere $\\mathbf{M}_{0}^{\\infty }=\\mathbf{M}_{a}^{\\infty }$ with $a=0$ and\n\\begin{equation*}\n\\left( \\frac{4-p}{2}\\right) ^{1\/(p-2)}<T_{f_{\\infty }}(v_{a}^{-}):=\\left(\n\\frac{\\left\\Vert v_{a}^{-}\\right\\Vert _{H^{1}}^{2}}{\\int_{\\mathbb{R\n^{N}}f_{\\infty }|v_{a}^{-}|^{p}dx}\\right) ^{1\/(p-2)}<1.\n\\end{equation*\nThen by Lemmas $\\ref{g6}-\\ref{g15}$, we have\n\\begin{equation*}\nJ_{a}^{\\infty }(v_{a}^{-})=\\sup_{0\\leq t\\leq t_{a}^{+}}J_{a}^{\\infty\n}(tv_{a}^{-}),\n\\end{equation*\nwhere $1<\\left( \\frac{2}{4-p}\\right) ^{1\/(p-2)}T_{f_{\\infty\n}}(v_{a}^{-})<t_{a}^{+}$. Using the above equality, one get\n\\begin{equation*}\nJ_{a}^{\\infty }(v_{a}^{-})>J_{a}^{\\infty }(T_{f_{\\infty\n}}(v_{a}^{-})v_{a}^{-})>J_{0}^{\\infty }(T_{f_{\\infty\n}}(v_{a}^{-})v_{a}^{-})\\geq \\alpha _{0}^{\\infty }.\n\\end{equation*}\n\n$(ii)$ Following the argument of Theorem 2.1 in \\cite{SZ}. By $(i),$ we\nobtain that Eq. $(E_{a})$ admits a positive solution $v_{a,1}^{-}\\in H^{1}\n\\mathbb{R}^{N}).$ Applying Theorem 4.1 in \\cite{HL1} gives \nv_{a,1}^{-}\\rightarrow 0$ as $|x|\\rightarrow \\infty .$ Then after\ntranslation, we can make $v_{a,1}^{-}$ satisf\n\\begin{equation}\nv_{a,1}^{-}>0,v_{a,1}^{-}(\\infty )=0,v_{a,1}^{-}(0)=\\max v_{a,1}^{-}(x).\n\\label{18-5}\n\\end{equation}\n\nNow we show that $v_{a,1}^{-}$ is unique under $(\\ref{18-5})$. Otherwise, we\nassume that $v_{a,2}^{-}$ is another positive solution satisfying $(\\re\n{18-5})$. Le\n\\begin{equation*}\nK_{1}=b+a\\int_{\\mathbb{R}^{N}}|\\nabla v_{a,1}^{-}|^{2}dx\\text{ and \nK_{2}=b+a\\int_{\\mathbb{R}^{N}}|\\nabla v_{a,2}^{-}|^{2}dx.\n\\end{equation*\nThen $v_{a,i}^{-}(i=1,2)$ is a solution of the proble\n\\begin{equation*}\n-\\Delta u+\\frac{1}{K_{i}}u=\\frac{1}{K_{i}}|u|^{p-2}u\\text{ in }\\mathbb{R\n^{N}.\n\\end{equation*\nLet $w_{i}(x)=v_{a,i}^{-}(\\sqrt{K_{i}}x).$ Then $w_{i}(x)$ is a solution o\n\\begin{equation}\n\\left\\{\n\\begin{array}{ll}\n-\\Delta w+w=|w|^{p-2}w\\text{ } & \\text{in }\\mathbb{R}^{N}, \\\\\nw>0,w(\\infty )=0,w(0)=\\max w(x). &\n\\end{array\n\\right.  \\label{18-6}\n\\end{equation\nIt follows from \\cite{K} that the solution of problem $(\\ref{18-6})$ is\nunique. So $w_{1}(x)\\equiv w_{2}(x)$ i.e., $v_{a,1}^{-}(\\sqrt{K_{1}\nx)=v_{a,2}^{-}(\\sqrt{K_{2}}x).$ Thus, we have\n\\begin{equation}\nv_{a,2}^{-}(x)=v_{a,1}^{-}\\left( \\sqrt{\\frac{K_{1}}{K_{2}}}x\\right) .\n\\label{18-8}\n\\end{equation\nThus, we hav\n\\begin{eqnarray*}\nK_{2} &=&b+a\\int_{\\mathbb{R}^{N}}|\\nabla v_{a,2}^{-}|^{2}dx \\\\\n&=&b+a\\left( \\frac{K_{2}}{K_{1}}\\right) ^{\\frac{N-2}{2}}\\int_{\\mathbb{R\n^{N}}|\\nabla v_{a,1}^{-}|^{2}dx \\\\\n&=&b+\\left( \\frac{K_{2}}{K_{1}}\\right) ^{\\frac{N-2}{2}}(K_{1}-b),\n\\end{eqnarray*\nwhich implies tha\n\\begin{equation}\n\\frac{(K_{2}-b)^{2}}{K_{2}^{N-2}}=\\frac{(K_{1}-b)^{2}}{K_{1}^{N-2}}.\n\\label{18-7}\n\\end{equation\nDefin\n\\begin{equation*}\ny(x)=\\frac{(x-b)^{2}}{x^{N-2}}\\text{ for }x>b.\n\\end{equation*\nA direct calculation shows that\n\\begin{equation*}\ny^{\\prime }(x)=\\frac{\\left[ (4-N)x+b(N-2)\\right] (x-b)}{x^{N-1}}>0\\text{ for\n}x>b\\text{ and }1\\leq N\\leq 4,\n\\end{equation*}\nwhich implies that $y(x)$ is strictly increasing when $x>b$ and $1\\leq N\\leq\n4.$ This indicates that $K_{1}=K_{2},$since $K_{1},K_{2}>b.$ So by $(\\re\n{18-8})$ one have $v_{a,1}^{-}=v_{a,2}^{-}.$ Since the unique solution \nw_{1}(x)$ of problem $(\\ref{18-6})$ is radially symmetric by \\cite{K} and \nw_{1}(x)=v_{a,1}^{-}(\\sqrt{K_{1}}x)$, we obtain that $v_{a,1}^{-}$ is also\nradially symmetric.\n\n$\\left( iii\\right) $ Note that $\\Lambda \\leq \\overline{a}_{\\ast }$ for \nN\\geq 5.$ Then by virtue of Theorem \\ref{t0-3} $(i),$ for each $0<a<\\Lambda $\nEq. $(E_{a})$ admits a positive ground state solution $v_{a}^{+}\\in H^{1}\n\\mathbb{R}^{N})$ such that $J_{a}^{\\infty }(v_{a}^{+})<0$ when $N\\geq 5.$\nClearly, $v_{a}^{+}\\in \\mathbf{M}_{a}^{\\infty }\\left( \\frac{D(p)(p-2)}{2p\n\\left( \\frac{2S_{p}^{p}}{f_{\\infty }(4-p)}\\right) ^{2\/(p-2)}\\right) ,$ since\n$J_{a}^{\\infty }(v_{a}^{+})<0.$ By Lemma \\ref{g1}, we further get \nv_{a}^{+}\\in \\mathbf{M}_{a}^{\\infty ,(2)}.$ Thus, there hold\n\\begin{equation*}\n\\left\\Vert v_{a}^{+}\\right\\Vert _{H^{1}}>\\sqrt{2}\\left( \\frac{2S_{p}^{p}}\nf_{\\infty }(4-p)}\\right) ^{1\/(p-2)}.\n\\end{equation*\nMoreover, from $(i)$ it follows that for each $0<a<\\Lambda ,$ Eq. $(E_{a})$\nadmits a positive solution $v_{a}^{-}\\in H^{1}(\\mathbb{R}^{N})$ satisfyin\n\\begin{equation*}\nJ_{a}^{\\infty }(v_{a}^{-})>\\frac{p-2}{4p}\\left( \\frac{S_{p}^{p}}{f_{\\infty }\n\\right) ^{2\/(p-2)}\\text{ and }\\left\\Vert v_{a}^{-}\\right\\Vert\n_{H^{1}}<\\left( \\frac{2S_{p}^{p}}{f_{\\infty }(4-p)}\\right) ^{1\/(p-2)}.\n\\end{equation*\nConsequently, we complete the proof of Theorem \\ref{t1}.\n\n\\section{Proofs of Theorems \\protect\\ref{t2} and \\protect\\ref{t3}}\n\nBy virtue of Theorem \\ref{t1} $(i)$, we know that Eq. $(E_{a}^{\\infty })$\nadmits a positive solution $v_{a}^{-}\\in \\mathbf{M}_{a}^{\\infty ,-}$ such\nthat\n\\begin{equation*}\nJ_{a}^{\\infty }(v_{a}^{-})=\\alpha _{a}^{\\infty ,-}\\text{ and }(4-p)\\int_\n\\mathbb{R}^{N}}f_{\\infty }|v_{a}^{-}|^{p}dx<2\\Vert v_{a}^{-}\\Vert\n_{H^{1}}^{2}.\n\\end{equation*\nAccording to $(\\ref{2-0}),$ one has\n\\begin{equation}\nT_{f_{\\infty }}(v_{a}^{-})=\\left( \\frac{\\left\\Vert v_{a}^{-}\\right\\Vert\n_{H^{1}}^{2}}{\\int_{\\mathbb{R}^{N}}f_{\\infty }|v_{a}^{-}|^{p}dx}\\right)\n^{1\/(p-2)}>\\left( \\frac{4-p}{2}\\right) ^{1\/(p-2)}.  \\label{5-1}\n\\end{equation\nMoreover, by Theorem \\ref{t1} $(ii)$, we obtain that Eq. $(E_{a}^{\\infty })$\nadmits a positive solution $v_{a}^{+}\\in \\mathbf{M}_{a}^{\\infty ,+}$ such\nthat\n\\begin{equation*}\nJ_{a}^{\\infty }(v_{a}^{+})=\\alpha _{a}^{\\infty ,+}\\ \\text{and }(4-p)\\int_\n\\mathbb{R}^{N}}f_{\\infty }|v_{a}^{+}|^{p}dx>2\\Vert v_{a}^{+}\\Vert\n_{H^{1}}^{2}.\n\\end{equation*\nSimilar to $(\\ref{5-1})$, we have\n\\begin{equation*}\nT_{f_{\\infty }}(v_{a}^{+})=\\left( \\frac{\\left\\Vert v_{a}^{+}\\right\\Vert\n_{H^{1}}^{2}}{\\int_{\\mathbb{R}^{N}}f_{\\infty }|v_{a}^{+}|^{p}dx}\\right)\n^{1\/(p-2)}<\\left( \\frac{4-p}{2}\\right) ^{1\/(p-2)}.\n\\end{equation*\nThen we have the following results.\n\n\\begin{lemma}\n\\label{m5}$(i)$ Suppose that $N\\geq 1.$ Then for each $0<a<\\Lambda ,$ there\nexists $1<\\left( \\frac{2}{4-p}\\right) ^{1\/(p-2)}T_{f_{\\infty\n}}(v_{a}^{-})<t_{a}^{\\infty ,-}$ such that\n\\begin{equation}\nJ_{a}^{\\infty }(v_{a}^{-})=\\sup_{0\\leq t\\leq t_{a}^{\\infty ,-}}J_{a}^{\\infty\n}(tv_{a}^{-})=\\alpha _{a}^{\\infty ,-}.  \\label{eqq36}\n\\end{equation\n\\newline\n$(ii)$ Suppose that $N\\geq 5.$ Then for each $0<a<\\Lambda ,$ there exists \n0<t_{a}^{\\infty ,+}<1$ such that\n\\begin{equation*}\nJ_{a}^{\\infty }(v_{a}^{+})=\\inf_{t\\geq t_{a}^{\\infty ,+}}J_{a}^{\\infty\n}(tv_{a}^{+})=\\alpha _{a}^{\\infty ,+}.\n\\end{equation*}\n\\end{lemma}\n\n\\begin{proof}\n$(i)$ Let\n\\begin{equation}\nb_{a}^{\\infty }(t)=t^{-2}\\left\\Vert v_{a}^{-}\\right\\Vert\n_{H^{1}}^{2}-t^{p-4}\\int_{\\mathbb{R}^{N}}f_{\\infty }|v_{a}^{-}|^{p}dx\\text{\nfor }t>0.  \\label{5-3}\n\\end{equation\nClearly, there holds\n\\begin{equation}\nb_{a}^{\\infty }(1)+a\\left( \\int_{\\mathbb{R}^{N}}|\\nabla\nv_{a}^{-}|^{2}dx\\right) ^{2}=0  \\label{5-4}\n\\end{equation\nfor all $0<a<\\Lambda .$ It is easy to verify that\n\\begin{equation*}\nb_{a}^{\\infty }(t_{f_{\\infty }}(v_{a}^{-}))=0,\\ \\lim_{t\\rightarrow\n0^{+}}b_{a}^{\\infty }(t)=\\infty \\text{ and }\\lim_{t\\rightarrow \\infty\n}b_{a}^{\\infty }(t)=0.\n\\end{equation*\nBy calculating the derivative of $b_{a}^{\\infty }(t)$ one ha\n\\begin{equation*}\n(b_{a}^{\\infty })^{\\prime }(t)=t^{-3}\\left( -2\\left\\Vert\nv_{a}^{-}\\right\\Vert _{H^{1}}^{2}+(4-p)t^{p-2}\\int_{\\mathbb{R}^{N}}f_{\\infty\n}|v_{a}^{-}|^{p}dx\\right) ,\n\\end{equation*\nwhich implies that $b_{a}^{\\infty }(t)$ is decreasing when $0<t<\\left( \\frac\n2}{4-p}\\right) ^{1\/(p-2)}T_{f_{\\infty }}(v_{a}^{-})$ and is increasing when \nt>\\left( \\frac{2}{4-p}\\right) ^{1\/(p-2)}T_{f_{\\infty }}(v_{a}^{-}).$ This\nindicates that\n\\begin{equation}\n\\inf_{t>0}b_{a}^{\\infty }(t)=b_{a}^{\\infty }\\left( \\left( \\frac{2}{4-p\n\\right) ^{1\/(p-2)}T_{f_{\\infty }}(v_{a}^{-})\\right) .  \\label{5-2}\n\\end{equation\nMoreover, we notice that\n\\begin{equation}\n\\left( \\frac{2}{4-p}\\right) ^{1\/(p-2)}T_{f_{\\infty }}(v_{a}^{-})>1.\n\\label{5-5}\n\\end{equation\nThus, it follows from $(\\ref{5-4})-(\\ref{5-5})$ that\n\\begin{equation}\n\\inf_{t>0}b_{a}^{\\infty }(t)<b_{a}^{\\infty }(1)=-a\\left( \\int_{\\mathbb{R\n^{N}}|\\nabla v_{a}^{-}|^{2}dx\\right) ^{2},  \\label{5-6}\n\\end{equation\nwhich shows that there exists $1<\\left( \\frac{2}{4-p}\\right)\n^{1\/(p-2)}T_{f_{\\infty }}(v_{a}^{-})<t_{a}^{\\infty ,-}$ such that\n\\begin{equation*}\nb_{a}^{\\infty }(t_{a}^{\\infty ,-})+a\\left( \\int_{\\mathbb{R}^{N}}|\\nabla\nv_{a}^{-}|^{2}dx\\right) ^{2}=0.\n\\end{equation*\nUsing a similar argument as in Lemma \\ref{g6}, we arrive at $(\\ref{eqq36})$\n\\newline\n$(ii)$ The proof is analogous to that of part $(i)$, and we omit it here.\n\\end{proof}\n\n\\begin{lemma}\n\\label{m3}Suppose that $N\\geq 1$ and conditions ${(D1)}-{(D2),(D4)}-{(D5)}$\nhold. Then for each $0<a<\\Lambda ,$ there exist two constants $t_{a}^{(1),-}$\nand $t_{a}^{(2),-}$ satisfying\n\\begin{equation*}\nT_{f}(v_{a}^{-})<t_{a}^{(1),-}<\\left( \\frac{2}{4-p}\\right)\n^{1\/(p-2)}T_{f}(v_{a}^{-})<t_{a}^{(2),-}\n\\end{equation*\nsuch that $t_{a}^{(i),-}v_{a}^{-}\\in \\mathbf{M}_{a}^{(i)}\\ (i=1,2),$ and\n\\begin{equation*}\nJ_{a}\\left( t_{a}^{(1),-}v_{a}^{-}\\right) =\\sup_{0\\leq t\\leq\nt_{a}^{(2),-}}J_{a}(tv_{a}^{-})<\\alpha _{a}^{\\infty ,-}\\text{ and \nJ_{a}\\left( t_{a}^{(2),-}v_{a}^{-}\\right) =\\inf_{t\\geq\nt_{a}^{(1),-}}J_{a}(tv_{a}^{-}),\n\\end{equation*\nwhere $T_{f}(v_{a}^{-})$ is as $(\\ref{2-0})$ with $u=v_{a}^{-}.$\n\\end{lemma}\n\n\\begin{proof}\nLet\n\\begin{equation}\nb_{a}(t)=t^{-2}\\left\\Vert v_{a}^{-}\\right\\Vert _{H^{1}}^{2}-t^{p-4}\\int_\n\\mathbb{R}^{N}}f(x)|v_{a}^{-}|^{p}dx\\text{ for }t>0.  \\label{eqq41}\n\\end{equation\nApparently, $tv_{a}^{-}\\in \\mathbf{M}_{a}$ if and only if\n\\begin{equation*}\nb_{a}(t)+a\\left( \\int_{\\mathbb{R}^{N}}|\\nabla v_{a}^{-}|^{2}dx\\right) ^{2}=0.\n\\end{equation*\nBy analyzing $(\\ref{eqq41}),$ we obtain\n\\begin{equation*}\nb_{a}(T_{f}(v_{a}^{-}))=0,\\ \\lim_{t\\rightarrow 0^{+}}b_{a}(t)=\\infty \\text{\nand }\\lim_{t\\rightarrow \\infty }b_{a}(t)=0.\n\\end{equation*\nA direct calculation shows that\n\\begin{equation*}\nb_{a}^{\\prime }(t)=t^{-3}\\left( -2\\left\\Vert v_{a}^{-}\\right\\Vert\n_{H^{1}}^{2}+(4-p)t^{p-2}\\int_{\\mathbb{R}^{N}}f(x)|v_{a}^{-}|^{p}dx\\right) ,\n\\end{equation*\nwhich implies that $b_{a}(t)$ is decreasing on $0<t<\\left( \\frac{2}{4-p\n\\right) ^{1\/(p-2)}T_{f}(v_{a}^{-})$ and is increasing on $t>\\left( \\frac{2}\n4-p}\\right) ^{1\/(p-2)}T_{f}(v_{a}^{-}).$ By virtue of condition $(D5)$ one\nhas\n\\begin{equation*}\nT_{f}(v_{a}^{-})\\leq T_{f_{\\infty }}(v_{a}^{-})<1\\text{ and }b_{a}(t)\\leq\nb_{a}^{\\infty }(t),\n\\end{equation*\nwhere $b_{a}^{\\infty }(t)$ is given in $(\\ref{5-3}).$ Using condition $(D5)$\nand $(\\ref{5-6})$ gives\n\\begin{eqnarray*}\n\\inf_{t>0}b_{a}(t) &=&b_{a}\\left( \\left( \\frac{2}{4-p}\\right)\n^{1\/(p-2)}T_{f}(v_{a}^{-})\\right) \\\\\n&\\leq &-\\frac{p-2}{4-p}\\left( \\frac{4-p}{2}\\right) ^{2\/(p-2)}\\left\\Vert\nv_{a}^{-}\\right\\Vert _{H^{1}}^{2}\\left( \\frac{\\left\\Vert\nv_{a}^{-}\\right\\Vert _{H^{1}}^{2}}{\\int_{\\mathbb{R}^{N}}f_{\\infty\n}|v_{a}^{-}|^{p}dx}\\right) ^{-2\/(p-2)} \\\\\n&=&\\inf_{t>0}b_{a}^{\\infty }(t)<-a\\left( \\int_{\\mathbb{R}^{N}}|\\nabla\nv_{a}^{-}|^{2}dx\\right) ^{2}.\n\\end{eqnarray*\nThis explicitly tells us that there are two constants $t_{a}^{(1),-}$ and \nt_{a}^{(2),-}$ satisfyin\n\\begin{equation*}\nT_{f}(v_{a}^{-})<t_{a}^{(1),-}<\\left( \\frac{2}{4-p}\\right)\n^{1\/(p-2)}T_{f}\\left( v_{a}^{-}\\right) <t_{a}^{(2),-}\n\\end{equation*\nsuch that\n\\begin{equation*}\nb_{a}\\left( t_{a}^{(i),-}\\right) +a\\left( \\int_{\\mathbb{R}^{N}}|\\nabla\nv_{a}^{-}|^{2}dx\\right) ^{2}=0\\text{ for }i=1,2.\n\\end{equation*\nThat is, $t_{a}^{(i),-}v_{a}^{-}\\in \\mathbf{M}_{a}$ $(i=1,2).$\n\nA direct calculation on the second order derivatives gives\n\\begin{eqnarray*}\nh_{a,t_{a}^{(1),-}v_{a}^{-}}^{\\prime \\prime }(1) &=&-2\\left\\Vert\nt_{a}^{(1),-}v_{a}^{-}\\right\\Vert _{H^{1}}^{2}+(4-p)\\int_{\\mathbb{R\n^{N}}f(x)\\left\\vert t_{a}^{(1),-}v_{a}^{-}\\right\\vert ^{p}dx \\\\\n&=&\\left( t_{a}^{(1),-}\\right) ^{5}b_{a}^{\\prime }\\left(\nt_{a}^{(1),-}\\right) <0\n\\end{eqnarray*\nand\n\\begin{eqnarray*}\nh_{a,t_{a}^{(2),-}v_{a}^{-}}^{\\prime \\prime }(1) &=&-2\\left\\Vert\nt_{a}^{(2),-}v_{a}^{-}\\right\\Vert _{H^{1}}^{2}+(4-p)\\int_{\\mathbb{R\n^{N}}f(x)\\left\\vert t_{a}^{(2),-}v_{a}^{-}\\right\\vert ^{p}dx \\\\\n&=&\\left( t_{a}^{(2),-}\\right) ^{5}b_{a}^{\\prime }\\left(\nt_{a}^{(2),-}\\right) >0.\n\\end{eqnarray*\nSo, we get $t_{a}^{(1),-}v_{a}^{-}\\in \\mathbf{M}_{a}^{-}$ and \nt_{a}^{(2),-}v_{a}^{-}\\in \\mathbf{M}_{a}^{+}.$\n\nNote that\n\\begin{equation*}\nt_{a}^{(1),-}<\\left( \\frac{2}{4-p}\\right) ^{\\frac{1}{p-2}}T_{f}(v_{a}^{-}\n\\leq \\left( \\frac{2}{4-p}\\right) ^{\\frac{1}{p-2}}T_{f_{\\infty\n}}(v_{a}^{-})<t_{a}^{\\infty },\n\\end{equation*\nwhere $t_{a}^{\\infty }$ is the same as described in Lemma \\ref{m5}. It\nfollows from Lemma \\ref{m5}$(i)$ and condition $\\left( D5\\right) $ that for\neach $0<a<\\Lambda ,\n\\begin{eqnarray*}\nJ_{a}\\left( t_{a}^{(1),-}v_{a}^{-}\\right) &=&J_{a}^{\\infty }\\left(\nt_{a}^{(1),-}v_{a}^{-}\\right) -\\frac{\\left( t_{a}^{\\left( 1\\right)\n,-}\\right) ^{p}}{p}\\int_{\\mathbb{R}^{N}}(f(x)-f_{\\infty })(v_{a}^{-})^{p}dx\n\\\\\n&\\leq &\\sup_{0\\leq t\\leq t_{a}^{\\infty }}J_{a}^{\\infty }(tv_{a}^{-})-\\frac\n\\left( t_{a}^{\\left( 1\\right) ,-}\\right) ^{p}}{p}\\int_{\\mathbb{R\n^{N}}(f(x)-f_{\\infty })(v_{a}^{-})^{p}dx \\\\\n&<&\\alpha _{a}^{\\infty ,-}<\\frac{p-2}{2p}D(p)\\left( \\frac{2S_{p}^{p}}\nf_{\\infty }(4-p)}\\right) ^{2\/(p-2)},\n\\end{eqnarray*\nwhich indicates that $t_{a}^{(1),-}v_{a}^{-}\\in \\mathbf{M}_{a}^{(1)}$ and \nJ_{a}\\left( t_{a}^{(1),-}v_{a}^{-}\\right) <\\alpha _{a}^{\\infty ,-}.$ Note\ntha\n\\begin{equation*}\nh_{a,v_{a}^{-}}^{\\prime }\\left( t\\right) =t^{3}\\left( b_{a}\\left( t\\right)\n+a\\left( \\int_{\\mathbb{R}^{N}}\\left\\vert \\nabla v_{a}^{-}\\right\\vert\n^{2}dx\\right) ^{2}\\right) .\n\\end{equation*\nThen we have $h_{a,v_{a}^{-}}^{\\prime }(t)>0$ for all $t\\in \\left(\n0,t_{a}^{(1),-}\\right) \\cup \\left( t_{a}^{(2),-},\\infty \\right) $ and \nh_{a,v_{a}^{-}}^{\\prime }(t)<0$ for all $t\\in \\left(\nt_{a}^{(1),-},t_{a}^{(2),-}\\right) $. Consequently, we arrive at\n\\begin{equation*}\nJ_{a}\\left( t_{a}^{(1),-}v_{a}^{-}\\right) =\\sup_{0\\leq t\\leq\nt_{a}^{(2),-}}J_{a}(tv_{a}^{-})\\text{ and }J_{a}\\left(\nt_{a}^{(2),-}v_{a}^{-}\\right) =\\inf_{t\\geq t_{a}^{\\left( 1\\right)\n,-}}J_{a}(tv_{a}^{-}).\n\\end{equation*\nThat is, $J_{a}\\left( t_{a}^{(2),-}v_{a}^{-}\\right) \\leq J_{a}\\left(\nt_{a}^{(1),-}v_{a}^{-}\\right) <\\alpha _{a}^{\\infty ,-}$, and so \nt_{a}^{(2),-}v_{a}^{-}\\in \\mathbf{M}_{a}^{(2),-}.