\documentclass[reqno]{amsart} \usepackage{hyperref} \AtBeginDocument{{\noindent\small \emph{Electronic Journal of Differential Equations}, Vol. 2017 (2017), No. 231, pp. 1--13.\newline ISSN: 1072-6691. URL: http://ejde.math.txstate.edu or http://ejde.math.unt.edu} \thanks{\copyright 2017 Texas State University.} \vspace{8mm}} \begin{document} \title[\hfilneg EJDE-2017/231\hfil Existence of positive ground state solutions ] {Existence of positive ground state solutions for a class of asymptotically periodic Schr\"odinger-Poisson systems} \author[D.-B. Wang, H.-F. Xie, W. Guan \hfil EJDE-2017/??\hfilneg] {Da-Bin Wang, Hua-Fei Xie, Wen Guan} \address{Da-Bin Wang (corresponding author)\newline Department of Applied Mathematics, Lanzhou University of Technology, Lanzhou, Gansu 730050, China} \email{wangdb96@163.com} \address{Hua-Fei Xie \newline Department of Applied Mathematics, Lanzhou University of Technology, Lanzhou, Gansu 730050, China} \email{xiehuafeilz@163.com} \address{Wen Guan \newline Department of Applied Mathematics, Lanzhou University of Technology, Lanzhou, Gansu 730050, China} \email{mathguanw@163.com} \dedicatory{Communicated by Claudianor O. Alves} \thanks{Submitted March 22, 2017. Published September 22, 2017.} \subjclass[2010]{35J20, 35J60, 35J65} \keywords{Schr\"odinger-Poisson systems; ground state solution; \hfill\break\indent variational methods} \begin{abstract} In this article, by using variational method, we study the existence of a positive ground state solution for the Schr\"odinger-Poisson system \begin{gather*} -\Delta u+V(x)u+K(x)\phi u=f(x,u),\quad x\in\mathbb{R}^3,\\ -\Delta\phi=K(x)u^2,\quad x\in\mathbb{R}^3, \end{gather*} where $V(x),K(x)$ and $f(x,u)$ are asymptotically periodic functions in $x$ at infinity. \end{abstract} \maketitle \numberwithin{equation}{section} \newtheorem{theorem}{Theorem}[section] \newtheorem{lemma}[theorem]{Lemma} \newtheorem{remark}[theorem]{Remark} \allowdisplaybreaks \section{Introduction and statement of results} For past decades, much attention has been paid to the nonlinear Schr\"odinger-Poisson system \begin{equation}\label{e1.1} \begin{gathered} i\hbar\frac{\partial \Psi}{\partial t}=-\frac{\hbar^2}{2m}\Delta \Psi + U(x)\Psi+\phi(x)\Psi- |\Psi|^{q-1}\Psi,\quad x\in \mathbb{R}^3,\; t\in \mathbb{R}\\ -\Delta \phi=|\Psi|^2,\quad x\in \mathbb{R}^3, \end{gathered} \end{equation} where $\hbar$ is the Planck constant. Equation \eqref{e1.1} derived from quantum mechanics. For this equation, the existence of stationary wave solutions is often sought, that is, the following form of solution $$ \Psi(x,t)=e^{it}u(x),x\in \mathbb{R}^3,\quad t\in\mathbb{R}. $$ Therefore, the existence of the standing wave solution of the equation \eqref{e1.1} is equivalent to finding the solution of the following system ($m=\frac{1}{2}$, $\hbar=1$, $V(x)=U(x)+1$) \begin{equation}\label{e1.2} \begin{gathered} -\Delta u + V(x)u+\phi u= |u|^{q-1}u,\quad x\in \mathbb{R}^3,\\ -\Delta \phi=u^2,\quad x\in \mathbb{R}^3. \end{gathered} \end{equation} As far as we know, the first result on Schr\"odinger-Poisson system was obtained in \cite{BF}. Thereafter, using the variational method, there is a series of work to discuss the existence, non existence, radially symmetric solutions, non-radially symmetric solutions and ground state to Schr\"odinger-Poisson system \eqref{e1.2} \cite{ASS,AR1,A,AP,BF,C,CV,DM,DM1,DW,HRC,HZ,HT,JZ,LPW,LPY,LWZ,LGF,R2,SM,SWF,V, WXZC,YZD,ZXZ1,ZZ}. To the best of our knowledge, Azzollini and Pomponio \cite{AP} firstly obtained the ground state solution to the Schr\"odinger-Poisson system \eqref{e1.2}. The conclusion they got was that if $V$ is a positive constant and $20$, $x\in \mathbb{R}^3$; \item[(2)] $\lim_{|x|\to \infty}|V(x)-V_{0}(x)|=0$, where $V_{0}$ satisfy $ V_{0}(x)=V_{0}(x+y)$ for all $x\in \mathbb{R}^3$ and all $y\in \mathbb{Z}^3$; \item[(3)] $V(x)\leq V_{0}(x)$ for all $x\in \mathbb{R}^3$, and there exists an open set $\Omega\subset \mathbb{R}^3$ with $m(\Omega)>0$ such that $V(x)< V_{0}(x)$ for all $x\in \Omega$. \end{itemize} Alves et al.