$ Consequently, this\ncompletes the proof.\n\\end{proof}\n\n\\begin{lemma}\n\\label{m3-2}Suppose that $N\\geq 5$ and conditions ${(D1)}-{(D4)}$ hold. Then\nfor each $0<a<\\Lambda ,$ there exist two constants $t_{a}^{(1),+}$ and \nt_{a}^{(2),+}$ satisfying\n\\begin{equation*}\nT_{f}(v_{a}^{+})<t_{a}^{(1),+}<\\left( \\frac{2}{4-p}\\right)\n^{1\/(p-2)}T_{f}(v_{a}^{+})<t_{a}^{(2),+}\n\\end{equation*\nsuch that $t_{a}^{(i),+}v_{a}^{+}\\in \\mathbf{M}_{a}^{(i)}\\ (i=1,2),$ and\n\\begin{equation*}\nJ_{a}\\left( t_{a}^{(2),+}v_{a}^{+}\\right) =\\inf_{t\\geq\nt_{a}^{(1)}}J_{a}(tv_{a}^{+})<\\alpha _{a}^{\\infty ,+}\\text{ and }J_{a}\\left(\nt_{a}^{(2)}v_{a}^{+}\\right) =\\inf_{t\\geq t_{a}^{(1)}}J_{a}(tv_{a}^{+}),\n\\end{equation*\nwhere $T_{f}(v_{a}^{+})$ is as $(\\ref{2-0})$ with $u=v_{a}^{+}.$\n\\end{lemma}\n\n\\begin{proof}\nThe proof is analogous to that of Lemma \\ref{m3}, and we omit it here.\n\\end{proof}\n\nFollowing \\cite{NT,T}, we have the following result.\n\n\\begin{lemma}\n\\label{g9}Suppose that $0<a<\\Lambda .$ Then for each $u\\in \\mathbf{M\n_{a}^{(j)}(j=1,2),$ there exist a constant $\\sigma >0$ and a differentiable\nfunction $t^{\\ast }:B_{\\sigma }(0)\\subset H^{1}(\\mathbb{R}^{N})\\rightarrow\n\\mathbb{R}^{+}$ such that $t^{\\ast }(0)=1\\ $and$\\ t^{\\ast }(v)(u-v)\\in\n\\mathbf{M}_{a}^{(j)}$ for all $v\\in B_{\\sigma }(0),$ and\n\\begin{eqnarray*}\n&&\\left\\langle (t^{\\ast })^{\\prime }(0),\\varphi \\right\\rangle \\\\\n&=&\\frac{2\\int_{\\mathbb{R}^{N}}(\\nabla u\\nabla \\varphi +u\\varphi\n)dx+4a\\left( \\int_{\\mathbb{R}^{N}}|\\nabla u|dx\\right) ^{2}\\int_{\\mathbb{R\n^{N}}\\nabla u\\nabla \\varphi dx-p\\int_{\\mathbb{R}^{N}}f(x)|u|^{p-2}u\\varphi d\n}{\\left\\Vert u\\right\\Vert _{H^{1}}^{2}-(p-1)\\int_{\\mathbb{R\n^{N}}f(x)|u|^{p}dx}\n\\end{eqnarray*\nfor all $\\varphi \\in H^{1}(\\mathbb{R}^{N}).$\n\\end{lemma}\n\n\\begin{proof}\nFor any $u\\in \\mathbf{M}_{a}^{(j)}$, we define the function $F_{u}:\\mathbb{R\n\\times H^{1}(\\mathbb{R}^{N})\\rightarrow \\mathbb{R}$ b\n\\begin{eqnarray*}\nF_{u}(t,v) &=&\\left\\langle J_{a}^{\\prime }(t(u-v)),t(u-v)\\right\\rangle \\\\\n&=&t^{2}\\left\\Vert u-v\\right\\Vert _{H^{1}}^{2}+at^{4}\\left( \\int_{\\mathbb{R\n^{N}}|\\nabla u-\\nabla v|^{2}dx\\right) ^{2}-t^{p}\\int_{\\mathbb{R\n^{N}}f(x)|u-v|^{p}dx.\n\\end{eqnarray*\nClearly, $F_{u}(1,0)=\\left\\langle J_{a}^{\\prime }(u),u\\right\\rangle =0$ an\n\\begin{eqnarray*}\n\\frac{d}{dt}F_{u}(1,0) &=&2\\left\\Vert u\\right\\Vert _{H^{1}}^{2}+4a\\left(\n\\int_{\\mathbb{R}^{N}}|\\nabla u|^{2}dx\\right) ^{2}-p\\int_{\\mathbb{R\n^{N}}f(x)|u|^{p}dx \\\\\n&=&-2\\left\\Vert u\\right\\Vert _{H^{1}}^{2}-\\left( p-4\\right) \\int_{\\mathbb{R\n^{N}}f(x)|u|^{p}dx \\\\\n&\\neq &0.\n\\end{eqnarray*\nApplying the implicit function theorem, there exist a constant $\\sigma >0$\nand a differentiable function $t^{\\ast }:B_{\\sigma }(0)\\subset H^{1}(\\mathbb\nR}^{N})\\rightarrow \\mathbb{R}$ such that $t^{\\ast }(0)=1$ and\n\\begin{eqnarray*}\n&&\\left\\langle (t^{\\ast })^{\\prime }(0),\\varphi \\right\\rangle \\\\\n&=&\\frac{2\\int_{\\mathbb{R}^{N}}(\\nabla u\\nabla \\varphi +u\\varphi\n)dx+4a\\left( \\int_{\\mathbb{R}^{N}}|\\nabla u|dx\\right) ^{2}\\int_{\\mathbb{R\n^{N}}\\nabla u\\nabla \\varphi dx-p\\int_{\\mathbb{R}^{N}}f(x)|u|^{p-2}u\\varphi d\n}{\\left\\Vert u\\right\\Vert _{H^{1}}^{2}-(p-1)\\int_{\\mathbb{R\n^{N}}f(x)|u|^{p}dx}\n\\end{eqnarray*\nfor all $\\varphi \\in H^{1}(\\mathbb{R}^{N}),$ and $F_{u}(t^{\\ast }(v),v)=0$\nfor all $v\\in B_{\\sigma }(0),$ which is equivalent t\n\\begin{equation*}\n\\left\\langle J_{a}^{\\prime }(t^{\\ast }(v)(u-v)),t^{\\ast\n}(v)(u-v)\\right\\rangle =0\\text{ for all }v\\in B_{\\sigma }(0).\n\\end{equation*\nAccording to the continuity of the map $t^{\\ast }$, for $\\sigma $\nsufficiently small we have\n\\begin{eqnarray*}\nh_{a,t^{\\ast }(v)(u-v)}^{\\prime \\prime }(1) &=&-2\\left\\Vert t^{\\ast }\\left(\nv\\right) \\left( u-v\\right) \\right\\Vert _{H^{1}}^{2}-(p-4)\\int_{\\mathbb{R\n^{N}}f(x)|t^{\\ast }(v)(u-v)|^{p}dx \\\\\n&<&0,\n\\end{eqnarray*\nan\n\\begin{equation*}\nJ_{a}(t^{\\ast }(v)(u-v))<\\frac{(p-2)D(p)}{2p}\\left( \\frac{2S_{p}^{p}}\nf_{\\infty }(4-p)}\\right) ^{2\/(p-2)}.\n\\end{equation*\nHence, $t^{\\ast }(v)(u-v)\\in \\mathbf{M}_{a}^{(j)}$ for all $v\\in B_{\\sigma\n}(0).$ This completes the proof.\n\\end{proof}\n\nBy $(\\ref{4-4})$ and Lemma $\\ref{g7}$, we define\n\\begin{equation*}\n\\alpha _{a}^{-}=\\inf_{u\\in \\mathbf{M}_{a}^{(1)}}J_{a}(u)=\\inf_{u\\in \\mathbf{\n}_{a}^{-}}J_{a}(u)\\text{ for }N\\geq 1.\n\\end{equation*}\n\n\\begin{proposition}\n\\label{g8}Suppose that $N\\geq 1.$ Then for each $0<a<\\Lambda ,$ there exists\na sequence $\\{u_{n}\\}\\subset \\mathbf{M}_{a}^{(1)}$ such tha\n\\begin{equation}\nJ_{a}(u_{n})=\\alpha _{a}^{-}+o(1)\\text{ and }J_{a}^{\\prime }(u_{n})=o(1\n\\text{ in }H^{-1}(\\mathbb{R}^{N}).  \\label{eqq20}\n\\end{equation}\n\\end{proposition}\n\n\\begin{proof}\nBy the Ekeland variational principle \\cite{E} and Lemma \\ref{g1}, we obtain\nthat there exists a minimizing sequence $\\{u_{n}\\}\\subset \\mathbf{M\n_{a}^{(1)}$ such tha\n\\begin{equation*}\nJ_{a}(u_{n})<\\alpha _{a}^{-}+\\frac{1}{n}\n\\end{equation*\nan\n\\begin{equation}\nJ_{a}(u_{n})\\leq J_{a}(w)+\\frac{1}{n}\\left\\Vert w-u_{n}\\right\\Vert _{H^{1}\n\\text{ for all }w\\in \\mathbf{M}_{a}^{(1)}.  \\label{22}\n\\end{equation\nApplying Lemma \\ref{g9} with $u=u_{n}$, there exists a function $t_{n}^{\\ast\n}:B_{\\epsilon _{n}}(0)\\rightarrow \\mathbb{R}$ for some $\\epsilon _{n}>0$\nsuch that $t_{n}^{\\ast }(w)(u_{n}-w)\\in \\mathbf{M}_{a}^{(1)}.$ For $0<\\delta\n<\\epsilon _{n}$ and $u\\in H^{1}(\\mathbb{R}^{N})$ with $u\\not\\equiv 0,$ we\nset\n\\begin{equation*}\nw_{\\delta }=\\frac{\\delta u}{\\left\\Vert u\\right\\Vert _{H^{1}}}\\text{ and \nz_{\\delta }=t_{n}^{\\ast }(w_{\\delta })(u_{n}-w_{\\delta }).\n\\end{equation*\nSince $z_{\\delta }\\in \\mathbf{M}_{a}^{(1)},$ it follows from $(\\ref{22})$\ntha\n\\begin{equation*}\nJ_{a}(z_{\\delta })-J_{a}(u_{n})\\geq -\\frac{1}{n}\\Vert z_{\\delta }-u_{n}\\Vert\n_{H^{1}}.\n\\end{equation*\nUsing the mean value theorem gives\n\\begin{equation*}\n\\left\\langle J_{a}^{\\prime }(u_{n}),z_{\\delta }-u_{n}\\right\\rangle +o\\left(\n\\left\\Vert z_{\\delta }-u_{n}\\right\\Vert _{H^{1}}\\right) \\geq -\\frac{1}{n\n\\Vert z_{\\delta }-u_{n}\\Vert _{H^{1}}\n\\end{equation*\nand\n\\begin{eqnarray}\n&&\\left\\langle J_{a}^{\\prime }(u_{n}),-w_{\\delta }\\right\\rangle\n+(t_{n}^{\\ast }(w_{\\delta })-1)\\left\\langle J_{a}^{\\prime\n}(u_{n}),u_{n}-w_{\\delta }\\right\\rangle  \\notag \\\\\n&\\geq &-\\frac{1}{n}\\Vert z_{\\delta }-u_{n}\\Vert _{H^{1}}+o(\\Vert z_{\\delta\n}-u_{n}\\Vert _{H^{1}}).  \\label{23}\n\\end{eqnarray\nNote that $t_{n}^{\\ast }(w_{\\delta })(u_{n}-w_{\\delta })\\in \\mathbf{M\n_{a}^{(1)}$. From $(\\ref{23})$ it leads to\n\\begin{eqnarray*}\n&&-\\delta \\left\\langle J_{a}^{\\prime }(u_{n}),\\,\\frac{u}{\\left\\Vert\nu\\right\\Vert _{H^{1}}}\\right\\rangle +\\frac{(t_{n}^{\\ast }(w_{\\delta })-1)}\nt_{n}^{\\ast }(w_{\\delta })}\\left\\langle J_{a}^{\\prime }(z_{\\delta\n}),t_{n}^{\\ast }(w_{\\delta })(u_{n}-w_{\\delta })\\right\\rangle \\\\\n&&+(t_{n}^{\\ast }(w_{\\delta })-1)\\left\\langle J_{a}^{\\prime\n}(u_{n})-J_{a}^{\\prime }(z_{\\delta }),u_{n}-w_{\\delta }\\right\\rangle \\\\\n&\\geq &-\\frac{1}{n}\\left\\Vert z_{\\delta }-u_{n}\\right\\Vert\n_{H^{1}}+o(\\left\\Vert z_{\\delta }-u_{n}\\right\\Vert _{H^{1}}).\n\\end{eqnarray*\nWe rewrite the above inequality as\n\\begin{eqnarray}\n\\left\\langle J_{a}^{\\prime }(u_{n}),\\frac{u}{\\left\\Vert u\\right\\Vert _{H^{1}\n}\\right\\rangle &\\leq &\\frac{\\left\\Vert z_{\\delta }-u_{n}\\right\\Vert _{H^{1}\n}{\\delta n}+\\frac{o(\\left\\Vert z_{\\delta }-u_{n}\\right\\Vert _{H^{1}})}\n\\delta }  \\notag \\\\\n&&+\\frac{(t_{n}^{\\ast }(w_{\\delta })-1)}{\\delta }\\langle J_{a}^{\\prime\n}(u_{n})-J_{a}^{\\prime }(z_{\\delta }),u_{n}-w_{\\delta }\\rangle .  \\label{24}\n\\end{eqnarray\nThere exists a constant $C>0$ independent of $\\delta $ such tha\n\\begin{equation*}\n\\left\\Vert z_{\\delta }-u_{n}\\right\\Vert _{H^{1}}\\leq \\delta +C(\\left\\vert\nt_{n}^{\\ast }\\left( w_{\\delta }\\right) -1\\right\\vert )\n\\end{equation*\nan\n\\begin{equation*}\n\\lim_{\\delta \\rightarrow 0}\\frac{\\left\\vert t_{n}^{\\ast }(w_{\\delta\n})-1\\right\\vert }{\\delta }\\leq \\left\\Vert (t_{n}^{\\ast })^{\\prime\n}(0)\\right\\Vert \\leq C.\n\\end{equation*\nLetting $\\delta \\rightarrow 0$ in $(\\ref{24})$ and using the fact that \n\\lim_{\\delta \\rightarrow 0}\\left\\Vert z_{\\delta }-u_{n}\\right\\Vert\n_{H^{1}}=0,$ we ge\n\\begin{equation*}\n\\left\\langle J_{a}^{\\prime }(u_{n}),\\frac{u}{\\left\\Vert u\\right\\Vert _{H^{1}\n}\\right\\rangle \\leq \\frac{C}{n},\n\\end{equation*\nwhich leads to $(\\ref{eqq20}).$\n\\end{proof}\n\nBefore proving Theorem \\ref{t2}, we also need the following compactness\nlemma which is an immediate conclusion of Proposition \\ref{g10}.\n\n\\begin{lemma}\n\\label{lem2} Suppose that $N\\geq 1$ and conditions ${(D1)}-{(D2),(D4)}-{(D5)}\n$ hold. Let $\\{u_{n}\\}\\subset \\mathbf{M}_{a}^{(1)}$ be a $(PS)_{\\beta }\n--sequence in $H^{1}(\\mathbb{R}^{N})$ for $J_{a}$ with $0<\\beta <\\alpha\n_{a}^{\\infty ,-}.$ Then there exist a subsequence $\\{u_{n}\\}$ and a nonzero \nu_{0}$ in $H^{1}(\\mathbb{R}^{N})$ such that $u_{n}\\rightarrow u_{0}$\nstrongly in $H^{1}(\\mathbb{R}^{N})$ and $J_{a}(u_{0})=\\beta .$ Furthermore, \nu_{0}$ is a nonzero solution of Eq. $(E_{a}).$\n\\end{lemma}\n\n\\textbf{We are now ready to prove Theorems \\ref{t2} and \\ref{t3}:} By\nProposition \\ref{g8}, there exists a sequence $\\{u_{n}\\}\\subset \\mathbf{M\n_{a}^{(1)}$ satisfying\n\\begin{equation*}\nJ_{a}(u_{n})=\\alpha _{a}^{-}+o(1)\\text{ and }J_{a}^{\\prime }(u_{n})=o(1\n\\text{ in }H^{-1}(\\mathbb{R}^{N}).\n\\end{equation*\nIt follows from Lemmas \\ref{lem2} and \\ref{m3} that Eq. $(E_{a})$ has a\nnontrivial solution $u_{a}^{-}\\in \\mathbf{M}_{a}^{-}$ such that \nJ_{a}(u_{a}^{-})=\\alpha _{a}^{-}.$ Thus, $u_{a}^{-}$ is a minimizer for \nJ_{a}$ on $\\mathbf{M}_{a}^{-}.$ since\n\\begin{equation*}\n\\alpha _{a}^{-}<\\alpha _{a}^{\\infty ,-}<\\frac{p-2}{2p}D(p)\\left( \\frac\n2S_{p}^{p}}{f_{\\infty }(4-p)}\\right) ^{2\/(p-2)},\n\\end{equation*\none has $u_{a}^{-}\\in \\mathbf{M}_{a}^{(1)}$. Similarly, we obtain \n|u_{a}^{-}|\\in \\mathbf{M}_{a}^{-}$ and $J_{a}(|u_{a}^{-}|)=J_{a}(u_{a}^{-})\n\\alpha _{a}^{-}$. According to Lemma \\ref{g2}, $u_{a}^{-}$ is a positive\nsolution of Eq. $(E_{a})$ when $N\\geq 1.$ Consequently, the proof of Theorem\n\\ref{t2} is complete.\n\nNote that $\\Lambda \\leq \\overline{a}_{\\ast }$ for $N\\geq 5.$ Then it follows\nfrom Theorem \\ref{t0-3}$(ii)$ that for each $0<a<\\Lambda ,$ Eq. $(E_{a})$\nadmits a positive ground state solution $u_{a}^{+}\\in H^{1}(\\mathbb{R}^{N})$\nsuch that $J_{a}(u_{a}^{+})<0.$ Clearly, $u_{a}^{+}\\in \\mathbf{M}_{a}\\left(\n\\frac{D(p)(p-2)}{2p}\\left( \\frac{2S_{p}^{p}}{f_{\\infty }(4-p)}\\right)\n^{2\/(p-2)}\\right) ,$ since $J_{a}(u_{a}^{+})<0.$ By virtue of Lemma \\ref{g1\n, we further have $u_{a}^{+}\\in \\mathbf{M}_{a}^{(2)},$ which implies tha\n\\begin{equation*}\n\\left\\Vert u_{a}^{+}\\right\\Vert _{H^{1}}>\\sqrt{2}\\left( \\frac{2S_{p}^{p}}\nf_{\\max }(4-p)}\\right) ^{1\/(p-2)}.\n\\end{equation*\nMoreover, by Theorem \\ref{t2} we obtain that for each $0<a<\\Lambda ,$ Eq. \n(E_{a})$ admits a positive solution $u_{a}^{-}\\in H^{1}(\\mathbb{R}^{N})$\nsatisfyin\n\\begin{equation*}\nJ_{a}(u_{a}^{-})>\\frac{p-2}{4p}\\left( \\frac{S_{p}^{p}}{f_{\\max }}\\right)\n^{2\/(p-2)}\\text{ and }\\left\\Vert u_{a}^{-}\\right\\Vert _{H^{1}}<\\left( \\frac\n2S_{p}^{p}}{f_{\\max }(4-p)}\\right) ^{1\/(p-2)}.\n\\end{equation*\nConsequently, we complete the proof of Theorem \\ref{t3}.\n\n\\section{Ground State Solutions}\n\n\\begin{lemma}\n\\label{g14}Suppose that $N=1$ and $f(x)\\in C(\\mathbb{R})$ is weakly\ndifferentiable satisfying\n\\begin{equation*}\n(p-1)(p-2)f\\left( x\\right) +2\\langle \\nabla f(x),x\\rangle \\geq 0.\n\\end{equation*\nLet $u_{0}$ be a nontrivial solution of Eq. $(E_{a}).$ Then $u_{0}\\in\n\\mathbf{M}_{a}^{-}.$\n\\end{lemma}\n\n\\begin{proof}\nSince $u_{0}$ is a nontrivial solution of Eq. $(E_{a})$, we have\n\\begin{equation}\n\\left\\Vert u_{0}\\right\\Vert _{H^{1}}^{2}+a\\left( \\int_{\\mathbb{R}}|\\nabla\nu_{0}|^{2}dx\\right) ^{2}-\\int_{\\mathbb{R}}f(x)|u_{0}|^{p}dx=0.  \\label{3-1}\n\\end{equation\nFollowing the argument of \\cite[Lemma 3.1]{DM1}, $u_{0}$ satisfies the\nPohozaev type identity corresponding to Eq. $(E_{a})$ as follows\n\\begin{equation}\n\\frac{1}{2}\\int_{\\mathbb{R}}(-|\\nabla u_{0}|^{2}+u_{0}^{2})dx-\\frac{a}{2\n\\left( \\int_{\\mathbb{R}}|\\nabla u_{0}|^{2}dx\\right) ^{2}=\\frac{1}{p}\\int_\n\\mathbb{R}}(f(x)+\\langle \\nabla f(x),x\\rangle )|u_{0}|^{p}dx.  \\label{3-4}\n\\end{equation\nThen it follows from $(\\ref{3-1})-(\\ref{3-4})$ and the assumption of $f(x)$\ntha\n\\begin{eqnarray*}\nh_{a,u_{0}}^{\\prime \\prime }(1) &=&-2\\int_{\\mathbb{R}}|\\nabla u_{0}|^{2}dx\n\\frac{(p+2)}{p}\\int_{\\mathbb{R}}f(x)|u_{0}|^{p}dx+\\frac{2}{p}\\int_{\\mathbb{R\n}\\langle \\nabla f(x),x\\rangle |u_{0}|^{p}dx \\\\\n&&-\\frac{4}{p}\\int_{\\mathbb{R}}f(x)|u_{0}|^{p}dx-\\frac{4}{p}\\int_{\\mathbb{R\n}\\langle \\nabla f(x),x\\rangle |u_{0}|^{p}dx-(p-2)\\int_{\\mathbb{R\n}f(x)|u_{0}|^{p}dx \\\\\n&=&-2\\int_{\\mathbb{R}}|\\nabla u_{0}|^{2}dx-\\frac{1}{p}\\int_{\\mathbb{R}^{N}\n\\left[ (p-1)(p-2)f(x)+2\\langle \\nabla f(x),x\\rangle \\right] |u_{0}|^{p}dx \\\\\n&<&0,\n\\end{eqnarray*\nwhich shows that $u_{0}\\in \\mathbf{M}_{a}^{-}.$ This completes the proof.\n\\end{proof}\n\n\\begin{lemma}\n\\label{g13}Suppose that $N=2$ and $f(x)\\in C(\\mathbb{R}^{2})$ is weakly\ndifferentiable satisfying\n\\begin{equation*}\n(p-2)f(x)+\\langle \\nabla f(x),x\\rangle \\geq 0.\n\\end{equation*}\nLet $u_{0}$ be a nontrivial solution of Eq. $(E_{a}).$ Then $u_{0}\\in\n\\mathbf{M}_{a}^{-}.$\n\\end{lemma}\n\n\\begin{proof}\nSince $u_{0}$ is a nontrivial solution of Eq. $(E_{a})$, there holds\n\\begin{equation}\n\\left\\Vert u_{0}\\right\\Vert _{H^{1}}^{2}+a\\left( \\int_{\\mathbb{R\n^{2}}|\\nabla u_{0}|^{2}dx\\right) ^{2}-\\int_{\\mathbb{R\n^{2}}f(x)|u_{0}|^{p}dx=0.  \\label{3-3}\n\\end{equation\nMoreover, $u_{0}$ satisfies the Pohozaev type identity corresponding to Eq. \n(E_{a})$ as follows\n\\begin{equation}\n\\frac{2}{p}\\int_{\\mathbb{R}^{2}}u_{0}^{2}dx-\\frac{1}{2}\\int_{\\mathbb{R\n^{2}}\\langle \\nabla f(x),x\\rangle |u_{0}|^{p}dx=\\int_{\\mathbb{R}^{2}}f\\left(\nx\\right) |u_{0}|^{p}dx.  \\notag\n\\end{equation\nUsing the above two equalities give\n\\begin{equation}\n\\frac{2}{p}\\int_{\\mathbb{R}^{2}}u_{0}^{2}dx-\\frac{1}{2}\\int_{\\mathbb{R\n^{2}}\\langle \\nabla f(x),x\\rangle |u_{0}|^{p}dx=\\left\\Vert u_{0}\\right\\Vert\n_{H^{1}}^{2}+a\\left( \\int_{\\mathbb{R}^{2}}|\\nabla u_{0}|^{2}dx\\right) ^{2}.\n\\label{3-6}\n\\end{equation\nThen it follows from $(\\ref{3-3})-(\\ref{3-6})$ and the assumption of $f(x)$\ntha\n\\begin{eqnarray*}\nh_{a,u_{0}}^{\\prime \\prime }(1) &=&2a\\left( \\int_{\\mathbb{R}^{2}}|\\nabla\nu_{0}|^{2}dx\\right) ^{2}-(p-2)\\int_{\\mathbb{R}^{2}}f\\left( x\\right)\n|u_{0}|^{p}dx \\\\\n&=&-2\\int_{\\mathbb{R}^{2}}|\\nabla u_{0}|^{2}dx-\\frac{2(p-2)}{p}\\int_{\\mathbb\nR}^{2}}u_{0}^{2}dx-\\int_{\\mathbb{R}^{2}}\\langle \\nabla f(x),x\\rangle\n|u_{0}|^{p}dx \\\\\n&&-(p-2)\\int_{\\mathbb{R}^{2}}f\\left( x\\right) |u_{0}|^{p}dx \\\\\n&=&-2\\int_{\\mathbb{R}^{2}}|\\nabla u_{0}|^{2}dx-\\frac{2(p-2)}{p}\\int_{\\mathbb\nR}^{2}}u_{0}^{2}dx-\\int_{\\mathbb{R}^{2}}\\left[ (p-2)f(x)+\\langle \\nabla\nf(x),x\\rangle \\right] |u_{0}|^{p}dx \\\\\n&<&0,\n\\end{eqnarray*\nwhich shows that $u_{0}\\in \\mathbf{M}_{a}^{-}.$ This completes the proof.\n\\end{proof}\n\n\\textbf{We are now ready to prove Theorem \\ref{t4}:} Let $u^{-}$ be the\npositive solution of Eq. $(E_{a})$ as described in Theorem \\ref{t1}$(i)$ or\n\\ref{t2}$(i)$. Then there holds $u^{-}\\in \\mathbf{M}_{a}^{-}$ and \nJ_{a}(u^{-})=\\inf_{u\\in \\mathbf{M}_{a}^{-}}J_{a}(u)=\\alpha _{a}^{-}.$ By\nLemma \\ref{g14} or \\ref{g13}, $u^{-}$ is a positive ground state solution of\nEq. $(E_{a}).$\n\n\\begin{lemma}\n\\label{g3}Suppose that $N=3$ and $f(x)\\equiv f_{\\infty }.$ Let $u_{0}$ be a\nnontrivial solution of Eq. $(E_{a}).$ Then for each $a>0$ with $\\sqrt{a^{2}+\n}+\\frac{2}{a}\\geq A_{0},$ there holds $u_{0}\\in \\mathbf{M}_{a}^{-},$ where \nA_{0}>0$ is defined as $(\\ref{1-3}).$\n\\end{lemma}\n\n\\begin{proof}\nLet $u_{0}$ be a nontrivial solution of Eq. $(E_{a})$. Then there holds\n\\begin{equation}\n\\left\\Vert u_{0}\\right\\Vert _{H^{1}}^{2}+a\\left( \\int_{\\mathbb{R\n^{3}}|\\nabla u_{0}|^{2}dx\\right) ^{2}=f_{\\infty }\\int_{\\mathbb{R\n^{3}}|u_{0}|^{p}dx.  \\label{6-9}\n\\end{equation\nMoreover, $u_{0}$ satisfies the Pohozaev type identity corresponding to Eq. \n(E_{a})$ as follow\n\\begin{equation}\n\\frac{p}{6}\\int_{\\mathbb{R}^{3}}(|\\nabla u_{0}|^{2}+3u_{0}^{2})dx+\\frac{ap}{\n}\\left( \\int_{\\mathbb{R}^{3}}|\\nabla u_{0}|^{2}dx\\right) ^{2}=f_{\\infty\n}\\int_{\\mathbb{R}^{3}}|u_{0}|^{p}dx.  \\label{6-10}\n\\end{equation\nBy Azzollini \\cite[Theorem 1.1]{Az2}, for each $a>0$ there exists a constan\n\\begin{equation*}\nt_{a}=\\frac{-a+\\sqrt{a^{2}+4}}{2}>0\n\\end{equation*\nsuch that $u_{0}(\\cdot )=w_{0}(t_{a}\\cdot ).$ Then it follows from $\\left(\n\\ref{1-8}\\right) ,(\\ref{6-9})-(\\ref{6-10})$ tha\n\\begin{eqnarray*}\nh_{a,u_{0}}^{\\prime \\prime }(1) &=&2a\\left( \\int_{\\mathbb{R}^{3}}|\\nabla\nu_{0}|^{2}dx\\right) ^{2}-\\left( p-2\\right) f_{\\infty }\\int_{\\mathbb{R\n^{3}}|u_{0}|^{p}dx \\\\\n&=&\\frac{\\int_{\\mathbb{R}^{3}}|\\nabla u_{0}|^{2}dx}{6}\\left[\na(-p^{2}+2p+12)\\int_{\\mathbb{R}^{3}}|\\nabla u_{0}|^{2}dx-p(p-2)\\right] \n\\frac{p(p-2)}{2}\\int_{\\mathbb{R}^{3}}u_{0}^{2}dx \\\\\n&=&\\frac{\\int_{\\mathbb{R}^{3}}|\\nabla u_{0}|^{2}dx}{6}\\left[\na(-p^{2}+2p+12)t_{a}^{-2}\\int_{\\mathbb{R}^{3}}|\\nabla w_{0}|^{2}dx-p(p-2\n\\right] -\\frac{p(p-2)}{2}\\int_{\\mathbb{R}^{3}}u_{0}^{2}dx \\\\\n&=&\\frac{\\int_{\\mathbb{R}^{3}}|\\nabla u_{0}|^{2}dx}{6}\\left[ \\frac\n2a(-p^{2}+2p+12)\\int_{\\mathbb{R}^{3}}|\\nabla w_{0}|^{2}dx}{2+a\\sqrt{a^{2}+4}\n-p(p-2)\\right] -\\frac{p(p-2)}{2}\\int_{\\mathbb{R}^{3}}u_{0}^{2}dx. \\\\\n&=&\\frac{\\int_{\\mathbb{R}^{3}}|\\nabla u_{0}|^{2}dx}{6}\\left[ \\frac{3\\left(\np-1\\right) (-p^{2}+2p+12)}{p\\left( \\frac{2}{a}+\\sqrt{a^{2}+4}\\right) }\\left(\n\\frac{S_{p}^{p}}{f_{\\infty }}\\right) ^{\\frac{2}{p-2}}-p(p-2)\\right] -\\frac\np(p-2)}{2}\\int_{\\mathbb{R}^{3}}u_{0}^{2}dx\n\\end{eqnarray*\nThis implies tha\n\\begin{equation*}\nf_{\\infty }\\geq \\left[ \\frac{3\\left( p-1\\right) (-p^{2}+2p+12)}\np^{2}(p-2)\\inf_{a>0}\\left( \\sqrt{a^{2}+4}+\\frac{2}{a}\\right) }\\right]\n^{2\/\\left( p-2\\right) }S_{p}^{p}\n\\end{equation*\n\\begin{equation*}\nh_{a,u_{0}}^{\\prime \\prime }(1)<0\\text{ if }\\sqrt{a^{2}+4}+\\frac{2}{a}\\geq\n\\inf_{a>0}\\sqrt{a^{2}+4}+\\frac{2}{a}\\geq A_{0}=\\frac{3\\left( p-1\\right)\n(-p^{2}+2p+12)}{p^{2}(p-2)}\\left( \\frac{S_{p}^{p}}{f_{\\infty }}\\right) ^\n\\frac{2}{p-2}},\n\\end{equation*\nand so for each $a>0$ with $\\sqrt{a^{2}+4}+\\frac{2}{a}\\geq A_{0},$ there\nholds $u_{0}\\in \\mathbf{M}_{a}^{-}$. This completes the proof.\n\\end{proof}\n\n\\begin{lemma}\n\\label{g5}Suppose that $N=4$ and $f(x)\\equiv f_{\\infty }$. Let $u_{0}$ be a\nnontrivial solution of Eq. $(E_{a}).$ Then for each $0<a\\leq \\overline{A\n_{0},$ there holds $u_{0}\\in \\mathbf{M}_{a}^{-},$ where $\\overline{A}_{0}>0$\nis defined as $(\\ref{1-4}).$\n\\end{lemma}\n\n\\begin{proof}\nLet $u_{0}$ be a nontrivial solution of Eq. $(E_{a})$. Then we have\n\\begin{equation}\n\\left\\Vert u_{0}\\right\\Vert _{H^{1}}^{2}+a\\left( \\int_{\\mathbb{R\n^{4}}|\\nabla u_{0}|^{2}dx\\right) ^{2}=f_{\\infty }\\int_{\\mathbb{R\n^{4}}|u_{0}|^{p}dx.  \\label{6-1}\n\\end{equation\nMoreover, $u_{0}$ satisfies the Pohozaev type identity corresponding to Eq. \n(E_{a})$ as follow\n\\begin{equation}\n\\int_{\\mathbb{R}^{4}}|\\nabla u_{0}|^{2}dx+a\\left( \\int_{\\mathbb{R\n^{4}}|\\nabla u_{0}|^{2}dx\\right) ^{2}+2\\int_{\\mathbb{R}^{4}}u_{0}^{2}dx\n\\frac{4}{p}f_{\\infty }\\int_{\\mathbb{R}^{4}}|u_{0}|^{p}dx  \\label{6-2}\n\\end{equation\nBy Azzollini \\cite[Theorem 1.1]{Az2}, for each $0<a<\\left( \\int_{\\mathbb{R\n^{4}}|\\nabla w_{0}|^{2}dx\\right) ^{-1}$ there exists a constan\n\\begin{equation*}\nt_{a}=\\sqrt{1-a\\int_{\\mathbb{R}^{4}}|\\nabla w_{0}|^{2}dx}>0\n\\end{equation*\nsuch that $u_{0}(\\cdot )=w_{0}(t_{a}\\cdot ).$ Then it follows from $(\\re\n{6-1})-(\\ref{6-2})$ tha\n\\begin{eqnarray*}\nh_{a,u_{0}}^{\\prime \\prime }(1) &=&2a\\left( \\int_{\\mathbb{R}^{4}}|\\nabla\nu_{0}|^{2}dx\\right) ^{2}-\\left( p-2\\right) f_{\\infty }\\int_{\\mathbb{R\n^{4}}|u_{0}|^{p}dx \\\\\n&=&\\frac{\\int_{\\mathbb{R}^{4}}|\\nabla u_{0}|^{2}dx}{4}\\left[\na(4-p)(p+2)\\int_{\\mathbb{R}^{4}}|\\nabla u_{0}|^{2}dx-p(p-2)\\right] -\\frac\np(p-2)}{4}\\int_{\\mathbb{R}^{4}}u_{0}^{2}dx \\\\\n&=&\\frac{\\int_{\\mathbb{R}^{4}}|\\nabla u_{0}|^{2}dx}{4}\\left[\na(4-p)(p+2)t_{a}^{-2}\\int_{\\mathbb{R}^{4}}|\\nabla w_{0}|^{2}dx-p(p-2)\\right]\n-\\frac{p(p-2)}{2}\\int_{\\mathbb{R}^{4}}u_{0}^{2}dx \\\\\n&=&\\frac{\\int_{\\mathbb{R}^{4}}|\\nabla u_{0}|^{2}dx}{4}\\left[ \\frac\na(4-p)(p+2)\\int_{\\mathbb{R}^{4}}|\\nabla w_{0}|^{2}dx}{1-a\\int_{\\mathbb{R\n^{4}}|\\nabla w_{0}|^{2}dx}-p(p-2)\\right] -\\frac{p(p-2)}{2}\\int_{\\mathbb{R\n^{4}}u_{0}^{2}dx \\\\\n&<&0\\text{ for all }0<a\\leq \\overline{A}_{0},\n\\end{eqnarray*\nwhich indicates that $u_{0}\\in \\mathbf{M}_{a}^{-}$ for all $0<a\\leq\n\\overline{A}_{0}.$ This completes the proof.\n\\end{proof}\n\n\\begin{lemma}\n\\label{g16}Suppose that $N=4$ and $f(x)\\equiv f_{\\infty }$. Let $u_{0}$ be a\nnontrivial solution of Eq. $(E_{a}).$ Then for each $a>A_{0}^{\\ast },$ there\nholds $u_{0}\\in \\mathbf{M}_{a}^{+},$ where $A_{0}^{\\ast }>0$ is defined as $\n\\ref{1-5}).$\n\\end{lemma}\n\n\\begin{proof}\nLet $u_{0}$ be a nontrivial solution of Eq. $(E_{a}).$ By $(\\ref{6-1})-(\\re\n{6-2}),$ we hav\n\\begin{equation}\n-(p-2)\\left\\Vert u_{0}\\right\\Vert _{H^{1}}^{2}+a(4-p)\\left( \\int_{\\mathbb{R\n^{4}}|\\nabla u_{0}|^{2}dx\\right) ^{2}=p\\int_{\\mathbb{R}^{4}}u_{0}^{2}dx-\n\\left\\Vert u_{0}\\right\\Vert _{H^{1}}^{2}.  \\label{D-1}\n\\end{equation\nApplying the Gagliardo-Nirenberg and Young inequalities leads t\n\\begin{eqnarray*}\n\\left\\Vert u_{0}\\right\\Vert _{H^{1}}^{2}+a\\left( \\int_{\\mathbb{R\n^{4}}|\\nabla u_{0}|^{2}dx\\right) ^{2} &\\leq &f_{\\infty }C^{p}\\left( \\int_\n\\mathbb{R}^{4}}|\\nabla u_{0}|^{2}dx\\right) ^{p-2}\\left( \\int_{\\mathbb{R\n^{4}}u_{0}^{2}dx\\right) ^{\\left( 4-p\\right) \/2} \\\\\n&\\leq &\\frac{p-2}{2}\\left( \\frac{4-p}{p}\\right) ^{\\left( 4-p\\right) \/\\left(\np-2\\right) }\\left( f_{\\infty }C^{p}\\right) ^{\\frac{2}{p-2}}\\left( \\int_\n\\mathbb{R}^{4}}|\\nabla u_{0}|^{2}dx\\right) ^{2} \\\\\n&&+\\frac{p}{2}\\int_{\\mathbb{R}^{4}}u_{0}^{2}dx,\n\\end{eqnarray*\nwhich implies tha\n\\begin{equation}\np\\int_{\\mathbb{R}^{4}}u_{0}^{2}dx-2\\left\\Vert u_{0}\\right\\Vert\n_{H^{1}}^{2}\\geq 2\\left( a-\\frac{p-2}{2}\\left( \\frac{4-p}{p}\\right) ^{\\left(\n4-p\\right) \/\\left( p-2\\right) }(f_{\\infty }C^{p})^{2\/\\left( p-2\\right)\n}\\right) \\left( \\int_{\\mathbb{R}^{4}}|\\nabla u_{0}|^{2}dx\\right) ^{2}.\n\\label{D-0}\n\\end{equation\nThus, by $(\\ref{2-2})$ and $(\\ref{D-1})-(\\ref{D-0}),$ for each \na>A_{0}^{\\ast }$ one ha\n\\begin{eqnarray*}\nh_{a,u_{0}}^{\\prime \\prime }(1) &=&-(p-2)\\left\\Vert u_{0}\\right\\Vert\n_{H^{1}}^{2}+a(4-p)\\left( \\int_{\\mathbb{R}^{4}}|\\nabla u_{0}|^{2}dx\\right)\n^{2} \\\\\n&=&p\\int_{\\mathbb{R}^{4}}u_{0}^{2}dx-2\\left\\Vert u_{0}\\right\\Vert\n_{H^{1}}^{2} \\\\\n&\\geq &2\\left( a-\\frac{p-2}{2}\\left( \\frac{4-p}{p}\\right) ^{\\frac{4-p}{p-2\n}(f_{\\infty }C^{p})^{\\frac{2}{p-2}}\\right) \\left( \\int_{\\mathbb{R\n^{4}}|\\nabla u_{0}|^{2}dx\\right) ^{2} \\\\\n&>&0,\n\\end{eqnarray*\nwhich indicates that $u_{0}\\in \\mathbf{M}_{a}^{+}.$ This complete the proof.\n\\end{proof}\n\n\\textbf{We are now ready to prove Theorem \\ref{t5}:} By Lemmas \\ref{g3}--\\re\n{g16}, we can arrive at the conclusions directly.\n\n\\section{Appendix}\n\nIn order to verify $(\\ref{18-4}),$ we use the concentration-compactness\nlemma \\cite{Li1,Li2}. First of all, we discuss the three possibilities on\nthe measures defined by a functional related to $J_{a}^{\\infty }.$\n\nFor $2<p<\\min \\{4,2^{\\ast }\\},$ let $\\{u_{n}\\}\\subset \\mathbf{M}_{a}^{\\infty\n,(1)}$ be a sequence such that\n\\begin{equation}\n\\lim_{n\\rightarrow \\infty }J_{a}^{\\infty }(u_{n})=\\alpha _{a}^{\\infty ,-}>0.\n\\label{7-1}\n\\end{equation\nDefine the functional $\\Phi _{a}^{\\infty }:H^{1}(\\mathbb{R}^{N})\\rightarrow\n\\mathbb{R}$ b\n\\begin{equation*}\n\\Phi _{a}^{\\infty }(u)=\\frac{1}{4}\\left\\Vert u\\right\\Vert _{H^{1}}^{2}-\\frac\n4-p}{4p}\\int_{\\mathbb{R}^{N}}f_{\\infty }|u|^{p}dx.\n\\end{equation*\nBy Lemma \\ref{g1}, for any $u\\in \\mathbf{M}_{a}^{\\infty ,(1)}$ one ha\n\\begin{equation*}\nJ_{a}^{\\infty }(u)=\\Phi _{a}^{\\infty }(u)>0.\n\\end{equation*\nNote that $\\{u_{n}\\}$ is bounded in $H^{1}(\\mathbb{R}^{N}),$ since \n\\{u_{n}\\}\\subset \\mathbf{M}_{a}^{\\infty ,(1)}.$ Then there exist a\nsubsequence $\\{u_{n}\\}$ and $u_{\\infty }\\in H^{1}(\\mathbb{R}^{N})$ such tha\n\\begin{eqnarray*}\nu_{n} &\\rightharpoonup &u_{\\infty }\\text{ weakly in }H^{1}(\\mathbb{R}^{N}),\n\\\\\nu_{n} &\\rightarrow &u_{\\infty }\\text{ strongly in }L_{loc}^{s}(\\mathbb{R\n^{N})\\text{ for }2\\leq s<2^{\\ast }.\n\\end{eqnarray*\nFor any $u_{n}\\in \\mathbf{M}_{a}^{\\infty ,(1)}$, we define the measure \ny_{n}(\\Omega )$ b\n\\begin{equation}\ny_{n}(\\Omega )=\\frac{1}{4}\\int_{\\Omega }(|\\nabla u_{n}|^{2}+u_{n}^{2})dx\n\\frac{4-p}{4p}\\int_{\\Omega }f_{\\infty }|u_{n}|^{p}dx.  \\label{7-4}\n\\end{equation\nSince $u_{n}\\in \\mathbf{M}_{a}^{\\infty ,\\left( 1\\right) },$ we have \n\\left\\Vert u_{n}\\right\\Vert _{H^{1}}<D_{1}.$ This implies that $y_{n}(\\Omega\n)$ are positive measures. Furthermore, using $(\\ref{7-1})$ give\n\\begin{equation*}\ny_{n}(\\mathbb{R}^{N})=\\Phi _{a}^{\\infty }(u_{n})=\\alpha _{a}^{\\infty\n,-}+o\\left( 1\\right)\n\\end{equation*\nand then, by P.L. Lions \\cite{Li1}, there are three possibilities:\n\n\\textbf{vanishing:} for all $r>0,\n\\begin{equation}\n\\lim_{n\\rightarrow \\infty }\\sup_{\\xi \\in \\mathbb{R}^{N}}\\int_{B_{r}(\\xi\n)}dy_{n}=0,  \\label{7-5}\n\\end{equation\nwhere $B_{r}(\\xi )=\\{x\\in \\mathbb{R}^{N}:|x-\\xi |<r\\};$\n\n\\textbf{dichotomy:} there exist a constant $\\alpha \\in (0,\\alpha\n_{a}^{\\infty ,-})$, two sequences $\\{\\xi _{n}\\}$ and $\\{r_{n}\\},$ with \nr_{n}\\rightarrow +\\infty $ and two nonnegative measures $y_{n}^{1}$ and \ny_{n}^{2}$ such tha\n\\begin{equation}\ny_{n}-(y_{n}^{1}+y_{n}^{2})\\rightarrow 0,\\text{ }y_{n}^{1}(\\mathbb{R\n^{N})\\rightarrow \\alpha ,\\text{ }y_{n}^{2}(\\mathbb{R}^{N})\\rightarrow \\alpha\n_{a}^{\\infty ,-}-\\alpha ,  \\label{7-6}\n\\end{equation\n\\begin{equation}\n\\text{\\textrm{supp }}(y_{n}^{1})\\subset B_{r_{n}}(\\xi _{n}),\\text{ \\textrm\nsupp }}(y_{n}^{2})\\subset \\mathbb{R}^{N}\\backslash B_{2r_{n}}(\\xi _{n});\n\\label{7-7}\n\\end{equation}\n\n\\textbf{compactness:} there exists a sequence $\\{\\xi _{n}\\}\\subset \\mathbb{R\n^{N}$ with the following property: for any $\\delta >0,$ there exists \nr=r(\\delta )>0$ such tha\n\\begin{equation}\n\\int_{B_{r}(\\xi _{n})}dy_{n}\\geq \\alpha _{a}^{\\infty ,-}-\\delta ,\\text{ for\nlarge }n.  \\label{7-8}\n\\end{equation}\n\n\\begin{theorem}\n\\label{g10} For $0<a<\\Lambda ,$ compactness holds for the sequence of\nmeasures $\\{u_{n}\\}$ defined in $(\\ref{7-4}).$\n\\end{theorem}\n\n\\begin{proof}\n$(I)$ Vanishing does not occur. Suppose the contrary. Then for all $r>0,$ $\n\\ref{7-5})$ holds. In particular, we deduce that there exists $\\bar{r}>0$\nsuch tha\n\\begin{equation*}\n\\lim_{n\\rightarrow \\infty }\\sup_{\\xi \\in \\mathbb{R}^{N}}\\int_{B_{\\bar{r\n}(\\xi )}u_{n}^{2}dx=0,\n\\end{equation*\nwhich implies that $u_{n}\\rightarrow 0$ strongly in $L^{s}(\\mathbb{R}^{N})$\nfor $2<s<2^{\\ast }$ by using \\cite[Lemma I.1]{Li2}. Thus, for \n\\{u_{n}\\}\\subset \\mathbf{M}_{a}^{\\infty ,(1)},$ it follows from Lemma \\re\n{g1} tha\n\\begin{eqnarray*}\n\\frac{p-2}{4p}\\left( \\frac{S_{p}^{p}}{f_{\\infty }}\\right) ^{\\frac{2}{p-2}}\n&<&J_{a}^{\\infty }(u_{n})=-\\frac{a}{4}\\left( \\int_{\\mathbb{R}^{N}}|\\nabla\nu_{n}|^{2}dx\\right) ^{2}+\\frac{f_{\\infty }(p-2)}{2p}\\int_{\\mathbb{R\n^{N}}|u_{n}|^{p}dx \\\\\n&\\leq &\\frac{f_{\\infty }(p-2)}{2p}\\int_{\\mathbb{R}^{N}}|u_{n}|^{p}d\n\\rightarrow 0,\n\\end{eqnarray*\nwhich is a contradiction.\\newline\n$(II)$ Dichotomy does not occur. Suppose by contradiction that there exist a\nconstant $\\alpha \\in (0,\\alpha _{a}^{\\infty ,-})$, two sequences $\\{\\xi\n_{n}\\}$ and $\\{r_{n}\\},$ with $r_{n}\\rightarrow +\\infty $ and two\nnonnegative measures $y_{n}^{1}$ and $y_{n}^{2}$ such that $(\\ref{7-6})-(\\re\n{7-7})$ holds. Let $\\rho _{n}\\in C^{1}(\\mathbb{R}^{N})$ be such that $\\rho\n_{n}\\equiv 1$ in $B_{r_{n}}(\\xi _{n}),$ $\\rho _{n}\\equiv 0$ in $\\mathbb{R\n^{N}\\backslash B_{2r_{n}}(\\xi _{n}),$ $0\\leq \\rho _{n}\\leq 1$ and $|\\nabla\n\\rho _{n}|\\leq 2\/r_{n}.$ We se\n\\begin{equation*}\nh_{n}:=\\rho _{n}u_{n},\\text{ }w_{n}:=(1-\\rho _{n})u_{n}.\n\\end{equation*\nIt is not difficult to verify tha\n\\begin{equation}\n\\liminf_{n\\rightarrow \\infty }\\Phi _{a}^{\\infty }(h_{n})\\geq \\alpha \\text{\nand }\\liminf_{n\\rightarrow \\infty }\\Phi _{a}^{\\infty }(w_{n})\\geq \\alpha\n_{a}^{\\infty ,-}-\\alpha .  \\label{7-10}\n\\end{equation\nMoreover, let us denote $\\Omega _{n}:=B_{2r_{n}}(\\xi _{n})\\backslash\nB_{r_{n}}(\\xi _{n}).$ Then there hold\n\\begin{equation*}\ny_{n}(\\Omega _{n})\\rightarrow 0\\text{ as }n\\rightarrow \\infty ,\n\\end{equation*\nnamel\n\\begin{equation*}\n\\int_{\\Omega _{n}}(|\\nabla u_{n}|^{2}+u_{n}^{2})dx\\rightarrow 0\\text{ as \nn\\rightarrow \\infty\n\\end{equation*\nan\n\\begin{equation*}\n\\int_{\\Omega _{n}}f_{\\infty }|u_{n}|^{p}dx\\rightarrow 0\\text{ as \nn\\rightarrow \\infty .\n\\end{equation*\nA direct calculation shows tha\n\\begin{equation*}\n\\int_{\\Omega _{n}}(|\\nabla h_{n}|^{2}+h_{n}^{2})dx\\rightarrow 0\\text{ as \nn\\rightarrow \\infty\n\\end{equation*\nand\n\\begin{equation*}\n\\int_{\\Omega _{n}}(|\\nabla w_{n}|^{2}+w_{n}^{2})dx\\rightarrow 0\\text{ as \nn\\rightarrow \\infty .