\ studied the ground state solutions to system \eqref{e1.3} in case the asymptotically periodic condition under conditions (1)--(3). In case $p\in(3,5)$, Cerami and Vaira \cite{CV} studied the existence of positive solutions for the following non-autonomous Schr\"odinger-Poisson system \begin{equation}\label{e1.4} \begin{gathered} -\Delta u + u+K(x)\phi(x) u= a(x)|u|^{p-1}u,\quad x\in \mathbb{R}^3,\\ -\Delta \phi=K(x)u^2,\quad x\in \mathbb{R}^3, \end{gathered} \end{equation} where $a,K$ are nonnegative functions such that $\lim_{|x|\to \infty}a(x)=a_{\infty}>0$, and $\lim_{|x|\to \infty}K(x)=0$. Zhang, Xu and Zhang \cite{ZXZ1} considered existence of positive ground state solution for the Schr\"odinger-Poisson system \begin{equation}\label{e1.5} \begin{gathered} -\Delta u + V(x)u+K(x)\phi u= f(x,u),\quad x\in \mathbb{R}^3,\\ -\Delta \phi=K(x)u^2,\quad x\in \mathbb{R}^3. \end{gathered} \end{equation} In their paper, $V$ and $K$ satisfy: \begin{itemize} \item $V,K\in L^{\infty}(\mathbb{R}^3)$, $\inf_{\mathbb{R}^3}V>0$, $\inf_{\mathbb{R}^3}K>0$, and $V-V_{p}, K-K_{p}\in\mathcal{F}$, where $V_{p}$ and $K_{p}$ satisfy $V_{p}(x+z)=V_{p}(x)$, $K_{p}(x+z)=K_{p}(x)$ for all $x\in\mathbb{R}^3$ and $z\in\mathbb{Z}^3$, here $\mathcal{F}=\{g\in L^{\infty}(\mathbb{R}^3): \forall \varepsilon>0\}$, the set $\{x\in \mathbb{R}^3:|g(x)| \geq \varepsilon\} \text{ has finite Lebesgue measure}\}$. \end{itemize} On the other hand, when $K=0$ the Schr\"odinger-Poisson system \eqref{e1.5} becomes the standard Schr\"odinger equation (replace $\mathbb{R}^3$ with $\mathbb{R}^N$) \begin{equation}\label{e1.6} -\Delta u + V(x)u= f(x,u), \quad x\in\mathbb{R}^N . \end{equation} The Schr\"odinger equation \eqref{e1.6} has been widely investigated by many authors, see \cite{ABC,BP,DF,DL,JT,LT,LLT,LWZ1,M,R,R1,T,T1,W1} and reference their. Especially, in~\cite{LT,LLT,M,T,T1}, they studied the nontrivial solution and ground state solution for problem \eqref{e1.6} in which $V$ or $f$ satisfy the asymptotically periodic condition. In the other context about asymptotically periodic condition, we refer the reader to \cite{LS,LS1,LS2,SV}~and reference their. Motivated by above results, in this paper we study positive ground state solutions to system \eqref{e1.5} under reformative condition about asymptotically periodic case of $V, K$ and $f$ at infinity. To state our main results, we assume that: \begin{itemize} \item[(A1)] $V,~V_{p}\in L^{\infty}(\mathbb{R}^3)$, $0\leq V(x)\leq V_{p}(x)$ and $V(x)-V_{p}(x)\in A_{0}$, where $A_{0}:=\{k(x): \text{ for any } \varepsilon >0, ~m\{x\in B_{1}(y): |k(x)|\geq \varepsilon\}\to 0 \text{ as } |y|\to\infty\}$ and $V_{p}$ satisfies $V_{0}:=\inf_{x\in \mathbb{R}^3}V_{p}>0$ and $V_{p}(x+z)=V_{p}(x)$ for all $x\in \mathbb{R}^3$ and $z\in \mathbb{Z}^3$. $K,~K_{p}\in L^{\infty}(\mathbb{R}^3)$, $00$ and $K_{p}(x+z)=K_{p}(x)$ for all $x\in \mathbb{R}^3$ and $z\in \mathbb{Z}^3$; \end{itemize} and $f\in C(\mathbb{R}^3\times\mathbb{R}^{+},\mathbb{R})$ satisfies \begin{itemize} \item[(A2)] $\lim_{s\to 0^{+}}\frac{f(x,s)}{s}=0$ uniformly for $ x\in \mathbb{R}^3$, \item[(A3)] $\lim_{s\to +\infty}\frac{f(x,s)}{s^{5}}=0$ uniformly for $x\in \mathbb{R}^3$, \item[(A4)] $\frac{f(x,s)}{s^3}$ is nondecreasing on $(0,+\infty)$, \item[(A5)] there exists $f_{p}\in C(\mathbb{R}^3\times \mathbb{R}^{+},\mathbb{R})$ such that \begin{itemize} \item[(i)] $f(x,s)\geq f_{p}(x,s)$ for all $(x,s)\in \mathbb{R}^3\times \mathbb{R}^{+}$ and $f(x,s)-f_{p}(x,s)\in A$, where $A:=\{h(x,s): \text{for any } \varepsilon >0,\, m\{x\in B_{1}(y): |h(x,s)|\geq \varepsilon\}\to 0 \text{as }|y|\to\infty \text{ uniformly for } |s| \text{bounded}\}$, \item[(ii)] $f_{p}(x+z,s)=f_{p}(x,s)$ for all $(x,s)\in \mathbb{R}^3 \times \mathbb{R}^{+}$ and $z\in \mathbb{Z}^3$, \item[(iii)] $\frac{f_{p}(x,s)}{s^3}$ is nondecreasing on $(0,+\infty)$, \item[(iv)] $\lim_{s\to +\infty}\frac{F_{p}(x,s)}{s^{4}}=+\infty$ uniformly for $x\in \mathbb{R}^3$, where \\ $F_{p}(x,s)=\int ^{s} _{0}f_{p}(x,t)dt$. \end{itemize} \end{itemize} \begin{remark} \rm (i) Functional sets $A_{0}$ in (A1) and $A$ in (A5) were introduced by \cite{LLT} in which Liu, Liao and Tang studied positive ground state solution to Schr\"odinger equation \eqref{e1.6}. (ii) Since $\mathcal{F}\subset A_{0}$, our assumptions on $V$ and $K$ are weaker than in \cite{ZXZ1}. Furthermore, in our paper $V(x)\geq 0$ but in \cite{ZXZ1} they assumed $V(x)>0$. (iii) In \cite{ZXZ1}, to obtain the positive ground state to system \eqref{e1.5}, they firstly consider the periodic system \begin{equation}\label{e1.7} \begin{gathered} -\Delta u+V_{p}(x)u+K_{p}(x)\phi u=f_{p}(x,u)\quad x\in\mathbb{R}^3,\\ -\Delta\phi=K_{p}(x)u^2\quad x\in\mathbb{R}^3. \end{gathered} \end{equation} Then a solution of system \eqref{e1.5} was obtained by applying inequality between the energy of periodic system \eqref{e1.7} and that of system \eqref{e1.5}. In this paper, we do not using methods that of ~\cite{ZXZ1} and we proof the Theorem \ref{thm1.1} directly. \end{remark} Since we are looking for a positive solution, we may assume that $f(x,s)=f_{p}(x,s)=0$ for all $(x,s)\in(\mathbb{R}^3\times \mathbb{R}^{-})$. The next theorems are the main results of the present paper. \begin{theorem} \label{thm1.1} Suppose that {\rm (A1)--(A5)} are satisfied. Then system \eqref{e1.5} has a positive ground state solution. \end{theorem} \begin{theorem} \label{thm1.2} Suppose that $V(x)\equiv V_{p}(x),~K(x)\equiv K_{p}(x)$ satisfy {\rm(A1)}, and $f(x,s)\equiv f_{p}(x,s)$ satisfies {\rm (A2)--(A5)}. Then system \eqref{e1.5} has a positive ground state solution. \end{theorem} \section{Variational framework and preliminary results} The letter $C$ and $C_{i}$ will be repeatedly used to denote various positive constants whose exact values are irrelevant. $B_{R}(z)$ denotes the open ball centered at $z$ with radius $R$. We denote the standard norm of $L^{p}$ by $|u|_{p}=(\int_{\mathbb{R}^3}|u|^{p}dx)^{1/p}$ and $|u|_{\infty}=\operatorname{ess\,sup}_{x\in \mathbb{R}^3}|u|$. The Sobolev space $H^{1}(\mathbb{R}^3)$ is endowed with the norm \[ \|u\|_{H}^2:=\int_{\mathbb{R}^3}(|\nabla u|^2+u^2)dx. \] The space $D^{1,2}(\mathbb{R}^3)$ is endowed with the standard norm \[ \|u\|^2_{D^{1,2}}:=\int_{\mathbb{R}^3}|\nabla u|^2dx. \] Let $E:=\{u\in L^{6}(\mathbb{R}^3):|\nabla u|\in L^2(\mathbb{R}^3) \text{ and } \int_{\mathbb{R}^3}V(x)u^2dx<\infty\}$ be the Sobolev space endowed with the norm \[ \|u\|^2:=\int_{\mathbb{R}^3}(|\nabla u|^2+V(x)u^2)dx. \] \begin{lemma}\cite{LLT} \label{lem1} Suppose {\rm (A1)} holds. Then there exists two positive constants $C_{1}$ and $C_{2}$ such that $C_{1}\|u\|_{H}^2\leq\|u\| \leq C_{2}\|u\|_{H}^2$ for all $u\in E$. Moreover, $E\hookrightarrow L^{p}(\mathbb{R}^3)$ for any $p\in[2,6]$ is continuous. \end{lemma} System \eqref{e1.5} can be transformed into a Schr\"odinger equation with a nonlocal term. In fact, for all $u\in E$ (then $u\in H^{1}(\mathbb{R}^3)$), considering the linear functional $L_{u}$ defined in $D^{1,2}(\mathbb{R}^3)$ by \[ L_{u}(v)=\int_{\mathbb{R}^3}K(x)u^2vdx. \] According to the H\"older inequality and lemma \eqref{lem1}, one has that \begin{equation}\label{0.1} |L_{u}(v)|\leq |K|_{\infty}|u|^2_{12/5}|v|_{6}\leq C\|u\|^2\|v\|_{D^{1,2}}. \end{equation} So, by the Lax-Milgram theorem exists an unique $\phi_{u}\in D^{1,2}(\mathbb{R}^3)$ such that $$ \int_{\mathbb{R}^3}\nabla \phi_{u}\cdot \nabla v\,dx =(\phi_{u},v)_{D^{1,2}}=L_{u}(v)=\int_{\mathbb{R}^3}K(x)u^2v\,dx, $$ for any $v\in D^{1,2}(\mathbb{R}^3)$ and $\|\phi_u\|_{D^{1,2}}\leq C \|u\|^2$. Namely, $\phi_{u}$ is the unique solution of $$ -\Delta \phi=K(x)u^2,~x\in \mathbb{R}^3. $$ Moreover, $\phi_{u}$ can be expressed as \[ \phi_{u}=C\int_{\mathbb{R}^3}\frac{K(y)u^2(y)}{|x-y|}dy. \] Substituting $\phi_{u}$ into the system \eqref{e1.5}, we obtain \begin{equation}\label{2} -\Delta u+V(x)u+K(x)\phi_{u}u=f(x,u),\quad x\in \mathbb{R}^3. \end{equation} By \eqref{0.1}, we get \begin{equation} \label{0.2} |\int_{\mathbb{R}^3}K(x)\phi_{u}u^2dx| \leq C\|u\|^{4}. \end{equation} So the energy functional $I:H^{1}(\mathbb{R}^3)\to \mathbb{R}$ corresponding to \eqref{2} is given by \[ I(u)=\frac{1}{2}\int_{\mathbb{R}^3}(|\nabla u|^2+V(x)u^2)dx +\frac{1}{4}\int_{\mathbb{R}^3}K(x)\phi_{u}u^2dx-\int_{\mathbb{R}^3}F(x,u)dx, \] where $F(x,s)=\int^{s}_{0}f(x,t)dt$. Moreover, under our condition, $I$ belongs to $C^{1}$, so the Fr\'echet derivative of $I$ is \[ \langle I'(u),v \rangle=\int_{\mathbb{R}^3}(\nabla u\cdot\nabla v +V(x)uv)dx+\int_{\mathbb{R}^3}K(x)\phi_{u}uvdx-\int_{\mathbb{R}^3}f(x,u)vdx \] and $(u,\phi)\in H^{1}(\mathbb{R}^3)\times D^{1,2}(\mathbb{R}^3)$ is a solution of system \eqref{e1.5} if and only if $u\in H^{1}(\mathbb{R}^3)$ is a critical point of $I$ and $\phi=\phi_{u}$. For all $u\in E$, let~$\tilde{\phi}_{u}\in D^{1,2}(\mathbb{R}^3)$ is unique solution of the following equation $$ -\Delta \phi=K_{p}(x)u^2,~x\in \mathbb{R}^3. $$ Moreover, $\widetilde{\phi}_{u}$ can be expressed as \[ \widetilde{\phi}_{u}=C\int_{\mathbb{R}^3}\frac{K_{p}(y)u^2(y)}{|x-y|}dy. \] Let \[I_{p}(u)=\frac{1}{2}\int_{\mathbb{R}^3}(|\nabla u|^2+V_{p}(x)u^2)dx+\frac{1}{4}\int_{\mathbb{R}^3}K_{p}(x)\widetilde{\phi}_{u}u^2dx-\int_{\mathbb{R}^3}F_{p}(x,u)dx, \] where $F_{p}(x,s)=\int^{s}_{0}f_{p}(x,t)dt$. Then $I_{p}$ is the energy functional corresponding to the equation \begin{equation}\label{3} -\Delta u+V_{p}(x)u+K_{p}(x)\widetilde{\phi}_{u}u=f_{p}(x,u), \quad x\in \mathbb{R}^3. \end{equation} It is easy to see that $(u,\phi)\in H^{1}(\mathbb{R}^3)\times D^{1,2}(\mathbb{R}^3)$ is a solution of periodic system \eqref{e1.7} if and only if $u\in H^{1}(\mathbb{R}^3)$ is a critical point of $I_{p}$ and $\phi=\widetilde{\phi}_{u}$. \begin{lemma}\label{lem2.7} Suppose {\rm (A1)} holds. Then \[ \int_{\mathbb{R}^3}K_{p}(x)\widetilde{\phi}_{u(\cdot+z)}u^2(\cdot+z)dx =\int_{\mathbb{R}^3}K_{p}(x)\widetilde{\phi}_{u}u^2dx,\quad \forall z\in \mathbb{Z}^3, u\in E. \] \end{lemma} \begin{lemma}\label{lem2.2} Suppose that {\rm (A2), (A4), (A5)} hold. Then \begin{itemize} \item[(i)] $\frac{1}{4}f(x,s)s\geq F(x,s)\geq 0 $ for all $(x,s)\in \mathbb{R}^3\times \mathbb{R}$, \item[(ii)] $\frac{1}{4}f_{p}(x,s)s\geq F_{p}(x,s)\geq 0$ for all $(x,s)\in \mathbb{R}^3\times \mathbb{R}$. \end{itemize} \end{lemma} The proof of the above lemma is similar to that in \cite{L}, so we omitted here. \begin{lemma}\label{lem2.3} Operator $I'$ is weakly sequentially continuous. Namely if $u_{n}\rightharpoonup u$ in $E$, $I'(u_{n})\rightharpoonup I'(u)$ in $E^{-1}$. \end{lemma} The proof of the above lemma is similar to that of in \cite{ZXZ1}, so we omitted here. \begin{lemma}[\cite{LLT}] \label{lem2.8} Suppose that {\rm (A2), (A3), (A5)(i)} hold. Assume that $\{u_{n}\}$ is bounded in $E$ and $u_{n}\to0$ in $L^{s}_{\rm loc}(\mathbb{R}^3)$, for any $s\in[2,6)$. Then up to a subsequence, one has \[ \int_{\mathbb{R}^3}(F(x,u_{n})-F_{p}(x,u_{n}))dx=o_{n}(1). \] \end{lemma} \begin{lemma}[\cite{LLT}] \label{lem2.81} Suppose that {\rm (A1), (A2), (A3) (A5)(i)} hold. Assume that $\{u_{n}\}$ is bounded in $E$ and $|z_{n}|\to\infty$. Then any $\varphi\in C^{\infty}_{0}(\mathbb{R}^3)$, one has \begin{gather*} \int_{\mathbb{R}^3}(V_{p}(x)-V(x))u_{n}\varphi(\cdot-z_{n})dx=o_{n}(1), \\ \int_{\mathbb{R}^3}(f(x,u_{n})-f_{p}(x,u_{n}))\varphi(\cdot-z_{n})dx=o_{n}(1). \end{gather*} \end{lemma} \begin{lemma}\label{lem2.9} Suppose that {\rm (A1), (A2), (A3), (A5)(i)} hold. Assume that $u_{n}\rightharpoonup0$ in $E$. Then up to a subsequence, one has \[ \int_{\mathbb{R}^3}(K(x)\phi_{u_{n}}u_{n}\varphi(\cdot-z_{n}) -K_{p}(x)\widetilde{\phi}_{u_{n}}u_{n}\varphi(\cdot-z_{n}))dx=o_{n}(1), \] where $|z_{n}|\to\infty$ and $\varphi\in C^{\infty}_{0}(\mathbb{R}^3)$. \end{lemma} \begin{proof} Set $h(x):=K(x)-K_{p}(x)$. By (A1), we have $h(x)\in A_{0}$. Then for any $\varepsilon>0$, there exists $R_{\varepsilon}>0$ such that \[ m\{x\in B_{1}(y):|h(x)|\geq \varepsilon\}<\varepsilon, \quad \text{for any }|y|\geq R_{\varepsilon}. \] We cover $\mathbb{R}^3$ by balls $B_{1}(y_{i})$, $i\in \mathbb{N}$. In such a way that each point of $\mathbb{R}^3$ is contained in at most $N+1$ balls. Without any loss of generality, we suppose that $|y_{i}|1\}}\frac{K_{p}(y)u_{n}(y)\varphi(y-z_{n})}{|x-y|}dy. \end{align*} By the H\"older inequality and the Sobolev embedding, we have \begin{align*} |H(x)| & \leq |K_{p}|_{\infty}|u_{n}|_{3}|\varphi|_{6} \Big(\int_{\{y:|x-y|\leq1\}}\frac{1}{|x-y|^2}dy\Big)^{1/2} \\ &\quad +|K_{p}|_{\infty}|u_{n}|_{2}|\varphi|_{4} \Big(\int_{\{y:|x-y|>1\}}\frac{1}{|x-y|^{4}}dy\Big)^{1/4}\\ & \leq C\Big(\int_{\{z:|z|\leq1\}}\frac{1}{|z|^2}dz\Big)^{1/2} +C\Big(\int_{\{z:|z|>1\}}\frac{1}{|z|^{4}}dz\Big)^{1/4}. \end{align*} So, $\sup_{x\in\mathbb{R}^3}|H(x)|<\infty$. Then, we obtain \begin{align*} E_{1} & = \int_{\mathbb{R}^3}H(x)h(x)u^2_{n}(x)dx\\ & \leq \int_{\{x:|h(x)|\geq\varepsilon\}}|H(x)h(x)u^2_{n}(x)|dx+\int_{\{x:|h(x)|<\varepsilon\}}|H(x)h(x)u^2_{n}(x)|dx\\ & :=Q_{1}+Q_{2} \end{align*} \begin{align*} Q_{1} & = \int_{\{x:|h(x)|\geq\varepsilon\}}|H(x)h(x)u^2_{n}(x)|dx \\ & = \int_{\{x:|h(x)|\geq\varepsilon, |x|>R_{\varepsilon}+1\}} |H(x)h(x)u^2_{n}(x)|dx \\ &\quad +\int_{\{x:|h(x)|\geq\varepsilon, |x|\leq R_{\varepsilon}+1\}} |H(x)h(x)u^2_{n}(x)|dx \\ & \leq \sum^{\infty}_{n_{\varepsilon}+1}\int_{\{x\in B_{1}(y_{i}) :|h(x)|\geq\varepsilon, |x|>R_{\varepsilon}+1\}}|H(x)h(x)u^2_{n}(x)|dx \\ &\quad +2\sup_{x\in\mathbb{R}^3}|H(x)\|K_{p}|_{\infty} \int_{B_{R_{\varepsilon}+1}}|u_{n}(x)|^2dx\\ & :=Q_{11}+Q_{12} \end{align*} \begin{align*} Q_{11} &=\sum^{\infty}_{n_{\varepsilon}+1}\int_{\{x\in B_{1}(y_{i}): |h(x)|\geq\varepsilon, |x|>R_{\varepsilon}+1\}}|H(x)h(x)u^2_{n}(x)|dx\\ & \leq 2\sup_{x\in\mathbb{R}^3}|H(x)\|K_{p}|_{\infty} \sum^{\infty}_{n_{\varepsilon}+1}\int_{\{x\in