\n\\end{equation*\nThus, we obtain tha\n\\begin{equation}\n\\left\\Vert u_{n}\\right\\Vert _{H^{1}}^{2}=\\left\\Vert h_{n}\\right\\Vert\n_{H^{1}}^{2}+\\left\\Vert w_{n}\\right\\Vert _{H^{1}}^{2}+o_{n}(1)  \\label{7-9}\n\\end{equation\nan\n\\begin{equation}\n\\int_{\\mathbb{R}^{N}}|u_{n}|^{p}dx=\\int_{\\mathbb{R}^{N}}|h_{n}|^{p}dx+\\int_\n\\mathbb{R}^{N}}|w_{n}|^{p}dx+o_{n}(1).  \\label{7-13}\n\\end{equation\nMoreover, using $(\\ref{7-9})$ leads t\n\\begin{eqnarray}\n\\left( \\int_{\\mathbb{R}^{N}}|\\nabla u_{n}|^{2}dx\\right) ^{2} &=&\\left( \\int_\n\\mathbb{R}^{N}}|\\nabla h_{n}|^{2}dx+\\int_{\\mathbb{R}^{N}}|\\nabla\nw_{n}|^{2}dx+o_{n}(1)\\right) ^{2}  \\notag \\\\\n&\\geq &\\left( \\int_{\\mathbb{R}^{N}}|\\nabla h_{n}|^{2}dx\\right) ^{2}+\\left(\n\\int_{\\mathbb{R}^{N}}|\\nabla w_{n}|^{2}dx\\right) ^{2}+o_{n}(1).  \\label{7-23}\n\\end{eqnarray\nIt follows from $(\\ref{7-9})-\\ref{7-13})$ tha\n\\begin{equation*}\n\\Phi _{a}^{\\infty }(u_{n})=\\Phi _{a}^{\\infty }(h_{n})+\\Phi _{a}^{\\infty\n}(w_{n})+o_{n}(1).\n\\end{equation*\nThus, there hold\n\\begin{equation*}\n\\alpha _{a}^{\\infty ,-}=\\lim_{n\\rightarrow \\infty }\\Phi _{a}^{\\infty\n}(u_{n})=\\lim_{n\\rightarrow \\infty }\\Phi _{a}^{\\infty\n}(h_{n})+\\lim_{n\\rightarrow \\infty }\\Phi _{a}^{\\infty }(w_{n}).\n\\end{equation*\nUsing the above equality, together with $(\\ref{7-10})$ yield\n\\begin{equation}\n\\lim_{n\\rightarrow \\infty }\\Phi _{a}^{\\infty }(h_{n})=\\alpha \\text{ and \n\\lim_{n\\rightarrow \\infty }\\Phi _{a}^{\\infty }(w_{n})=\\alpha _{a}^{\\infty\n,-}-\\alpha .  \\label{7-24}\n\\end{equation\nFurthermore, by $(\\ref{7-9})-(\\ref{7-23})$ we ge\n\\begin{eqnarray}\n0 &=&\\left\\langle (J_{a}^{\\infty })^{\\prime }(u_{n}),u_{n}\\right\\rangle\n\\notag \\\\\n&\\geq &\\left\\langle J_{a}^{\\infty })^{\\prime }(h_{n}),h_{n}\\right\\rangle\n+\\left\\langle J_{a}^{\\infty })^{\\prime }(w_{n}),w_{n}\\right\\rangle +o_{n}(1).\n\\label{7-18}\n\\end{eqnarray\nWe have to distinguish two cases as follows:\n\nCase $(i):$ Up to a subsequence, $\\left\\langle J_{a}^{\\infty })^{\\prime\n}(h_{n}),h_{n}\\right\\rangle \\leq 0$ or $\\left\\langle J_{a}^{\\infty\n})^{\\prime }(w_{n}),w_{n}\\right\\rangle \\leq 0.$\n\nWithout loss of generality, we only consider the case of $\\left\\langle\nJ_{a}^{\\infty })^{\\prime }(h_{n}),h_{n}\\right\\rangle \\leq 0$, since the case\nof $\\left\\langle \\left( J_{a}^{\\infty }\\right) ^{\\prime }\\left( w_{n}\\right)\n,w_{n}\\right\\rangle \\leq 0$ is similar. Note that for all $n\\geq 1,$ there\nhold\n\\begin{equation}\n\\max \\left\\{ \\left\\Vert h_{n}\\right\\Vert _{H^{1}},\\left\\Vert\nw_{n}\\right\\Vert _{H^{1}}\\right\\} <D_{1}<\\left( \\frac{2S_{p}^{p}}{f_{\\infty\n}(4-p)}\\right) ^{1\/\\left( p-2\\right) }  \\label{7-17}\n\\end{equation\nThen for each $a\\leq \\Lambda ,$ we hav\n\\begin{eqnarray*}\n\\frac{p}{4-p}\\left( \\frac{2a(4-p)}{p-2}\\right) ^{\\left( p-2\\right)\n\/2}\\left\\Vert h_{n}\\right\\Vert _{H^{1}}^{p} &=&\\frac{p}{4-p}\\left( \\frac\n2a(4-p)}{p-2}\\right) ^{\\left( p-2\\right) \/2}\\left\\Vert h_{n}\\right\\Vert\n_{H^{1}}^{p-2}\\left\\Vert h_{n}\\right\\Vert _{H^{1}}^{2} \\\\\n&<&\\frac{2pS_{p}^{p}}{f_{\\infty }(4-p)^{2}}\\left( \\frac{2a(4-p)}{p-2}\\right)\n^{\\left( p-2\\right) \/2}\\left\\Vert h_{n}\\right\\Vert _{H^{1}}^{2} \\\\\n&\\leq &\\left\\Vert h_{n}\\right\\Vert _{H^{1}}^{2}+a\\left( \\int_{\\mathbb{R\n^{N}}\\left\\vert \\nabla h_{n}\\right\\vert ^{2}dx\\right) ^{2} \\\\\n&\\leq &\\int_{\\mathbb{R}^{N}}f_{\\infty }|h_{n}|^{p}dx\\text{ for }N=1,2,3.\n\\end{eqnarray*\nBy Lemmas $\\ref{g6}-\\ref{g15},$ for any $n\\geq 1$, there exist\n\\begin{equation*}\nT_{f_{\\infty }}(h_{n})<t_{a,n}^{-}<\\sqrt{D(p)}\\left( \\frac{2}{4-p}\\right)\n^{1\/(p-2)}T_{f_{\\infty }}(h_{n})\n\\end{equation*\nsuch that $t_{a,n}^{-}h_{n}\\in \\mathbf{M}_{a}^{\\infty ,-}$, where\n\\begin{equation*}\nT_{f_{\\infty }}(h_{n})=\\left( \\frac{\\left\\Vert h_{n}\\right\\Vert _{H^{1}}^{2\n}{\\int_{\\mathbb{R}^{N}}f_{\\infty }|h_{n}|^{p}dx}\\right) ^{1\/(p-2)}>0.\n\\end{equation*\nWe now prove that $t_{a,n}^{-}\\leq 1.$ Suppose the contrary. Then \nt_{a,n}^{-}>1.$ Since $t_{a,n}^{-}h_{n}\\in \\mathbf{M}_{a}^{\\infty ,-},$ we\nhav\n\\begin{equation*}\na\\left( \\int_{\\mathbb{R}^{N}}|\\nabla h_{n}|^{2}dx\\right)\n^{2}=-(t_{a,n}^{-})^{-2}\\left\\Vert h_{n}\\right\\Vert\n_{H^{1}}^{2}+(t_{a,n}^{-})^{p-4}\\int_{\\mathbb{R}^{N}}f_{\\infty\n}|h_{n}|^{p}dx.\n\\end{equation*\nUsing the above equality give\n\\begin{eqnarray}\n0 &\\geq &\\left\\langle (J_{a}^{\\infty })^{\\prime }(h_{n}),h_{n}\\right\\rangle\n=\\left\\Vert h_{n}\\right\\Vert _{H^{1}}^{2}+a\\left( \\int_{\\mathbb{R\n^{N}}|\\nabla h_{n}|^{2}dx\\right) ^{2}-\\int_{\\mathbb{R}^{N}}f_{\\infty\n}|h_{n}|^{p}dx  \\notag \\\\\n&=&\\left\\Vert h_{n}\\right\\Vert _{H^{1}}^{2}-(t_{a,n}^{-})^{-2}\\left\\Vert\nh_{n}\\right\\Vert _{H^{1}}^{2}+(t_{a,n}^{-})^{p-4}\\int_{\\mathbb{R\n^{N}}f_{\\infty }|h_{n}|^{p}dx  \\notag \\\\\n&&-\\int_{\\mathbb{R}^{N}}f_{\\infty }|h_{n}|^{p}dx  \\notag \\\\\n&=&[1-(t_{a,n}^{-})^{-2}]\\left\\Vert h_{n}\\right\\Vert\n_{H^{1}}^{2}+[(t_{a,n}^{-})^{p-4}-1]\\int_{\\mathbb{R}^{N}}f_{\\infty\n}|h_{n}|^{p}dx.  \\label{7-16}\n\\end{eqnarray\nNote that\n\\begin{equation}\n\\int_{\\mathbb{R}^{N}}f_{\\infty }|h_{n}|^{p}dx<\\frac{2}{4-p\n(t_{a,n}^{-})^{2-p}\\left\\Vert h_{n}\\right\\Vert _{H^{1}}^{2}  \\label{7-15}\n\\end{equation\nby $(\\ref{2-2}).$ It follows from $(\\ref{7-16})-(\\ref{7-15})$ tha\n\\begin{eqnarray}\n0 &\\geq &[1-(t_{a,n}^{-})^{-2}]\\left\\Vert h_{n}\\right\\Vert _{H^{1}}^{2}\n\\frac{2}{4-p}[(t_{a,n}^{-})^{-2}-(t_{a,n}^{-})^{2-p}]\\left\\Vert\nh_{n}\\right\\Vert _{H^{1}}^{2}  \\notag \\\\\n&=&\\left[ 1+\\frac{p-2}{4-p}(t_{a,n}^{-})^{-2}-\\frac{2}{4-p\n(t_{a,n}^{-})^{2-p}\\right] \\left\\Vert h_{n}\\right\\Vert _{H^{1}}^{2}  \\notag\n\\\\\n&=&(t_{a,n}^{-})^{-2}\\left[ (t_{a,n}^{-})^{2}-\\frac{2}{4-p\n(t_{a,n}^{-})^{4-p}+\\frac{p-2}{4-p}\\right] \\left\\Vert h_{n}\\right\\Vert\n_{H^{1}}^{2},  \\label{7-21}\n\\end{eqnarray\nwhich implies tha\n\\begin{equation*}\n(t_{a,n}^{-})^{2}-\\frac{2}{4-p}(t_{a,n}^{-})^{4-p}+\\frac{p-2}{4-p}\\leq 0.\n\\end{equation*\nHowever, we observe that for $2<p<\\min \\{4,2^{\\ast }\\},\n\\begin{equation*}\nt^{2}-\\frac{2}{4-p}t^{4-p}+\\frac{p-2}{4-p}>0\\text{ for }t>1.\n\\end{equation*\nThis is a contradiction. Thus, $t_{a,n}^{-}\\leq 1.$\n\nNext, let us consider the functional $\\Phi _{a}^{\\infty }(th_{n})$ defined b\n\\begin{equation*}\n\\Phi _{a}^{\\infty }(th_{n})=\\frac{t^{2}}{4}\\left\\Vert h_{n}\\right\\Vert\n_{H^{1}}^{2}-\\frac{(4-p)t^{p}}{4p}\\int_{\\mathbb{R}^{N}}f_{\\infty\n}|h_{n}|^{p}dx\\text{ for }t>0.\n\\end{equation*\nA direct calculation shows that there exists a constan\n\\begin{equation*}\nt_{a}^{\\infty }(h_{n})=\\left( \\frac{2\\left\\Vert h_{n}\\right\\Vert _{H^{1}}^{2\n}{(4-p)f_{\\infty }\\int_{\\mathbb{R}^{N}}|h_{n}|^{p}dx}\\right) ^{1\/(p-2)}>0\n\\end{equation*\nsuch that $\\Phi _{a}^{\\infty }(th_{n})$ is increasing on $(0,t_{a}^{\\infty\n}(h_{n}))$ and is decreasing on $(t_{a}^{\\infty }(h_{n}),\\infty ).$\nFurthermore, by the Sobolev inequality and $(\\ref{7-17})$ we hav\n\\begin{equation*}\nt_{a}^{\\infty }(h_{n})\\geq \\left( \\frac{2}{(4-p)f_{\\infty\n}S_{p}^{-p}\\left\\Vert h_{n}\\right\\Vert _{H^{1}}^{p-2}}\\right) ^{1\/(p-2)}>1.\n\\end{equation*\nThis indicates that $\\Phi _{a}^{\\infty }(t_{a,n}^{-}h_{n})\\leq \\Phi\n_{a}^{\\infty }(h_{n}).$ Hence, for all $n\\geq 1,$ there hold\n\\begin{equation*}\n\\alpha _{a}^{\\infty ,-}\\leq J_{a}^{\\infty }(t_{a,n}^{-}h_{n})=\\Phi\n_{a}^{\\infty }(t_{a,n}^{-}h_{n})\\leq \\Phi _{a}^{\\infty }(h_{n})\\rightarrow\n\\alpha <\\alpha _{a}^{\\infty ,-},\n\\end{equation*\nwhich is a contradiction.\\newline\nCase $(ii):$ Up to a subsequence, $\\left\\langle (J_{a}^{\\infty })^{\\prime\n}(h_{n}),h_{n}\\right\\rangle >0$ and $\\left\\langle (J_{a}^{\\infty })^{\\prime\n}(w_{n}),w_{n}\\right\\rangle >0.$\n\nBy $(\\ref{7-18})$ one has $\\left\\langle (J_{a}^{\\infty })^{\\prime\n}(h_{n}),h_{n}\\right\\rangle =o_{n}(1)$ and $\\left\\langle (J_{a}^{\\infty\n})^{\\prime }(w_{n}),w_{n}\\right\\rangle =o_{n}(1).$ If $t_{a,n}^{-}\\leq\n1+o_{n}(1),$ then we can repeat the argument of Case $(i)$ and arrive at the\ncontradiction. Suppose that\n\\begin{equation*}\n\\lim_{n\\rightarrow \\infty }t_{a,n}^{-}=t_{a,\\infty }^{-}>1.\n\\end{equation*\nSimilar to the argument of $(\\ref{7-16})$, we hav\n\\begin{eqnarray*}\no_{n}(1) &=&\\left\\langle (J_{a}^{\\infty })^{\\prime\n}(h_{n}),h_{n}\\right\\rangle \\\\\n&=&[1-(t_{a,n}^{-})^{-2}]\\left\\Vert h_{n}\\right\\Vert\n_{H^{1}}^{2}+[(t_{a,n}^{-})^{p-4}-1]\\int_{\\mathbb{R}^{N}}f_{\\infty\n}|h_{n}|^{p}dx.\n\\end{eqnarray*\nSimilar to the argument of $(\\ref{7-21}),$ one ge\n\\begin{equation}\no_{n}(1)\\geq (t_{a,n}^{-})^{-2}\\left[ (t_{a,n}^{-})^{2}-\\frac{2}{4-p\n(t_{a,n}^{-})^{4-p}+\\frac{p-2}{4-p}\\right] \\left\\Vert h_{n}\\right\\Vert\n_{H^{1}}^{2},  \\notag\n\\end{equation\nwhich shows that\n\\begin{equation*}\n\\left\\Vert h_{n}\\right\\Vert _{H^{1}}^{2}\\rightarrow 0\\text{ as }n\\rightarrow\n\\infty ,\n\\end{equation*\nwhere we have used the fact of $(t_{a,n}^{-})^{2}-\\frac{2}{4-p\n(t_{a,n}^{-})^{4-p}+\\frac{p-2}{4-p}>0.$ The\n\\begin{equation*}\n\\int_{\\mathbb{R}^{N}}|h_{n}|^{p}dx\\rightarrow 0\\text{ as }n\\rightarrow\n\\infty .\n\\end{equation*\nHence, $\\Phi _{a}^{\\infty }(h_{n})\\rightarrow 0$ as $n\\rightarrow \\infty ,$\nwhich contradicts $(\\ref{7-24})$. Therefore, the dichotomy cannot occur.\n\\end{proof}\n\n\\section*{Acknowledgments}\n\nJ. Sun is supported by the National Natural Science Foundation of China\n(Grant No. 11671236). T.F. Wu is supported in part by the Ministry of\nScience and Technology, Taiwan (Grant 108-2115-M-390-007-MY2), the\nMathematics Research Promotion Center, Taiwan and the National Center for\nTheoretical Sciences, Taiwan.\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\n\\label{sec:intro}\nA wireless relay network is one in which a set of\nrelay nodes assist a source node transmit information to a\ndestination node. Practically the wireless nodes can only support\nhalf-duplex communication \\cite{Laneman04}, i.e., no nodes can\nreceive and transmit information simultaneously on the same\nfrequency band. Different cooperative transmission schemes for\nsystems with half-duplex nodes have been proposed in the literature.\nFundamentally, these schemes consist of two basic steps. First, the\nsource transmits to the destination, and the relay listens and\n``captures'' \\cite{Host05} the transmission from the source at the\nsame time. Next, the relays send processed source information to the\ndestination while the source may still transmit to the destination\ndirectly. Variants of these techniques have been proposed and have\nbeen shown to yield good performance under different circumstances\n\\cite{Avestimehr06,Azarian05,Laneman04}.\n\nAssuming channel state information (CSI) at the nodes, an\nopportunistic decode-and-forward (DF) protocol for half-duplex relay\nchannels is proposed in \\cite{Gunduz07}.\nIn \\cite{Ong07}, the authors present\nrouting algorithms to optimize the rate from a source to a\ndestination, based on the DF technique that uses regular block\nMarkov encoding and windowed decoding \\cite{Xie04,Kramer03}, for the\nGaussian full-duplex multiple-relay channel. The achievable rate of\n\\cite{Xie04} for the Gaussian physically degraded full-duplex\nmulti-relay channel has been established as the capacity of this\nchannel in \\cite{Reznik04}. In \\cite{Ong06}, it is shown that the\ncut-set bound on the capacity of the Gaussian single source-multiple\nrelay-single destination mesh network can be achieved using the\ncompress-and-forward (CF) method, as the relay powers go to\ninfinity.\n\nSome simpler cooperative diversity methods based on network path\nselection have been recently reported \\cite{Bletsas06,Beres06}.\nThese selection methods include: (i) the max-min selection method\n\\cite{Bletsas06}, wherein the relay node with the maximum of the\nminimum of the source-relay and relay-destination channel gains is\nselected; (ii) the harmonic mean selection method \\cite{Bletsas06},\nwherein the relay node with the highest harmonic mean of the\nsource-relay and relay-destination channel gains is selected; and\n(iii) the selection scheme of \\cite{Beres06}, in which the relay\nthat can correctly decode the information from the source and has\nthe best relay-destination channel is selected. These methods\nachieve a DMT of $d(r)=(N-1)(1-2r)$ for\nan $N$ node relay network and multiplexing gain $0<r<0.5$. This is\nclose to what the distributed space-time coding protocol\n\\cite{Laneman03} achieves, when $N$ is large. Unfortunately, these\nnetwork selection protocols perform poorly in high-rate scenarios\n($r>0.5$).\n\nWe have proposed a cooperative diversity design based on a flow\noptimization approach for a three-node network in \\cite{Wong06}. In\nthis design, the source node broadcasts two distinct flows to the\ndestination and the relay node respectively during the relay's\nlisten period.  Then the relay forwards this information using the\nDF approach while the source may also send another flow of\ninformation to the destination during the relay's transmit period. \nThis scheme is shown to achieve the\noptimal diversity order for the three-node relay channel and yield\nperformance very close to optimal full-duplex relaying in both\nlow- and high-rate situations.\n\nHere, we apply this cooperative transmission design to a general\nrelay network, wherein wireless links are present between each\npair of nodes in the network.\nAs in \\cite{Wong06}, assuming CSI is available at all nodes we use\nbroadcasting (BC), multiple access (MA) and time sharing (TS)\ntechniques to formulate a flow theoretic convex optimization problem\nbased on the channel conditions. Instead of considering a total\npower constraint for all the transmitting nodes as in~\\cite{Wong06},\nwe subject each node to a maximum transmit power constraint. This\nyields a more reasonable system model for a general wireless\nrelay network, especially when the number of nodes in the relay network is\nlarge. The resulting relaying protocol will be referred to as the\nflow-optimized (FO) protocol. To obtain a more practical cooperative\ndesign we develop a generalized-link selection (GLS) protocol, in\nwhich we select the best relay node out of the available ones to\nform an equivalent three-node relay network to transmit the information\nfrom the source to the destination. The benefit of this, over other\nnetwork path selection strategies, becomes evident when the rate\nrequirement is high. It is shown that the simple GLS protocol is\noptimal in terms of the DMT\n\\cite{Zheng03} and yields acceptable performance even when the rate\nrequirement is high.\n\nRecently, in~\\cite{Yuksel07}, the authors have shown that compress-and-forward (CF) relaying\nachieves the optimal DMT for the three-node, half-duplex network, and that DF relaying\ncan achieve the optimal DMT of the four-node full-duplex network.\nIn this work, we show that the optimal DMT can be achieved for a general $N$-node ($N\\geq3$)\nhalf-duplex network using the FO or GLS protocols. Here, it should be clarified\nthat we consider that the wireless\nlinks between each node-pair experience\nindependent Rayleigh fading, and this corresponds to the definition\nof \\emph{non-clustered} networks in~\\cite{Yuksel07}.\nThe performances of the FO and GLS protocols are\nevaluated numerically in terms of their outage probabilities for\nfour- and five-node relay networks for uniform and non-uniform\naverage power gains. The numerical results motivate the use of the\nGLS protocol for situations where computation complexity is an issue\nand show a remarkable improvement over the max-min selection method\nof \\cite{Bletsas06}. The proposed designs, based on BC and MA alone,\nare sub-optimal in general. For a fair appraisal of the proposed\nprotocols, we compare the proposed protocols to an upper bound on\nthe maximum rate, derived using the max-flow-min-cut theorem\n\\cite[Thm.~14.10.1]{Cover91}.\n\n\\section{A General Design Using A Flow-theoretic Approach}\n\\label{sec:gdesgn}\n\nWe consider an $N$-node wireless relay network with a link joining each\npair of nodes. Each such wireless link is described by a bandpass\nGaussian channel with bandwidth $W$ and one-sided noise spectral\ndensity $N_0$. We denote the power gain of the link from node $i$ to\nnode $j$ as $Z_{ij}$. The link power gains are assumed to be\nindependent and identically distributed (i.i.d.) exponential random\nvariables with unit mean. This corresponds to the case of\nindependent Rayleigh fading channels with unit average power gains.\nMoreover, we assume that each node has a maximum power limit of $P$\nand can only support half-duplex transmission. Note that this model\ncan be easily generalized to the case where channels may have\nnon-uniform average power gains (for which numerical examples are\npresented in Section~\\ref{sec:num_ex}), and where different nodes\nmay have different maximum power constraints. More specifically, the\nlatter case can be converted into the uniform maximum power\nconstraints case by absorbing the non-uniformity in the transmit\npowers into the average power gains of the corresponding links.  In\nthe sequel, we characterize the system in terms of the transmit\nsignal-to-noise ratio (SNR), $S = \\frac{P}{N_0W}$, at the input of\nthe links.  Time is divided into unit intervals, and BC and MA are\napplied with a TS strategy that is optimized to maximize the\nspectral efficiency (which we call ``rate'' hereafter for\nconvenience). To avoid interference between concurrent\ntransmissions, a time interval is divided into slots:\n\n\\begin{itemize}\n\\item During the first slot, the source may BC to all the other\n  nodes in the network\n\n\\item During the subsequent slots, a relay may BC to all other nodes\n  (except the source node), or it may receive flows from all other\n  nodes (except from the destination) through MA.\n\n\\item During the very last slot, the source and the relays may send\n  information flows to the destination using MA.\n\n\n\\end{itemize}\nNote that the forwarding of information by the relays is based on the DF approach.\nFor practicality consideration, it is assumed that the phases of the\nsimultaneously transmitted signals from different nodes are not\nsynchronized. In general, for the above transmission protocol, there would be a maximum of $2(N-2)+ 2 = 2N-2$ time slots of lengths $t_1, t_2, \\cdots, t_{2N-2}$ respectively.\n\nNext, we describe the optimization problem using a graph-theoretic\nformulation.  Define a graph $G=(V,E)$, where $V$ is the set of\nnodes, $E$ is the set of all links joining the nodes in the graph,\nand associate the vector $\\underbar{r}$ to represent the flow rates\nassociated with each link in $E$. Thus, the number of elements in\n$\\underbar{r}$ equals the cardinality of $E$.  For convenience, we\nwrite $G=(V,E,\\underbar{r})$.  Now denote the source by\n$\\mathcal{S}$, the destination by $\\mathcal{D}$, and the relay nodes\nby $\\mathcal{R}_1, \\ldots, \\mathcal{R}_{N-2}$. The slotting of a\nunit time interval, as described above, yields simpler graphs for\neach time slot, that we call \\emph{basic graphs}. A basic graph is\neither one in which a particular node may BC to several nodes, or in\nwhich several nodes transmit via MA to a particular node. Thus for a\nbasic graph, we need to include only the links between the nodes\nthat may participate during the concerned time slot. For example,\nassume that the relay $\\mathcal{R}_1$ broadcasts to all nodes other than the\nsource, during the $i$-th time slot. The basic graph is given by\n$G_i=(V,E_i,\\underbar{r}_i)$ where $V=\\{\\mathcal{S}, \\mathcal{R}_1,\n\\cdots, \\mathcal{R}_{N-2}, \\mathcal{D}\\},~\nE_i=\\{\\mathcal{R}_1\\mathcal{R}_2, \\cdots,\n\\mathcal{R}_1\\mathcal{R}_{N-2}, \\mathcal{R}_1\\mathcal{D}\\},~\n\\underbar{r}_i =\n\\left(\\frac{x_{\\mathcal{R}_1\\mathcal{R}_2}^{i}}{t_i}~ \\cdots\n~\\frac{x_{\\mathcal{R}_1\\mathcal{R}_{N-2}}^{i}}{t_i}\n~\\frac{x_{\\mathcal{R}_1\\mathcal{D}}^{i}}{t_i}\\right)^T$, where\n$x_{AB}^{i}$ is the flow from node $A$ to node $B$ during the $i$-th\ntime slot.\n\nIn general, the proposed design involves TS between the basic graphs\nto yield the following equivalent graph $G$ corresponding to a unit\ninterval (see \\cite{Wu05} for a similar idea):\n\\begin{equation}\nG = \\left( V, ~\\bigcup_i E_i, ~\\sum_i\nt_i\\underbar{r}_i\\right)\n= t_1 G_1 + t_2 G_2 + \\ldots + t_{2N-2} G_{2N-2}.\n\\label{eqn:G_equiv}\n\\end{equation}\n\\noindent where the number of elements in each vector\n$\\underbar{r}_i$ is extended to $\\vert\\bigcup_i E_i\\vert$ by\ninserting zeros appropriately. The second equality in\n(\\ref{eqn:G_equiv}) implies that $G$ can be viewed as a linear\ncombination of the basic graphs $G_i$s, with the equivalent set of\nedges given by the union of the sets $E_i$, and the equivalent flow\nrate vector given by the linear combination of the individual flow\nrate vectors $\\underbar{r}_i$.\n\nTo maximize the data rate from the source to the destination through\nthe relay network, we need to consider each cut that partitions $V$\ninto sets $V^s$ and $V^d$ with $\\mathcal{S}\\in V^s$ and\n$\\mathcal{D}\\in V^d$.\nClearly, there can be $2^{N-2}$ such possible cuts for the $N$-node\nrelay network. Let these cuts and the corresponding cut sets be denoted by\n$\\mathcal{C}_k$, $V^s_k$, and $V^d_k$, respectively, for\n$k=1,2,\\cdots,2^{N-2}$.  Further, for the graph $G$, for any two\nnodes $A\\in V^s_k$ and $B\\in V^d_k$, there exists a \\emph{cut edge}\n$AB$ that crosses the cut. Denote the total flow through cut edge $AB$ in a\nunit time interval by $x_{AB}=\\sum_{i=1}^{2N-2}x_{AB}^{i}$. Now\nrecall from network flow theory~\\cite{Ahuja93} that the maximum flow\nrate from the source to the destination is specified by the minimal\ncut of the equivalent graph (\\ref{eqn:G_equiv}). Consequently, we\narrive at the following convex flow optimization problem that can be\nsolved using standard optimization techniques:\n\\begin{equation}\n  \\max\\min \\left(\\sum_{A\\in V^s_1,B\\in V^d_1}x_{AB}, \\sum_{A\\in\n      V^s_2,B\\in V^d_2}x_{AB}, \\cdots, \\sum_{A\\in V^s_{2^{N-2}},B\\in\n      V^d_{2^{N-2}}}x_{AB} \\right)\\label{eqn:main_opt_prob1}\n\\end{equation}\n\\noindent over all flow allocations $x_{AB}^{i}$ and all time slot\nlengths $t_i$, subject to\n\\begin{itemize}\n\\item the \\emph{non-negativity constraints}: $x_{AB}^{i}$, $t_i \\geq\n  0$ for all cut edges $AB$ and $i=1,2,\\cdots,2N-2$,\n\\item the \\emph{total-time constraint}: $t_1+\\ldots+t_{2N-2} =1$,\n\\item the \\emph{power (capacity) constraints}:\n\\begin{itemize}\n\\item for a BC slot the flow rates should lie in the capacity region\n  of the BC channel with the transmitting node having a power\n  constraint of $P$,\n\\item for an MA slot the flow rates should lie in the capacity region\n  of the MA channel with a maximum power constraint $P$ for each\n  transmitting node,\n\\end{itemize}\n\\item the \\emph{flow constraints}: considering steady state operation,\n  the total information flow out of a relay should equal the flow into\n  the relay in each unit time interval.\n\\end{itemize}\n\n\\noindent Note that the dependence of the objective function on the channel gains and the\ntime slot lengths is implicitly expressed through the capacity constraints.\nDenote the cut separating $\\mathcal{S}$ from all the other nodes and\nthe cut separating $\\mathcal{D}$ from all nodes as\n$\\mathcal{C}_{\\mathcal{S}}$ and $\\mathcal{C}_{\\mathcal{D}}$,\nrespectively.  Then we observe that the cost function in\n(\\ref{eqn:main_opt_prob1}) above can be further simplified\nto $\\max~\\min\\left\\{x(\\mathcal{C}_{\\mathcal{S}}),~x(\\mathcal{C}_{\\mathcal{D}})\\right\\}$,\nwhere\n\\begin{equation}\n\\displaystyle x(\\mathcal{C}_{\\mathcal{S}}) = x_{\\mathcal{SD}} +\n\\sum_{j=1}^{N-2}x_{\\mathcal{S}\\mathcal{R}_j}\\mathrm{~~~ and ~~~}\nx(\\mathcal{C}_{\\mathcal{D}}) = x_{\\mathcal{SD}} +\n\\sum_{i=1}^{N-2}x_{\\mathcal{R}_i\\mathcal{D}}\\label{eqn:BC_cut}\n\\end{equation}\nare the total flows across the above-mentioned cuts\n$\\mathcal{C}_{\\mathcal{S}}$ and $\\mathcal{C}_{\\mathcal{D}}$,\nrespectively.  To see this, consider the cut $\\mathcal{C}$ with\n$V^s=\\{\\mathcal{S},\\mathcal{R}_1,\\cdots,\\mathcal{R}_l\\}$, and\n$V^d=\\{\\mathcal{R}_{l+1},\\cdots,\\mathcal{R}_{N-2},\\mathcal{D}\\}$ for\nsome $l\\in\\{1,2,\\cdots,N-2\\}$. The total flow across this cut is\ngiven by\n\\begin{equation}\n\\displaystyle x(\\mathcal{C}) = x_{\\mathcal{SD}} +\n\\sum_{j=l+1}^{N-2}x_{\\mathcal{S}\\mathcal{R}_j} +\n\\sum_{i=1}^l\\left(x_{\\mathcal{R}_i\\mathcal{D}} +\n\\sum_{j=l+1}^{N-2}x_{\\mathcal{R}_i\\mathcal{R}_j}\\right).\\label{eqn:flow_thru_C}\n\\end{equation}\n\n\\noindent Now, consider node $i$ for $i\\in\\{1,2,\\cdots,l\\}$.\nAccording to the flow constraint for node $i$,\n\\begin{equation}\n\\displaystyle x_{\\mathcal{R}_i\\mathcal{D}} +\n\\sum_{j=l+1}^{N-2}x_{\\mathcal{R}_i\\mathcal{R}_j} + \\sum_{k=1,k\\neq\ni}^{l}x_{\\mathcal{R}_i\\mathcal{R}_k} = x_{\\mathcal{S}\\mathcal{R}_i}\n+ \\sum_{j=l+1}^{N-2}x_{\\mathcal{R}_j\\mathcal{R}_i} + \\sum_{k=1,k\\neq\ni}^{l}x_{\\mathcal{R}_k\\mathcal{R}_i}.\\label{eqn:node_i_flow}\n\\end{equation}\nSumming (\\ref{eqn:node_i_flow}) over all $i\\in\\{1,2,\\cdots,l\\}$ we get\n\\begin{equation}\n\\sum_{i=1}^l\\left(x_{\\mathcal{R}_i\\mathcal{D}}\n+ \\sum_{j=l+1}^{N-2}x_{\\mathcal{R}_i\\mathcal{R}_j}\\right)=\n\\sum_{i=1}^l\\left(x_{\\mathcal{S}\\mathcal{R}_i} +\n\\sum_{j=l+1}^{N-2}x_{\\mathcal{R}_j\\mathcal{R}_i}\\right).\\label{eqn:nodes_i2l_flow}\n\\end{equation}\nSince\n$\\sum_{i=1}^l\\sum_{j=l+1}^{N-2}x_{\\mathcal{R}_j\\mathcal{R}_i}\\geq 0$,\ncombining (\\ref{eqn:BC_cut}), (\\ref{eqn:flow_thru_C}) and\n(\\ref{eqn:nodes_i2l_flow}) gives $x(\\mathcal{C}) \\geq\nx(\\mathcal{C}_{\\mathcal{S}})$.  Similarly, we have\n$x(\\mathcal{C}) \\geq x(\\mathcal{C}_{\\mathcal{D}})$. Thus the cost\nfunction in (\\ref{eqn:main_opt_prob1}) reduces to the above-mentioned form.\n\n\\section{Generalized-link Selection and Its Optimality}\\label{sec:optimality}\n\nIn this section, we present the GLS protocol and establish the\noptimality of the FO and GLS protocols in terms of the\nDMT.  This is accomplished in three\nsteps. First, we apply the FO protocol to the three-node relay network.\nNext, we propose the GLS protocol based on a selection strategy that\nis sub-optimal to the FO protocol of Section~\\ref{sec:gdesgn}.\nFinally, the optimality of the GLS protocol, and thereby, that of the FO\nprotocol, is established.\n\n\\subsection{The Three-node Relay Network} \\label{subsec:3node}\n\nThe three-node relay network consists of a source ($\\mathcal{S}$), a relay\n($\\mathcal{R}$), and a destination ($\\mathcal{D}$). We specialize\nthe general design described in the previous section to this\nthree-node relay network.  A unit time interval is divided into two time\nslots of lengths $t_1$ and $t_2$ with $t_1 + t_2 = 1$.\nDuring the first time slot, $\\mathcal{S}$ sends (via BC) two flows\nof rates $x_{\\mathcal{SD}}^{1}\/t_1=x_1\/t_1$ and\n$x_{\\mathcal{SR}}^{1}\/t_1=x_2\/t_1$ to $\\mathcal{D}$ and\n$\\mathcal{R}$, respectively, resulting in the basic graph $G_1$.\nDuring the second time slot, $\\mathcal{R}$ and $\\mathcal{S}$ send\n(via MA) two flows of rates $x_{\\mathcal{RD}}^{2}\/t_2=x_4\/t_2$ and\n$x_{\\mathcal{SD}}^{2}\/t_2=x_3\/t_2$ to $\\mathcal{D}$, respectively,\nresulting in the basic graph $G_2$.\nCombining the two basic graphs yields the\nequivalent graph as $G = t_1G_1 + t_2G_2$. Note that the information\nflow of rate $x_4\/t_2$ sent by $\\mathcal{R}$ during the MA time slot\nis from the flow of rate $x_2\/t_1$ it received during the BC time\nslot.\nThus, we have the flow constraint $x_4=x_2$. The rate for this\nnetwork is specified by the min-cut which is clearly $\\min\\{(x_1 +\nx_2 + x_3),(x_1+x_4+x_3)\\}$. Hence, the flow optimization problem is\ngiven by:\n\\begin{equation}\n\\max~\\min~\\{(x_1 + x_2 + x_3),(x_1+x_4+x_3)\\} \\label{eqn:3node_opt}\n\\end{equation}\nover flow allocations $x_1,~x_2,~x_3,~x_4$, and time slot lengths\n$t_1,~t_2$, subject to\n\\begin{itemize}\n\\item {\\it non-negativity constraints: } $x_1, x_2, x_3, x_4 \\geq\n  0,\\quad t_1, t_2 \\geq 0$,\n\\item {\\it total-time constraint: } $t_1+t_2=1$,\n\\item {\\it power constraints: } $S_{BC}\\leq S,~x_1\\leq t_1C(Z_{\\mathcal{SD}}S),~x_2\\leq t_1C(Z_{\\mathcal{SR}}S)$ for the BC slot,\\\\\n   \n  $\\qquad x_3\\leq t_2C(Z_{\\mathcal{SD}}S),~ x_4\\leq\n  t_2C(Z_{\\mathcal{RD}}S),~ x_3 + x_4\\leq\n  t_2C(Z_{\\mathcal{SD}}S+Z_{\\mathcal{RD}}S)$ for the MA slot,\n\\item {\\it flow constraint: } $x_2 = x_4$,\n\\end{itemize}\nwhere $C(x)=\\log(1+x)$,  and $S_{BC}$, the minimum SNR required for\nthe source to broadcast at rates $x_1\/t_1$ and $x_2\/t_1$ to the\ndestination and the relay, respectively, in the first time slot with\n$0<t_1\\leq1$, is given by (see~\\cite[Lemma 3.1]{Wong06} for proof)\n\\[\nS_{BC}= \\left \\{\n\\begin{array}{ll}\n\\frac{1}{Z_{\\mathcal{S}\\mathcal{D}}} ( e^{x_1\/t_1} -1) +\n\\frac{1}{Z_{\\mathcal{S}\\mathcal{R}}} e^{x_1\/t_1} ( e^ {x_2\/t_1} -1)\n\\mathrm{~ for ~} Z_{\\mathcal{S}\\mathcal{R}}> Z_{\\mathcal{S}\\mathcal{D}}, \\\\\n\\frac{1}{Z_{\\mathcal{S}\\mathcal{R}}} ( e^{x_2\/t_1} -1) +\n\\frac{1}{Z_{\\mathcal{S}\\mathcal{D}}} e^{x_2\/t_1} ( e^ {x_1\/t_1} -1)\n\\mathrm{~ for ~} Z_{\\mathcal{S}\\mathcal{R}} \\leq\nZ_{\\mathcal{S}\\mathcal{D}}.\n\\end{array}\n\\right .\n\\]\nFor $t_1=0$, $S_{BC}=0$. Note that for the BC slot, the last two constraints are redundant when $t_1>0$, and complements the first constraint when $t_1=0$.\n\nThe solution of this flow optimization problem is given in\nAppendix~\\ref{app:opt_soln}. As mentioned in Section~\\ref{sec:intro},\nthe above optimization problem formulation is different from that\nin~\\cite{Wong06} wherein the sum of the source and relay powers,\nrequired to achieve a certain data rate, is minimized. More\nspecifically, when considering individual power constraints for each\nnode, we cannot use part $2$ of~\\cite[Lemma 3.1]{Wong06} to describe\nthe power constraints for the MA slot. This is because doing so would\nrestrict the flows $x_2$ and $x_3$ such that the sum of powers\nexpended at $\\mathcal{S}$ and $\\mathcal{R}$ is minimized.  