B_{1}(y_{i}): |h(x)|\geq\varepsilon, |x|>R_{\varepsilon}+1\}}|u^2_{n}(x)|dx \\ & \leq C\sum^{\infty}_{n_{\varepsilon}+1}\Big(m\{x\in B_{1}(y): |h(x)|\geq \varepsilon\}\Big)^{2/3} \\ &\quad\times \Big(\int_{\{x\in B_{1}(y_{i}): |h(x)|\geq\varepsilon, |x|>R_{\varepsilon}+1\}}|u^{6}_{n}(x)|dx\Big)^{1/3}\\ & \leq C_{1}\varepsilon^{2/3}\sum^{\infty}_{n_{\varepsilon}+1} \int_{\{x\in B_{1}(y_{i}):|h(x)|\geq\varepsilon, |x|>R_{\varepsilon}+1\}} (|\nabla u_{n}|^2+u^2_{n})dx\\ & \leq C_{1}(N+1)\varepsilon^{2/3}\int_{\mathbb{R}^3} (|\nabla u_{n}|^2+u^2_{n})dx \leq C_{2}\varepsilon^{2/3}. \end{align*} Letting $\varepsilon\to0$, we obtain $Q_{11}\to0$. Since $u_{n}\rightharpoonup0$, one has that $Q_{12}\to0$. So, $Q_{1}=Q_{11}+Q_{12}\to0$. \begin{align*} Q_{2} & =\int_{\{x:|h(x)|<\varepsilon\}}|H(x)h(x)u^2_{n}(x)|dx\\ & \leq \varepsilon \sup_{x\in \mathbb{R}^3}|H(x)| \int_{\mathbb{R}^3}|u^2_{n}(x)|dx \leq C\varepsilon. \end{align*} Let $\varepsilon\to0$, we have $Q_{2}\to0$. Therefore, from the above fact we get that $E_{1}\to 0$. In the same way, we can prove $E_{2}\to 0$ and $E_{3}\to 0$. \end{proof} We define $\mathcal{N}:=\{u\in E\setminus\{0\}:(I'(u),u)=0\}$. Then $\mathcal{N}$ is a Nehari type associate to $I$, and set $c:=inf_{u\in \mathcal{N}} I$. Let $F:=\{u\in E: u^{+}\neq0\}$, where $u^{\pm}=\max\{\pm u,0\}$. In fact $$ \mathcal{N}=\{u\in F:(I'(u),u)=0\}. $$ \begin{lemma}\label{lem2.10} Suppose that {\rm (A1)--(A5)} hold. For any $u\in F$, there is a unique $t_{u}>0$ such that $t_{u}u\in \mathcal{N}$. Moreover, the maximum of $I(tu)$ for $t\geq0$ is achieved. \end{lemma} \begin{proof} Define $g(t):=I(tu)$, $t\geq0$. Using (A2), (A3) and (A5), we can prove that $g(0)=0$, $g(t)>0$ for $t$ small and $g(t)<0$ for $t$ large. In fact, by (A2) and (A3), for all $\varepsilon>0$ there exists a $C_{\varepsilon}>0$ such that \[ |f(x,s)|\leq \varepsilon|s|+C_{\varepsilon}|s|^{5},\quad |F(x,s)|\leq \frac{\varepsilon}{2}|s|^2+\frac{C_{\varepsilon}}{6}|s|^{6}, \quad s\in \mathbb{R}. \] Then \begin{align*} g(t) & = \frac{t^2}{2}\|u\|^2+\frac{t^{4}}{4}\int_{\mathbb{R}^3}K(x)\phi_{u}u^2dx -\int_{\mathbb{R}^3}F(x,tu)dx \\ & = \frac{t^2}{2}\|u\|^2+\frac{t^{4}}{4}\int_{\mathbb{R}^3}K(x)\phi_{u}u^2dx -\int_{\mathbb{R}^3}F(x,tu)dx \\ & \geq \frac{t^2}{2}\|u\|^2-\varepsilon t^2\int_{\mathbb{R}^3}|u|^2dx -C_{\varepsilon}t^{6}\int_{\mathbb{R}^3}|u|^{6}dx\\ & \geq \frac{t^2}{2}\|u\|^2-C\varepsilon t^2\|u\|^2-C_{\varepsilon}t^{6}\|u\|^{6}. \end{align*} Hence, $g(0)=0$, $g(t)>0$ for $t$ small. Set $\Omega:=\{x\in \mathbb{R}^3:u(x)>0\}$, by using Fatou lemma and (A5), we have \[ \liminf_{t\to +\infty}\int_{\Omega}\frac{F(x,tu)}{(tu)^{4}}u^{4}dx \geq \liminf_{t\to +\infty}\int_{\Omega}\frac{F_{p}(x,tu)}{(tu)^{4}}u^{4}dx=+\infty. \] Hence \begin{align*} &\limsup_{t\to +\infty}\frac{g(t)}{t^{4}} \\ &=\limsup_{t\to +\infty}\frac{1}{2t^2}\|u\|^{4} +\frac{1}{4}\int_{\mathbb{R}^3}K(x)\phi_{u}u^2dx -\liminf_{t\to +\infty}\int_{\mathbb{R}^3}\frac{F(x,tu)}{t^{4}}dx\\ & =\limsup_{t\to +\infty}\frac{1}{2t^2}\|u\|^{4} +\frac{1}{4}\int_{\mathbb{R}^3}K(x)\phi_{u}u^2dx -\liminf_{t\to +\infty}\int_{\Omega}\frac{F(x,tu)}{(tu)^{4}}u^{4}dx =-\infty, \end{align*} which deduces $g(t)\to -\infty$ as $t\to +\infty$. Therefore, there exists a $t_{u}$ such that $I(t_{u}u)=\max_{t>0}I(tu)$ and $t_{u}u\in \mathcal{N}$. Suppose that there exist $t'_{u}>t_{u}>0$ such that $t'_{u}u,\, t_{u}u\in \mathcal{N}$. Then, We have \[ \frac{1}{(t'_{u})^2}\|u\|^2+\int_{\mathbb{R}^3}K(x)\phi_{u}u^2dx =\int_{\mathbb{R}^3}\frac{f(x,t'_{u}u)u^{4}}{(t'_{u}u)^3}dx \] and this identity is also