On the\nother hand, in the present problem, the power constraints only dictate\nthat the flow-rates should lie in the MA capacity region specified by\nthe maximum power available at each transmitting node, for the\nparticular fading state. With this modification in the constraint for\nthe MA slot, the solution approach to the above problem needs to be\nmarkedly different from that in~\\cite{Wong06} as shown in Appendix~\\ref{app:opt_soln}.\nThe maximum information rate from the source $\\mathcal{S}$ to the\ndestination $\\mathcal{D}$ for different cases is summarized below:\n\n\\noindent a) $Z_{\\mathcal{SD}}\\geq Z_{\\mathcal{SR}}$: The maximum\nrate is $X(S) = C(Z_{\\mathcal{SD}}S)$ with direct transmission\nfrom $\\mathcal{S}$ to $\\mathcal{D}$.\n\n\\noindent b) $Z_{\\mathcal{SD}}<Z_{\\mathcal{SR}}$: The maximum rate\nis $X(S) = \\max_{0\\leq t_2\\leq t_{2\\textnormal{max}}}\nt_1\\log\\left(\\frac{1+Z_{\\mathcal{SD}}S}\n{1+\\frac{Z_{\\mathcal{SD}}}{Z_{\\mathcal{SR}}}\\left[\\left(1+\\frac{Z_{\\mathcal{RD}}S}{1+Z_{\\mathcal{SD}}S}\\right)^{t_2\/t_1}-1\\right]}\\right)\n+ t_2C(Z_{\\mathcal{SD}}S+Z_{\\mathcal{RD}}S)$ with\n$t_{2\\textnormal{max}} =\nC(Z_{\\mathcal{SR}}S)\/\\left[C(Z_{\\mathcal{SR}}S)+C(Z_{\\mathcal{RD}}S+Z_{\\mathcal{SD}}S)-C(Z_{\\mathcal{SD}}S)\\right]$\nand $t_1 = 1 - t_2$.\n\nThus for a given power limit (i.e. a given $S$) at the nodes,\nrelaying is advantageous only when\n$Z_{\\mathcal{SD}}<Z_{\\mathcal{SR}}$. Further, the optimal solution\nalways allocates a non-zero flow to the direct link. Also, the\nrelay-destination link gain $Z_{\\mathcal{RD}}$ does not influence\nthe strategy of transmission (i.e. whether to use only the direct\nlink or both the relay and direct links), but only the amount of\ninformation through the relay link.\n\n\\subsection{Generalized-link Selection} \\label{subsec:sub_opt}\nFor the general $N$-node relay network, the flow optimization solution can\nbe computationally demanding even for moderate values of $N$.\nThe GLS protocol described below provides a simple sub-optimal\ndesign to address this complexity issue.  In essence, the GLS\nprotocol identifies the \\emph{best} relay path out of the possible\n$N-2$ relay paths and considers only the chosen relay along with the\nsource and destination to form a three-node relay network, which we call a\n\\emph{generalized-link} from the source to the destination, for\ninformation transmission. In other words, the aim is to choose the\nbest relay such that the equivalent three-node relay network obtained\n(containing the source, destination and the chosen relay) gives the\nmaximum rate over all possible equivalent three-node networks\ncontaining the source and destination.  More precisely, we need to\nconsider the following possibilities:\n\\begin{enumerate}\n\\item $Z_{\\mathcal{SD}}\\geq Z_{\\mathcal{S}\\mathcal{R}_i}$ for all\n  $i\\in I = \\{1,2,\\cdots,N-2\\}$: From the results of the optimization\n  problem (\\ref{eqn:3node_opt}), it is clear that the maximum rate\n  would be $C(Z_{\\mathcal{SD}}S)$ with direct transmission of all data\n  from the source to the destination without using any relay.\n\n\\item There exists a $k\\in I$ such that\n  $Z_{\\mathcal{S}\\mathcal{R}_k}>Z_{\\mathcal{SD}}$: Let the set of all\n  such node indices be $K$ and for all $i\\in I\\setminus K$,\n  $Z_{\\mathcal{SD}}\\geq Z_{\\mathcal{S}\\mathcal{R}_i}$. For this case,\n  choose the node $\\mathcal{R}_k'$ as the relay such that\n  $k'=\\arg\\max_{k\\in K} X_k(S)$, where $X_k(S)$ is the maximum rate\n  for the three-node relay network with the source $\\mathcal{S}$, the relay\n  $\\mathcal{R}_k$ and destination $\\mathcal{D}$.\n\\end{enumerate}\nIn terms of the worst-case computational complexities for the FO and GLS\nprotocols, it can be seen that, for an $N$-node relay network with $N>3$, the FO protocol\ninvolves a max-min optimization over $2(N^2-2N+2)$ variables (all possible flows and time slot\nlengths), subject to $N-1$ non-linear and $2(N^2-N+1)$ linear constraints, whereas\nthe GLS protocol involves a maximum of $N-2$ maximizations of a non-linear concave function over two variables,\nsubject to two linear constraints, followed by finding the maximum of $N-2$ real numbers with\na worst-case complexity of $O(N-2)$. Moreover, for $N>3$, for the FO protocol, the BC slots\npotentially involve $(N-1)$- and $(N-2)$-level superposition coding (SPC) or dirty paper coding (DPC)\nimplementations for $\\mathcal{S}$ and the relays respectively, while the MA slots at\nthe relays and $\\mathcal{D}$ may involve a maximum of $(N-3)$ and $(N-2)$\ninterference cancelation (IC) operations respectively. On the other hand, the GLS protocol\ninvolves a maximum of $2$-level SPC\/DPC and one IC operation for the BC and MA slots respectively,\nfor any $N>3$.\n\n\\subsection{Diversity-multiplexing tradeoff} \\label{subsec:dm_tradeoff}\n\nAs in \\cite{Zheng03}, the multiplexing gain $r = \\lim_{S \\rightarrow\n  \\infty} \\frac{R(S)}{\\log S}$ where $S$ is the SNR and $R(S)$ is the\nrate at an SNR level of $S$.\nFollowing \\cite{Zheng03}, we parameterize the system, in terms of the\nSNR $S$ and the multiplexing gain, $0<r<1$, with the rate increasing\nwith the SNR as $R = r\\log(S)$.  With the parameterization $(r,S)$,\nthe diversity order achieved by the transmission scheme is given by\n\\begin{equation}\nd(r) = \\lim_{S \\rightarrow \\infty} \\frac{-\\log P_{e}\n(r,S)}{\\log S}\\label{eqn:div_def}\n\\end{equation}\nwhere $P_{e}(r,S)$ is the average probability of error when the SNR is\n$S$ and multiplexing gain $r$.\nThe following theorem, whose proof is outlined in\nAppendix~\\ref{app:pf_dm_sub_opt}, establishes the optimality of the\nthe GLS protocol (and hence the FO protocol) in terms of the\nDMT:\n\\begin{theorem} \\label{thm:dm_sub_opt}\n  The GLS and FO protocols achieve the optimal DMT\n  $d(r)=(N-1)(1-r)$ for all $0<r<1$, for the $N$-node\n  half-duplex wireless relay network.\n\\end{theorem}\n\n\\section{Numerical Examples}\\label{sec:num_ex}\n\nUsing the outage probability as the performance metric, we compare\nthe FO and GLS protocols against the max-min selection method of\n\\cite{Bletsas06}, as it provides the best performance amongst\npreviously proposed path selection methods, and an outage\nprobability lower bound derived using the max-flow-min-cut theorem\nof \\cite[Thm. 14.10.1]{Cover91}.\n\nFor the four-node relay network, there can be 6 possible time slots in the\nFO protocol as shown in Fig.~\\ref{fig:4_node_FO}.\nTo derive an upper bound on the achievable rate (and thereby a lower\nbound on the outage probability), we use max-flow-min-cut type\nbounds for half-duplex communication. There are four possible time\nslots as shown in Fig.~\\ref{fig:4_node_lb}, with the first BC slot\nand the last MA slot at the destination same as in the FO protocol,\nbut now, the source and a relay may transmit simultaneously to the\nother relay and the destination during each of the intermediate\nslots over interference channels. We use the max-flow-min-cut\ntheorem to upper bound the maximum information flow in these two\ntime slots.\n\nFor the five-node relay network, there can be 8 possible time slots in the\nFO protocol - four BC slots for the source and the three relays to\ntransmit information, and four MA slots for the three relays and the\ndestination to receive information respectively. Similar to the\nfour-node relay network case, for the max-flow-min-cut bound, there are\n$8$ possible time slots with the first BC slot and the last MA slot\nat the destination being the same as for the FO protocol, and\nmulti-source-multi-destination transmissions during the six\nintermediate slots.\n\n\nWith the above division of time slots, the formalization of the\nproblem is done as in the previous sections, and we use the\noptimization routine of \\cite{Lawrence97} to obtain the maximum\nachievable rates and upper bounds for different values of required\nrates. In Figs.~\\ref{fig:case_1u_R1} and \\ref{fig:case_6u_R6}, we\nplot the outage probabilities of the various schemes with the\nrequired rate $R$ at $1$bit\/s\/Hz and $6$bits\/s\/Hz respectively, for\nthe four-node relay network. Figs.~\\ref{fig:5node_case_1u_R1} and\n\\ref{fig:5node_case_6u_R6} present the same for the five-node\nrelay network. When compared to the FO protocol, the GLS protocol suffers\na loss of around  $1.0$dB, and around $1.5$dB (when $R$ is either\n$1$bit\/s\/Hz or $6$bits\/s\/Hz), at an outage probability of $10^{-4}$,\nfor the four- and five-node relay networks respectively. On the other\nhand, the performance degradation for the max-min selection method\nof \\cite{Bletsas06}, as compared to the FO protocol or even the GLS\nprotocol, is more than $12$dB at an outage probability of\n$7.0\\times10^{-2}$, when $R=6$bits\/s\/Hz for the four-node relay network,\nand an exactly similar situation can be observed for the five-node\nrelay network. Moreover, for the four-node relay network, the FO protocol is\nwithin $2.14$dB (when $R=1$bit\/s\/Hz) to within $7.05$dB (when\n$R=6$bits\/s\/Hz) of the lower bound on the outage probability when\nthe outage probability is $10^{-4}$. For the five-node relay network, the\ncorresponding differences are approximately $3$dB and $9.6$dB\nrespectively.\n\n\nThe performances of the different protocols for the four-node\nrelay network with non-uniform average power gains are presented in\nFigs.~\\ref{fig:case_1a_R1} and~\\ref{fig:case_6a_R6}, and\nFigs.~\\ref{fig:case_1b_R1} and~\\ref{fig:case_6b_R6} for cases A and\nB respectively, with the average power gains as stated in the\nfigures. In case A, the\nsource-relay links are, on average, better than the direct link, and\none relay node is, on average, a better candidate to forward the\ninformation to $\\mathcal{D}$. On the other hand, in case B,\nno one relay has very good source-relay and\nrelay-destination links, whereas the inter-relay link is, on\naverage, very good. This promotes increased inter-relay flows when\nusing the FO protocol, and thereby highlights the limitations of the\nGLS protocol. The differences between the outage performances of the\nFO and GLS protocols, at an outage probability of $10^{-4}$, are\n$1.2$dB or $1.0$dB, and $2.0$dB or $1.3$dB (when $R=1$bits\/s\/Hz or\n$R=6$bits\/s\/Hz), for cases A and B respectively. Thus, the gap between\nthe FO and GLS protocols decreases as the required\ndata rate increases. When the required rate is\nhigh, the coding gain offered by a protocol heavily relies on the\nefficient use of the direct link, and since the usage of the direct\nlink is similar for both the FO and the GLS protocols, the\nperformance gap narrows as the required data rate increases. On the\nother hand, at the same outage probability, the difference between\nthe outage performance of the FO protocol and the lower bound\nincreases from $1.5$dB to $7$dB, and from $1.9$dB to $6.0$dB as the\nrequired rate increases from $1$bit\/s\/Hz to $6$bits\/s\/Hz, for cases\nA and B respectively. Overall, these results demonstrate trends\nsimilar to the uniform average power gain case, and confirm the\ngenerality of the proposed protocols.\n\n\\section{Conclusions}\\label{sec:conclusions}\n\nWe proposed a cooperative transmission design for a general\nmulti-node half-duplex wireless relay network. \nIt is based on optimizing information flows, using the basic\ncomponents of BC and MA, to maximize the transmission rate from the\nsource to the destination, subject to maximum power constraints at\nindividual nodes. We also proposed the simpler GLS protocol,\nthat combines relay selection, and flow\noptimization for a three-node relay network.\nThese protocols were shown to achieve the optimal\nDMT for a general relay network. Simulation results for the four- and five-node\nrelay networks for uniform and non-uniform average power gains demonstrate\nthat the performance of the much simpler GLS protocol is close to\n that of the FO protocol.\nThis suggests that the GLS protocol can be used in systems with\nlow-complexity requirements. We also note that the proposed FO and\nGLS protocols can be used in wireless networks with topologies more\ncomplicated that the wireless relay network considered here. For example,\napplication of similar ideas to a parallel relay network in which\nthere is no direct connection between the source and the destination\nis considered in \\cite{Tam07}.\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\n\nTwo-dimensional topological insulators (2DTI) have attracted broad interest for novel applications due to their characteristic dissipationless transport properties \\cite{RMPSinova2015, JPCLKou2017}. While graphene was the first predicted 2DTI, its topological gap is too small to be observed experimentally \\cite{PRLKane2005,PRBYao2007}. Recently, a new class of large gap 2D topological materials, the metal-organic frameworks (MOFs), emerged as an ideal playground to achieve different properties. That is mainly due to a rich tunability of MOFs via different combinations of metals and organic ligands to form designed materials with selected properties \\cite{ChemCommunMiguel2016, NATCHEMColson2013, PCCPChen2015, LANGMaeda2016, CCRZhao2018,  PCCPGutzler2016}. For instance, it has been shown that Mn, Fe and Co-bis(dithiolene) are ferromagnetic half-metals, while n-doped Ru-bis(dithiolene) has a large gap Quantum Anomalous Hall phase \\cite{JPCCChakravarty2016,PCCPdeLima2018}. On the other hand, by keeping iron as the metal and changing the organic ligand to a coronene based molecule, a ferromagnetic semicondutor phase has been found \\cite{NatureDong2018}. Currently, a great number of MOFs are predicted to host topological phases \\cite{PRLLiu2013, PRB90Zhao, NanoscaleZhou2015, NanoLettersZhang2016, PRBYamada2016, PRBZhang2016, APLHsu2018, NanoscaleNi2018, JPCCSun2018}. Within these materials, nickel and palladium-bis(dithiolene) monolayers were experimentally synthesized \\cite{JACSSakamoto2013, CPCSakamoto2015}, and theoretically shown to host a 2DTI phase for n-doped systems \\cite{NANOFeng2013}. Although the monolayer system need to be n-doped to achieve the 2DTI phase, in a recent work we have shown that for bilayers of nickel and platinum-bis(dithiolene), the topological gap falls onto the Fermi level \\cite{PRBdeLima2017}.\n\t\nThe development of novel nano-devices based on 2DTIs is linked to the experimental control of its properties, and manipulation of its charge, spin and valley degrees of freedom \\cite{SCIENCEWolf2001, NATURESchaibley2016}. Recently, it has been proposed a bending strain perturbation approach to control spin currents in 2DTIs \\cite{NatCommHuang2017}. Meanwhile, ensuing research has shown layer localization control of topological states by an applied external electric field in MOF bilayer nanoribbons \\cite{PRBdeLima2017} and bilayer graphene grain boundaries \\cite{2DMJaskolski2018}. Furthermore, the exploration of van de Waals heterostructures allows for designed properties by combining the stacking of two-dimensional materials \\cite{NatureGeim2013,SCIENCENovoselov2016}. Within the heterostructures, the number of stacked layers and the stacking order has great impact in the electronic properties \\cite{PRBKim2011, SCIREPAraujo2018, 2DMSuchun2018}. Therefore, in order to control the electronic properties of MOFs multilayer, a clear understanding of its stacking is required.