true if $t'_{u}$ is replaced by $t_{u}$. Therefore, \[ \Big(\frac{1}{(t'_{u})^2}-\frac{1}{(t_{u})^2}\Big)\|u\|^2 =\int_{\mathbb{R}^3}(\frac{f(x,t'_{u}u)}{(t'_{u}u)^3} -\frac{f(x,t_{u}u)}{(t_{u}u)^3})u^{4}dx, \] which is absurd in view of (A4) and $t'_{u}>t_{u}>0$. \end{proof} \begin{remark} \label{0.01} \rm As in \cite{R1,W1}, we have \[ c=\inf_{u\in\mathcal{N}}I(u)=\inf_{u\in F}\max_{t>0}I(tu) =\inf_{\gamma(t)\in \Gamma}\max_{t\in [0,1]}I(\gamma(t))>0 \] where \[ \Gamma:=\{\gamma\in C([0,1],E): \gamma(0)=0,I(\gamma(1))<0\}. \] \end{remark} \begin{lemma}\label{lem2.5} Suppose that {\rm (A1), (A2)--(A5)} hold. Then there exists a nonnegative and bounded sequence $\{u_{n}\}\in E$ such that \[ I(u_{n})\to c\quad \text{and} \quad \|I'(u_{n})\|_{E^{-1}}\to 0. \] \end{lemma} \begin{proof} From the proof of Lemma \ref{lem2.10}, it is easy to see that $I$ satisfies the mountain pass geometry. By \cite{S2}, there exists an $\{u_{n}\}$ such that $I(u_{n})\to c$ and $(1+\|u_{n}\|)\|I'(u_{n})\|_{E^{-1}}\to 0$. By Lemma \eqref{lem2.2}, we have \begin{align*} c & =I(u)-\frac{1}{4}\langle I'(u_{n}),u_{n}\rangle+o_{n}(1)\\ & =\frac{1}{4}\|u_{n}\|^2 +\int_{\mathbb{R}^3}(\frac{1}{4}f(x,u_{n})u_{n}-F(x,u_{n}))dx+o_{n}(1)\\ & \geq \frac{1}{4}\|u_{n}\|^2+o_{n}(1). \end{align*} Therefore, $\{u_{n}\}$ is bounded. Moreover, we have \[ \langle I'(u_{n}),u^{-}_{n}\rangle=\|u^{-}_{n}\|^2 +\int_{\mathbb{R}^3}K(x)\phi_{u_{n}}(u^{-}_{n})^2dx=o_{n}(1). \] Then $\|u^{-}_{n}\|^2=o(1)$ and $\int_{\mathbb{R}^3}K(x)\phi_{u^{-}_{n}}(u^{-}_{n})^2=o(1)$. Therefore, we can infer that $I(u^{+}_{n})\to c$ and $\|I'(u^{+}_{n})\|_{E^{-1}}\to 0$. Hence, we may always assume that $\{u_{n}\}$ is nonnegative and the prove is fished. \end{proof} \begin{lemma}\label{lem2.6} Suppose that {\rm (A1)--(A5)} hold. If $u\in \mathcal{N}$ and $I(u)=c$, then $u$ is a solution of system \eqref{e1.5}. \end{lemma} \begin{proof} The proof is similar to that of \cite{LLT,LWW}. Suppose by contradiction, that $u$ is not a solution of system \eqref{e1.5}. Hence, there exists $\varphi\in E$ such that \[ \langle I'(u),\varphi\rangle<-1. \] Choose $\varepsilon\in (0,1)$ small enough such that for all $|t-1|\leq \varepsilon$ and $|\sigma|\leq \varepsilon$, \[ \langle I'(tu+\sigma\varphi),\varphi\rangle\leq -\frac{1}{2}. \] Let $\zeta(t)\in [0,1]$ satisfies $\zeta(t)=1$ for $|t-1|\leq \frac{\varepsilon}{2}$ and $\zeta(t)=0$ for $|t-1|\geq \varepsilon$. for all $t>0$, let $\gamma(t)$ be a curve such that $\gamma(t)=tu$ for $|t-1|\geq \varepsilon$ and $\gamma(t)=tu+\varepsilon\zeta(t)\varphi$ for $|t-1|<\varepsilon$. Obviously, $\gamma(t)$ is a continuous curve, furthermore, $\|\gamma(t)\|>0$ for $|t-1|<\varepsilon$ in which $\varepsilon$ small enough. Next we will prove $I(\gamma(t))0$. In fact, if $|t-1|\geq \varepsilon$, $I(\gamma(t))=I(tu)0$ and $\nu(\gamma(1+\varepsilon))=\nu((1+\varepsilon)u)<0$. According to the continuity of $t\to \nu(\gamma(t))$, there exists $t'\in (1-\varepsilon,1+\varepsilon)$ such that $\nu(\gamma(t'))=0$. Thus $\gamma(t')\in \mathcal{N}$ and $I(\gamma(t'))0}I_{p}(tu)$. \begin{remark} \label{00001} \rm For any $u\in F$, by Lemma \ref{lem2.10}, there exists $t_{u}>0$ such that $t_{u}u\in \mathcal{N}$ and then $I(t_{u}u)\geq c$. Using $V(x)\leq V_{p}(x)$ and $F(x,s)\geq F_{p}(x, s)$, we have $c\leq I(t_{u}u)\leq I_{p}(t_{u}u)\leq \max_{t>0}I_{p}(tu)$. Then we obtain $c\leq c_{p}$. \end{remark} \section{Proof of main results} \begin{proof} According to Lemma \ref{lem2.5}, there exist a nonnegative and bounded sequence $\{u_{n}\}\in E$ such that $I(u_{n})\to