\n\nIn this paper, we study the stability and control of the electronic properties of the MOF {(NiC$_4$S$_4)_3$} stacking, from a monolayer to the bulk limit. Our {\\it first-principles} results show that the mirror symmetric AA stacking is the most stable configuration. For a finite AA stacking, each layer corresponds to a 2DTI system with helical edge states weakly coupled to its counterpart in adjacent layers. Here, the edge states are obtained from a tight-binding model of the stacking, fitted to match the {\\it first-principles} data. These features allow us to propose a novel electric field control of the layer localization of the edge states. For an infinite AA stacking, in the bulk limit, the edge states from each layer combine to form a topological V-gutter-like surface states dispersion along the stacking direction. These surface states are highly anisotropic and tunable, introducing a new feature to the {design of future multifunctional spintronic devices \\cite{ADVMATGuo, NATURESoumyanarayanan2016}.}\n\n\\section{Results and Discussions}\n\n\n\\subsection{Stacking geometry}\n  \nThe Nickel bis(dithiolene) monolayer presents a planar geometry, composed by Ni atoms connected by carbon-sulfur molecules (C$_6$S$_6$), which combine to form a kagome lattice [dashed lines in Fig.\\,\\ref{fig:str}(a)]. In order to analyze the energetic stability of multilayered {(NiC$_4$S$_4)_3$} systems, we have considered various stacking geometries. For instance, in {(NiC$_4$S$_4)_3$} bilayers (2ML), the stacking geometry is characterized by the alignment of sites X and Y of the different layers (2ML-XY),  with X and Y = A, B, G, and H, as indicated in Fig.\\,\\ref{fig:str}(a). Within this nomenclature, 2ML-AA [Fig.\\,\\ref{fig:str}(b)] represents the eclipsed structure, 2ML-AB [Fig.\\,\\ref{fig:str}(c)] dislocates one layer along the zigzag direction, and 2ML-GH [Fig.\\,\\ref{fig:str}(d)] mimics the Bernal stacking of graphene.\n\n\\begin{figure}\n\\includegraphics[width=\\columnwidth]{fig-1.png}\n\\caption{\\label{fig:str} MOF {(NiC$_4$S$_4)_3$} monolayer atomic structure (a) with C, S and Ni atoms in brown, orange and red, respectively. Staking geometry for the eclipsed AA (b) AB (c) and GH (d), with the top layer in black, and bottom layer in red.}\n\\end{figure}\n \nIn a previous study \\cite{PRBdeLima2017}, we have found the 2ML-AA configuration to be the most stable. There, the calculations were performed using the non-local  vdW-DF2 \\cite{PhysRevB.82.081101} correction to describe the vdW interactions. Indeed, the energetic preference of 2ML-AA was also verified in Refs.~\\cite{JPCCShojaei2014, JPCMChacham2017} using different approaches to describe the long range vdW interactions \\cite{grimmeJCompChem2006, PhysRevLett.92.246401, klimesJPhysC2010}. In contrast, while Ref.~\\cite{2DMSuchun2018} shows an energetic preference for the 2ML-GH configuration using the vdW-D3 \\cite{grimme2010consistent}, our results with this functional corroborates the previous works, showing a preference for the 2ML-AA case.\n\nNotice that, despite the weak chemical interactions (essentially vdW) between the stacked layers, the electronic band structure of multilayered {(NiC$_4$S$_4)_3$} systems strongly depends on the stacking geometry. Indeed, the 2ML-AA and -GH configurations present significantly different band structures near the Fermi level, see Appendix \\ref{app:band-stacking}. Moreover, here we are interested in the topological insulator properties of {(NiC$_4$S$_4)_3$} MOFs \\cite{NANOFeng2013}, and such a dependence  will reflect on the characteristics of the edge states in bilayer \\cite{PRBdeLima2017}, and multilayer {(NiC$_4$S$_4)_3$}.\n  \nA detailed analysis of the binding energy $E^b$ \\footnote{The binding energy ($E^b$) of {(NiC$_4$S$_4)_3$} bilayer systems is defined as, $ E^b = E^{\\rm ML} - \\frac{1}{2}E^{\\text{\\rm 2ML-XY}}, $ where $E^{\\text{\\rm 2ML-XY}}$ and $E^{\\rm ML}$  are the total energies of the {(NiC$_4$S$_4)_3$} 2ML-XY, and (free standing) monolayer. Similar definition was applied  for the 3ML-XYZ and bulk-XY systems.} and geometric properties of different stackings is shown in Table\\,\\ref{tab:par}. Here, the total energy calculations were performed considering full relaxation of the atomic positions and lattice vectors, until the force in each atom was less than 5 meV\/{\\AA}. Moreover, we compare results obtained using different functionals, {\\it viz.}: LDA \\cite{Pe}, GGA-PBE \\cite{PBE}, local vdW-D3 correction \\cite{grimme2010consistent}, and  the non-local vdW-DF2 correction \\cite{PhysRevB.82.081101}. The wave functions were expanded in a plane-wave basis with cutoff energy of 450 eV. For BZ integration a $4\\times4\\times6$ ($4\\times4\\times1$) k-mesh for bulk (finite stacking) systems was considered.\n{We have also considered the GGA+U method to correct the on-site repulsion of Ni d-orbitals. As shown in Appendix \\ref{app:DFT+U}, these corrections do not affect significantly our results.}\n\n\n\\begin{table}[h]\n\\caption{\\label{tab:par}Lattice parameter $a$ ({\\AA}), averaged inter-layer \ndistance $d$ ({\\AA}), corrugation ${\\delta z} = \\sqrt{\\avg{z^2} - \n\\avg{z}^2}$ ({\\AA}) and binding energy ($E^b$ in eV\/layer) of the bulk \nand layered systems.}\n\\begin{ruledtabular}\n\\begin{tabular}{c|cccc}\n\\multicolumn{5}{c}{PBE} \\\\\n\\hline\nSystem &$a$&$d$&$\\delta z$&$E^b$\\\\ \n\\hline\n1ML     & 14.62 &  -   & 0.00 & 0.000 \\\\\n2ML-GH  & 14.63 & 4.23 & 0.01 & 0.037 \\\\ \n2ML-AB  & 14.62 & 3.99 & 0.05 & 0.033 \\\\\n2ML-AA  & 14.62 & 4.69 & 0.01 & 0.022 \\\\ \n3ML-AAA & 14.64 & 3.95 & 0.00 & 0.004 \\\\ \nbulk-GH & 14.63 & 3.79 & 0.00 & 0.055 \\\\ \nbulk-AB & 14.63 & 3.73 & 0.00 & 0.002 \\\\ \nbulk-AA & 14.63 & 3.93 & 0.00 & 0.009 \\\\ \n\\hline\n\\multicolumn{5}{c}{LDA} \\\\\n\\hline\nSystem &$a$&$d$&$\\delta z$&$E^b$\\\\ \n\\hline\n1ML     & 14.37 &  -   & 0.00 & 0.000 \\\\ \n2ML-GH  & 14.42 & 2.96 & 0.11 & 0.468 \\\\ \n2ML-AB  & 14.36 & 3.16 & 0.06 & 0.429 \\\\ \n2ML-AA  & 14.44 & 2.96 & 0.08 & 1.051 \\\\ \n3ML-AAA & 14.43 & 3.11 & 0.04 & 1.138 \\\\ \nbulk-GH & 14.39 & 3.13 & 0.00 & 0.985 \\\\ \nbulk-AB & 14.40 & 3.29 & 0.00 & 0.888 \\\\ \nbulk-AA & 14.45 & 3.12 & 0.00 & 1.800 \\\\ \n\\hline\n\\multicolumn{5}{c}{vdW-D3} \\\\\n\\hline\nSystem &$a$&$d$&$\\delta z$&$E^b$\\\\ \n\\hline\n1ML     & 14.62 &   -  & 0.00 & 0.000 \\\\ \n2ML-GH  & 14.61 & 3.26 & 0.11 & 0.679 \\\\ \n2ML-AB  & 14.60 & 3.34 & 0.15 & 0.661 \\\\ \n2ML-AA  & 14.64 & 3.54 & 0.01 & 0.860 \\\\ \n3ML-AAA & 14.64 & 3.57 & 0.00 & 1.206 \\\\ \nbulk-GH & 14.62 & 3.31 & 0.00 & 1.501 \\\\ \nbulk-AB & 14.60 & 3.46 & 0.00 & 1.404 \\\\ \nbulk-AA & 14.65 & 3.35 & 0.00 & 1.968 \\\\ \n\\hline\n\\multicolumn{5}{c}{vdW-DF2} \\\\\n\\hline\nSystem &$a$&$d$&$\\delta z$&$E^b$\\\\\n\\hline\n1ML     & 14.83 &   -  & 0.00 & 0.000 \\\\ \n2ML-GH  & 14.83 & 3.33 & 0.11 & 0.645 \\\\\n2ML-AB  & 14.82 & 3.34 & 0.21 & 0.602 \\\\\n2ML-AA  & 14.86 & 3.64 & 0.01 & 0.803 \\\\ \n3ML-AAA & 14.85 & 3.65 & 0.00 & 1.107 \\\\ \nbulk-GH & 14.84 & 3.46 & 0.00 & 1.360 \\\\ \nbulk-AB & 14.82 & 3.56 & 0.00 & 1.267 \\\\\nbulk-AA & 14.87 & 3.56 & 0.00 & 1.744 \\\\\n\\end{tabular}\n\\end{ruledtabular}\n\\end{table}\n\n\nStarting with the 2ML case, the PBE results show larger interlayer distances $d$, and lower values of binding energies when compared with the other functionals, thus indicating that the vdW interactions play an important role on the energetic stability and the equilibrium geometry of the multilayered {(NiC$_4$S$_4)_3$} systems. When vdW corrections are included, we find that 2ML-AA stackings are more stable than 2ML-GH \\footnote{Although our calculations have {more base functions and denser k-grid} than that of Ref.~\\cite{2DMSuchun2018}, we have also run calculations with exact the same parameters listed in this reference, but even in this case we find that the 2ML-AA stacking is more stable.}, with a $E^b$ difference of $\\sim$0.15\\,eV\/layer. The 2ML-AA stacking shows $E^b =$ 0.80 (0.86)\\,eV\/layer, and interlayer distance $d=$ 3.64 (3.54)\\,{\\AA}, within the vdW-DF2 (vdW-D3) functional. In general, the bond distances obtained from the vdW-DF2 approach are larger than those obtained by the vdW-D3 approximation. Nonetheless, both vdW corrections capture the enhancement of the vertical corrugation ($\\delta z$) due to the reduction of the interlayer distance in 2ML-AB, $\\delta z$=0.1$\\rightarrow$0.21  and 0.1$\\rightarrow$0.15\\,{\\AA}, for vdW-DF2 and -D3, respectively. The energetic preference for the 2ML-AA configurations has been also verified within the LDA approximation. However, a comparison between the LDA and vdW results show that, within the LDA approach, the $E^b$ difference between 2ML-AA and -AB stackings is larger (0.62\\,eV\/layer), and the 2ML-AA configuration shows a smaller interlayer distance (2.96\\,{\\AA}), which leads to a larger corrugation $\\delta z$=0.08\\,{\\AA}.\n\n\nFor trilayers and the bulk systems, further binding energy results, shown in Table \\ref{tab:par}, also indicate the energetic preference for the AA stacking (3ML-AAA and bulk-AA). In comparison with the 2ML-AA case, the binding energy of the bulk-AA system increases by $\\sim$1\\,eV\/layer, followed by a reduction of the interlayer distance by  0.08\\,{\\AA}, $d$=3.64$\\rightarrow$3.56\\,{\\AA},  (0.19\\,{\\AA}, 3.54$\\rightarrow$3.35\\,{\\AA}) using the vdW-DF2 (vdW-D3) approximation. Within the LDA approach, we have also found the bulk-AA stacking to be the most stable. However, in contrast with the (vdW) results above, at the equilibrium geometry the interlayer distance increases by 0.16\\,{\\AA} when compared with the bilayer 2ML-AA, 2.96$\\rightarrow$3.12\\,{\\AA}. These findings suggest that the vdW and LDA approximations present the same qualitative picture regarding the energetic stability of multilayered {(NiC$_4$S$_4)_3$} systems. However, the LDA approach fails to describe their equilibrium geometries. On the other hand, the PBE functional fail to describe the interlayer interaction, with the interlayer distance overestimated and smaller binding energies. Hereafter we will consider the most stable eclipsed (AA) geometry, obtained using the vdW-DF2 approach.\n\n\n\n\\subsection{Electronic properties}\n\nTo analyze the electronic structure of the AA stacking, we first consider the DFT results without SOC, as shown in Fig.~\\ref{fig:band} for 1ML, 2ML, 3ML, and bulk {(NiC$_4$S$_4)_3$} systems. Later on, in Section \\ref{sec:edge}, we will introduce the SOC to discuss the topological edge states. \n\nFor the single layer {(NiC$_4$S$_4)_3$}, the typical kagome bands (KBs), indicated by red lines in Fig.\\,\\ref{fig:band}(a1), are characterized by two energy bands [$E^1(\\bm k)$ and $E^2(\\bm k)$] with linear dispersion at the K-point, yielding a Dirac cone, connected to a nearly flat band [$E^3(\\bm k)$] at the $\\Gamma$-point. The orbital projection, Fig.\\,\\ref{fig:band}(b1), shows that the KBs are {mostly composed by C-p$_z$ and S-p$_z$ orbitals, while near the center of the Brilloin zone, and along the M-K direction, the Ni-d$_{xz}$ and -d$_{yz}$ orbitals also contribute to the formation of those energy bands.} For the 2ML-AA case, one finds two sets of KBs, Fig.\\,\\ref{fig:band}(a2). There, one set of KBs is fully unoccupied, while the other is partially occupied, with the Dirac point lying slightly below the Fermi level. Such a metallic character is also seen in 3ML, Fig.\\,\\ref{fig:band}(a3), where three sets of KBs are seen. The orbital projection on these energy bands, Figs.\\,\\ref{fig:band}(b2) and \\ref{fig:band}(b3), reveal that, despite showing similar energy dispersion, the KBs present distinct orbital distribution. Namely, the contribution of the Ni-d$_{xz}$\/d$_{yz}$ [blue circles in Fig.\\,\\ref{fig:band}(b1)-(b3)] orbitals are larger on high energy KBs, whereas the C-p$_z$ and S-p$_z$ orbitals [green circles in Fig.\\,\\ref{fig:band}(b1)-(b3)] are dominant on the low energy KBs. Such an orbital distribution has been strengthened in the {(NiC$_4$S$_4)_3$} bulk system. As shown in Figs.\\,\\ref{fig:band}(a4) and \\ref{fig:band}(b4), the contribution of the Ni-d (C-p$_z$ and S-p$_z$) orbitals becomes dominant on the fully occupied (empty) KBs at high  (low) energy, which are localized along the A-L-H-A ($\\Gamma$-M-K-$\\Gamma$) directions, see inset of Fig.\\,\\ref{fig:band}(a4).\n\n\\begin{figure*}\n\\includegraphics[width=\\textwidth]{fig-2.png}\n\\caption{\\label{fig:band} DFT band structure without SOC for 1ML (a1) 2ML (a2) 3ML (a3) and bulk (a4) {(NiC$_4$S$_4)_3$} stacking. The red lines indicate the kagome bands. In (b1)-(b4) the projected orbital contributions are show on top of the band structures from the top panels.}\n\\end{figure*}\n\n\\subsubsection{Effective tight-binding model}\n{Within the eclipsed AA stacking geometry, the interlayer coupling is mostly given by weak vdW interactions between kagome sites aligned along the stacking direction $z$. Given this scenario, we propose a tight-binding (TB) model, for the stacking of $N$ layers, considering first neighbors coupling between monolayers. The resulting Hamiltonian reads}\n\\begin{equation}\nH_N = h_{3}({\\bm k}) \\otimes \\mathbb{I}_{N} + \\Delta \\; \\mathbb{I}_{3} \\otimes \\sum_{j=1}^{N-1} \\ket{j+1} \\bra{j} + c.c.,\n\\end{equation}\nwhere $\\ket{j}$ labels the subspace of each layer $j$, $\\mathbb{I}_{N}$ is a $N\\times N$ identity matrix within this subspace{, and ${\\bm k} = (k_x,\\,k_y)$ is the in-plane momentum. The $N$-layer Hamiltonian $H_N$ above is written in terms of the monolayer's $3\\times3$ Hamiltonian $h_{3}({\\bm k})$, which gives the characteristic KBs dispersion. The vdW interlayer coupling is set by $\\Delta$, for which the diagonal form ($\\mathbb{I}_{3}$) labels the direct coupling between equivalent atoms in the AA stacking geometry.} The second term of $H_N$ is a tridiagonal Toeplitz matrix \\cite{trimatrix}, yielding a closed form for the eigenvalues as\n\\begin{equation}\n\\lambda^{n}_{\\nu}({\\bm k}) = E^n({\\bm k}) - 2\\,|\\Delta|\\, \\cos \\left( \n\\frac{\\nu \\pi}{N+1} \\right), \\label{eigenval}\n\\end{equation}\nwith $E^n(\\bm k)$ being the kagome monolayer dispersion [Fig.\\ref{fig:band}(a1)], where $n$ labels each of the three KBs for each set $\\nu = 1, \\dots, N$. The eigenvectors are $\\ket{n,\\,{\\bm k}} \\otimes \\ket{\\nu}$, where the first term is an eigenstate of $h_3(\\bm{k})$. To understand the second quantum number $\\nu$ and the eigenstate component $\\ket{\\nu}$, it is interesting to notice that the tridiagonal Toeplitz form of the $\\Delta$ coupling is equivalent to the Hamiltonian of a scalar $N$-sites chain, \\textit{i.e.,} a quantum well along the stacking direction. Namely, $\\ket{\\nu} = \\Omega\\, [\\phi_{1,\\nu}, \\cdots, \\phi_{j,\\nu}, \\cdots, \\phi_{N,\\nu}]$, where $\\Omega = \\sqrt{2\/(N+1)}$ is a normalization factor, and\n\\begin{equation}\n\\phi_{j,\\nu} = \\sin \\left( \\frac{j \\nu \\pi}{N+1} \\right) \\equiv \\sin\\left(k_\\nu z_j\\right)\n\\end{equation}\nis the contribution of the state $\\nu$ to the $j$-th layer. In the second form of $\\phi_{j,\\nu}$ above, we use the quantum well analogy to introduce $k_\\nu = \\nu \\pi\/L_N$ as the quantized $k_z$ momentum along the stacking direction $z_j = j d$, where $d$ is the interlayer distance, and $L_N = (N+1)d$ is the effective stacking length \\footnote{In this analogy, the quantum well chain has $N+2$ sites distant by $d$, yielding a length $L_N = (N+1)d$. A hard-wall boundary condition makes $\\phi_{0,\\nu} = \\phi_{N+1,\\nu} = 0$, such that the real layers lie within $1 \\leq j \\leq N$.}. This analogy is useful to interpret the layer localization and the electric field effects below.\n\nTo obtain an estimate for $\\Delta$, we compare the {\\it first-principles} band structures with the eigenvalues $\\lambda^{n}_{\\nu}({\\bm k})$ of our TB model. The {\\it first-principles} energy dispersion for 2ML and 3ML, Fig.~\\ref{fig:band}, show that each set of KBs are nearly rigidly shifted from one another. This matches the results of the TB model, where $\\lambda^n_{\\nu+1}({\\bm k}) - \\lambda^n_\\nu({\\bm k})$ is independent of $\\bm{k}$. For 2ML [Fig.\\,\\ref{fig:band}(a2)], the two sets ($\\nu = 1$ and 2) of KBs are distant by $\\lambda^n_2({\\bm k}) - \\lambda^n_1({\\bm k}) = 2|\\Delta| \\approx 0.56$~eV, thus $|\\Delta| \\approx 0.28$~eV. For the 3ML case we also find $|\\Delta|\\approx 0.28$\\,eV.\n\nTaking the bulk limit of $N\\rightarrow\\infty$ layers, we can explore the quantum well analogy to express $\\lambda^{n}_{\\nu}({\\bm k}) \\rightarrow \\lambda^{n}_{\\nu}({\\bm k}, k_z) = E^n({\\bm k}) - 2\\,|\\Delta|\\, \\cos(k_z d)$, assuming that now $k_z$ is a continuous variable, since $L_N \\rightarrow \\infty$. The $\\cos(k_z d)$ dispersion is clearly seen in Fig.~\\ref{fig:band}(a4) connecting the topmost and bottommost KBs along the $\\Gamma$-A direction ($\\parallel k_z$). From this dispersion, we obtain $|\\Delta|=0.31$\\,eV. Such an increase of $\\Delta$ (0.28\\,$\\rightarrow$\\,0.31\\,eV), although small, is a result of the reduction of the interlayer distance in the bulk system with respect to the few layer systems, 2ML or 3ML in Table\\,\\ref{tab:par}.\n\n\\begin{figure}\n\\includegraphics[width=\\columnwidth]{fig-3.png}\n\\caption{\\label{fig:bulk-strain} (a) Variation of coupling $|\\Delta|$ with the  interlayer distance $d$. Illustrative band structure for the bulk system with equilibrium $d$ (b), with $d$ increased by $2\\%$ (c), and decreased by $2\\%$ (d).}\n\\end{figure}\n \nTo further explore the band width changes, and support our TB model, in Fig.\\,\\ref{fig:bulk-strain}(a) we show $|\\Delta|$ as a function of the interlayer spacing $d$ for the {(NiC$_4$S$_4)_3$} bulk system. This distance can be controlled by an external pressure (uniaxial strain), thus allowing us to tune $\\Delta$. Since the $\\Delta$ coupling reflects the hybridization of neighboring layers, it decreases as $d$ increases. As discussed above, $\\Delta$ is extracted from the bands connecting KBs along the $\\Gamma$-A direction. At the equilibrium interlayer distance, $d$=3.56\\,{\\AA}, the energy difference between the upper- and bottommost KBs is about 1.24\\,eV, Fig.\\,\\ref{fig:bulk-strain}(b). Meanwhile, for $d$=3.63 (3.49)\\,{\\AA}, this energy split reduces (increases) to 1.12 (1.44)\\,eV [Figs.\\,\\ref{fig:bulk-strain}(c) and \\ref{fig:bulk-strain}(d)]. That is, a $\\sim 2$\\% uniaxial strain induces a $\\sim 10$\\% change in $\\Delta$.\n\n\\subsubsection{Layer localization}\n\n\\begin{figure}\n\\includegraphics[width=\\columnwidth]{fig-4.png}\n\\caption{\\label{fig:localiz} (a) DFT band structure projected into each layer of a 5ML {(NiC$_4$S$_4)_3$} stacking. (b) Layer localization contributions $|\\phi_{j,\\nu}|^2$ extracted from the DFT and TB model without (with) external electric field in the left (right) panel. (c) Layer localization $|\\phi_{j,\\nu}|^2$ for a 10L slab from the TB model without (with) electric field in the left (right) panel.}\n\\end{figure}\n\nThe quantum well analogy, introduced above, enricher our discussion in the layer localization of the kagome states for finite AA stackings. First, in order to  verify the validity of this model, we performed {\\it first-principles} DFT calculations of the KBs for a 5ML slab. Panels L1 to L5 in Fig.\\,\\ref{fig:localiz}(a) show the band structures projected on each layer $j$, and indicates the sets $\\nu$ of KBs for clarity. Even though the band structure is dense at low energies, it is possible to see that: (i) the highest and the lowest KBs (${\\nu\\text{=1, and 5}}$) are mostly localized in the inner layers (L2, L3, L4); (ii) L3 does not contribute to ${\\nu\\text{=2, and 4}}$, which localizes mostly on the outer layers; (iii) the $\\nu=3$ KBs contributes mostly to L1, L3 and L5. These features are more clearly seen in the circles of the left panel in Fig.\\,\\ref{fig:localiz}(b), which compares the layer projections $|\\phi_{j,\\nu}|^2$ extracted from the DFT data [Fig.\\,\\ref{fig:localiz}(a)], with the TB solutions. Indeed, these projections match remarkably well, thus validating the TB model.\n\nIn the presence of an external perpendicular electric field $\\bm{E}_{ext} = E_{ext}\\hat{z}$, the projections $|\\phi_{j,\\nu}|^2$ tend to shift to the outer layers. This is shown in Fig.\\,\\ref{fig:localiz}(b) and (c), for a 5ML and 10ML slab respectively. There, we compare the eigenstate $\\phi_{j,\\nu}$ distributions among the layers in the absence (left panels) and presence (right panels) of $E_{ext}$. For a 5ML slab, the DFT data on the right panel of Fig.~\\ref{fig:localiz}(b) is obtained for a nominal $E_{ext} = 0.5$~eV\/\\AA. However, due to charge rearrangement between the layers, the actual electric field is inhomogeneous across the layers. Nonetheless, the DFT data matches qualitatively well with the TB model for an effective uniform field $E_{ext} = 0.2 \\Delta\/d \\approx 0.02$~eV\/\\AA. Thus, the localization of each state follows the on-site energy shift due to $E_{ext}$, \\textit{e.g.}, the lower energy KBs shifts towards L1. \n\nThe same behavior is extrapolated to thicker slabs, for instance a 10ML slab as shown in Fig.\\,\\ref{fig:localiz}(c). Therefore $E_{ext}$ allows one to control the layer-localization of the KBs projection. In turn, combining this feature with the tunability of the Fermi energy (\\textit{e.g.}, doping), one obtains an interesting engineering of the layer localization of topological edge states, which we discuss in the next section.\n\n\\subsection{Edge states}\n\\label{sec:edge}\n\nThe SOC opens an energy gap at the degenerated points ($\\Gamma$ and K) of each set of KBs, yielding a QSH phase \\cite{NANOFeng2013, PRBdeLima2017}. Interestingly, the weak vdW coupling between the layers of our {(NiC$_4$S$_4)_3$} AA stacking leads to a series of nearly independent edge states in each layer. To model this layered QSH phase, we add the SOC to $h_3(\\bm{k})$ of our TB model, since the large number of atoms involved in ribbon geometries makes {\\it first-principles} calculations unattainable. The on-site, hopping, and SOC parameters within the single layer, and the $\\Delta$ coupling between layers were obtained by fitting the {\\it first-principles} DFT band structure of the bulk system. Further details can be found in the Appendix \\ref{app:TB-par} and in Refs.\\, \\cite{PRBdeLima2017, PCCPdeLima2018}.\n\n\\subsubsection{Edge states on a finite stacking}\n\nIn this section we consider a 10ML AA stacking in a ribbon geometry. Nevertheless, the following discussion applies to any finite stacking. First, in the absence of electric fields, we calculate the spin-polarized edge states and present its projections into the topmost (L10), middle (L5), and bottommost (L1) layers in Figs.~\\ref{fig:tb-ribbon-ext}(b1)--\\ref{fig:tb-ribbon-ext}(b3). These projections agree with the layer resolved distribution of the KBs discussed previously. For instance, the $\\nu=1$ state (low energy) is mostly located in the middle layer. The same is true for the $\\nu=10$ (high energy) state, since their KBs projections $\\phi_{j, \\nu}$ are similar, see left panel of Fig.\\,\\ref{fig:localiz}(c). On the other hand, states within $2 \\leq \\nu \\leq 9$ are spread along many layers, as illustrated in Fig.~\\ref{fig:tb-ribbon-ext}(a). These results tell us that the vdW coupling between layers reflects into weakly coupled QSH edge states. Such edge states distribution allows for an unexploited control of the edge currents. For instance, tuning the Fermi level to the $\\nu=1$ state, the helical currents will flow mostly through middle of the slab, while for $\\nu=2$ state is distributed along its extremities, see top panel of Fig.\\,\\ref{fig:tb-ribbon-ext}(a).\n\n\n\\begin{figure}[h!]\n\\includegraphics[width=0.95\\columnwidth]{fig-5.png}\n\\caption{\\label{fig:tb-ribbon-ext} TB nanoribbon of multilayer {(NiC$_4$S$_4)_3$} with external electric field ($E_{ext}$). (a) visualization of edge states distribution for some  eigenvalues ($\\nu$) without (with) $E_{ext}$ in the top (bottom) panels. Layered projected side surface states for $E_{ext} = 0.0$ (b), $=0.5$ (c) and $=1.0\\,\\Delta\/d_z$ (d). Here, (b1)-(d1) is the projected band structure in the bottom most layer 1L, (b2)-(d2) in the middle 5L layer and (b3)-(d3) the top most layer 10L. The color bar indicate the $\\langle S_z \\rangle$ value and the size of the line is proportional to the layer contribution.}\n\\end{figure}\n\nOn the other hand, in the presence of $E_{ext}$, the layer localization of the edge states follows that of the KBs, see right panel of Fig.\\,\\ref{fig:localiz}(c). This allows for a fine tuning of the edge states contributions near the Fermi level. To illustrate, Figs.~\\ref{fig:tb-ribbon-ext}(c1)-(c3) show the edge state dispersions for an external field $E_{ext} = 0.5 \\Delta\/d_z \\approx 0.05$\\,eV\/{\\AA}. Here, we can see the lower (higher) energy edge states beginning to localize in the bottom (top) layer. Increasing the external field, $E_{ext} = 0.5 \\rightarrow 1.0 \\Delta\/d_z$, the localization effect is enhanced, compare Figs,\\,\\ref{fig:tb-ribbon-ext}(c3) and (d3). That is, the field changes the on-site energies of each layer, pushing the $\\nu=1$ state towards the bottom layer, as depicted in Fig.~\\ref{fig:tb-ribbon-ext}(a). Consequently, the states near the Fermi level [shaded area in Figs.~\\ref{fig:tb-ribbon-ext}(b)-(d).] are mostly located in the bottom layer. If the field is reversed, $E_{ext} \\rightarrow -E_{ext}$, the states near the Fermi level will switch the localization to the top layer. This tuning of the edge states, by $E_{ext}$ and\/or changing the Fermi level, allow for an additional degree of freedom that can be useful to design electronic (\\textit{layertronic}) devices based on topologically protected edge currents in metal organic frameworks. \n\n\\subsubsection{Edge states on the bulk stacking}\n\nThe electronic confinement effects, present in the few layers systems, no longer takes place in the bulk phase. Here, the weakly coupled edge states from each layer are evenly spread along the side surfaces of the bulk staking, as shown schematically  in Fig.\\,\\ref{fig:tb-surface}(a1). Nonetheless, these side surface states carry the characteristics of the individual QSH edge states of the monolayers.\n\nAs discussed previously, the infinite superposition of the KBs gives rise to an continuum energy dispersion $\\propto -2|\\Delta|\\cos(k_z d)$, where $k_z$ is the wave vector parallel to the stacking direction $\\Gamma$-A. This bulk dispersion is projected in the 2D BZ of the side surface [Fig.\\,\\ref{fig:tb-surface}(a2) and \\ref{fig:tb-surface}(b)], yielding a continuum of linear crossings at the edge of the Brillouin zone (X--M $\\parallel k_z$), as shown in Fig.\\,\\ref{fig:tb-surface}(c). \n\n\n\\begin{figure}\n\\includegraphics[width=\\columnwidth]{fig-6.png}\n\\caption{\\label{fig:tb-surface} Distribution of the side surface (SS) states (a1) where the dashed arrow indicate the periodic directions; bulk and SS Brillouin zone (a2); spin projected side surface states (b); zoom showing the band dispersion as a function of $(k_x,\\,k_z)$ for the topological states close to the XM line (c). From (d) to (f) the spin projected side surface band structure for $k_z$ fixed varying $k_x$ as highlighted in blue in (a2). For the band plots the size of the lines are proportional to the sides surface sites contribution to the state $|\\braket{{\\rm SS}|n,\\,{\\bm k}}|^2$ while the color indicate the $S_z$ mean value.}\n\\end{figure}\n\nInterestingly, this peculiar V-gutter-like edge state dispersion [Fig.\\,\\ref{fig:tb-surface}(c)] is highly anisotropic. An effective model for these states can be cast as a individual 2DTI edge states coupled along the stacking direction, wich reads\n\\begin{equation}\n\tE_{XM}(k_x, k_z) = \\hbar v_F \\sigma_z k_x - 2|\\Delta| \\sigma_0 \\cos (k_z d),\n\\end{equation}\nwhere $\\sigma_0$ is the $2\\times2$ identity matrix, $\\sigma_z$ is third Pauli matrix in spin space, $v_F$ is the velocity of the edge states along $k_x$, and $\\Delta$ is the same interlayer coupling from the bulk model. Along $k_x$, the states are helical with constant velocities $v_x = \\pm v_F$ for spin up and down, while along $k_z$ the velocity is spin independent, but changes with $k_z$ or Fermi energy, $v_z = (|\\Delta|d\/\\hbar)\\sin(k_z d)$. This anisotropy is a consequence of the weak vdW coupling between the stacked layers. Thus, as shown in Fig.\\,\\ref{fig:bulk-strain}, the dynamics along $z$ can be further controlled by changing $\\Delta$ with the uniaxial strain.\n\n\n\\section{Conclusion}\n\nIn this paper, we show that electronic layer localization can be engineered in multilayer MOFs by controlling the Fermi level (\\textit{e.g.,} doping) and an external electric field $E_{ext}$. This is possible due to the van der Waals (vdW) coupling between adjacent layers on eclipsed AA stackings. Here, we consider the {(NiC$_4$S$_4)_3$} MOFs, showing that AA stacking is the most stable for this material. Nonetheless, our results remain valid for other MOFs that also allow for eclipsed stackings. The interlayer vdW coupling reflects into weakly coupled topological edge states in each monolayer of the stackings. Consequently, we have shown that the layer localization of edge states can be controlled by $E_{ext}$, applied along the stacking direction, thus providing a novel degree of freedom to the design of TI based electronics.\n\nFurthermore, extrapolating the system to an infinite stacking of nanoribbons, the weak vdW coupling between layers leads to a new V-gutter-like Dirac states in the side surfaces. This new surface states are highly anisotropic, showing a trivial velocity along the stacking direction, while along the monolayer edge a helical topological current is always present for a wide range of energy.\n\n \n\\section{ACKNOWLEDGMENTS}\n\nThe authors acknowledge   financial   support   from   the Brazilian  agencies  \nCNPq, and FAPEMIG, and the CENAPAD-SP and Laborat\u00f3rio Nacional de Computa\u00e7\u00e3o Cient\u00edfica (LNCC-SCAFMat) for computer time.\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}