c$ and $\|I'(u_{n})\|_{E^{-1}}\to 0$. Then there exists $u\in E$ such that, up to a subsequence, $u_{n}\rightharpoonup u$ in $E$, $u_{n}\to u$ in $L^2_{\rm loc}(\mathbb{R}^3)$ and $u_{n}(x)\to u(x)$ a.e. in $\mathbb{R}^3$. By lemma \ref{lem2.3}, we have that \[ 0=\langle I'(u_{n}),v \rangle+o_{n}(1)=\langle I'(u),v\rangle,~\forall v\in E, \] that is $u$ is a solution of system \eqref{e1.5}. We next distinguish the following two case to prove system \eqref{e1.5} have a nonnegative ground state solution. Case 1: $u\neq0$. Then $I(u)\geq c$. By Lemma \ref{lem2.2} and the Fatou lemma, we obtain \begin{align*} c & =\liminf_{n\to \infty}(I(u_{n})-\frac{1}{4}\langle I'(u_{n}),u_{n}\rangle) \\ & =\liminf_{n\to \infty}\Big(\frac{1}{4}\|u_{n}\|^2+\int_{\mathbb{R}^3} (\frac{1}{4}f(x,u_{n})u_{n}-F(x,u_{n}))dx\Big)\\ & \geq \frac{1}{4}\|u\|^2+\int_{\mathbb{R}^3}(\frac{1}{4}f(x,u)u-F(x,u))dx\\ & =I(u)-\frac{1}{4}\langle I'(u),u\rangle =I(u). \end{align*} Therefore, $I(u)=c$ and $I'(u)=0$. Case 2: $u=0$. Let \[ \beta:=\limsup_{n\to \infty}\sup_{z\in \mathbb{R}^3}\int_{B_{1}(z)}u^2_{n}dx. \] If $\beta=0$, by using the Lions lemma \cite{L1,L2}, we have $u_{n}\to 0$ in $L^{q}(\mathbb{R}^3)$ for all $q\in (2,6)$. From the conditions of (A2) and (A3), for all $\varepsilon>0$ there exists $C_{\varepsilon}>0$ such that $\frac{1}{2}f(x,u)u-F(x,u) \leq \varepsilon(|u|^2+|u|^{6})+C_{\varepsilon}|u|^{\alpha}$ for any $(x,s)\in \mathbb{R}^3\times \mathbb{R}$ and $\alpha\in (2,6)$. Let $\varepsilon$ small enough, we have that \begin{align*} c & =I(u_{n})-\frac{1}{2}\langle I'(u_{n}),u_{n}\rangle+o_{n}(1) \\ & =-\frac{1}{4}\int_{\mathbb{R}^3}K(x)\phi_{u_{n}}u^2_{n}dx +\int_{\mathbb{R}^3}(\frac{1}{2}f(x,u_{n})u_{n}-F(x,u_{n}))dx+o_{n}(1)\\ & \leq -\frac{1}{4}\int_{\mathbb{R}^3}K(x)\phi_{u_{n}}u^2_{n}dx +\int_{\mathbb{R}^3}(\varepsilon(|u_{n}|^2+|u_{n}|^{6}) +C_{\varepsilon}|u_{n}|^{\alpha})dx+o_{n}(1) \leq 0, \end{align*} which is a contradiction with $c>0$. So $\beta>0$. Up to a subsequence, there exists $R>0$ and $\{z_{n}\}\subset \mathbb{Z}^3$ such that \[ \int_{B_{R}}u_{n}(x+z_{n})^2dx=\int_{B_{R}(z_{n})}u_{n}^2dx>\frac{\beta}{2}. \] Set $w_{n}:=u_{n}(x+z_{n})$. Hence, there exists a nonnegative function $w\in E$ such that, up to a subsequence, $w_{n}\rightharpoonup w$ in $E$, $w_{n}\to w$ in $L^2_{\rm loc}(\mathbb{R}^3)$ and $w_{n}(x)\to w(x)$ a.e. in $\mathbb{R}^3$. Obviously, $w\neq 0$. If $\{z_{n}\}$ is bounded, $\exists~R'$ such that \[ \int_{B_{R'}(0)}u^2_{n}dx\geq \int_{B_{R}(z_{n})}u^2_{n}dx\geq \frac{\beta}{2}, \] which contradicts with the fact $u_{n}\to 0$ in $L^2_{\rm loc}(\mathbb{R}^3)$. Hence $\{z_{n}\}$ is unbounded. Up to a subsequence, we have $z_{n}\to \infty$. For all $\varphi\in C_{0}^{\infty}(\mathbb{R}^3)$, by Lemmas \ref{lem2.81} and \ref{lem2.9}, we have \begin{align*} 0 & = \langle I'(u_{n},\varphi(\cdot-z_{n}))\rangle+o_{n}(1) \\ & =\int_{\mathbb{R}^3}(\nabla u_{n}\cdot\nabla \varphi(\cdot-z_{n})dx +V(x)u_{n}\varphi(\cdot-z_{n}))dx+\int_{\mathbb{R}^3}K(x) \phi_{u_{n}}u_{n}\varphi(\cdot-z_{n})dx \\ &\quad -\int_{\mathbb{R}^3}f(x,u_{n})\varphi(\cdot-z_{n})dx+o_{n}(1)\\ & =\int_{\mathbb{R}^3}(\nabla u_{n}\cdot\nabla \varphi(\cdot-z_{n}) +V_{p}(x)u_{n}\varphi(\cdot-z_{n}))dx +\int_{\mathbb{R}^3}K_{p}(x)\widetilde{\phi}_{u_{n}}u_{n}\varphi(\cdot-z_{n})dx\\ &\quad -\int_{\mathbb{R}^3}f_{p}(x,u_{n})\varphi(\cdot-z_{n})dx+o_{n}(1)\\ & =\int_{\mathbb{R}^3}(\nabla w_{n}\cdot\nabla \varphi+V_{p}(x)w_{n} \varphi)dx+\int_{\mathbb{R}^3}K_{p}(x)\widetilde{\phi}_{w_{n}}w_{n}\varphi dx \\ &\quad -\int_{\mathbb{R}^3}f_{p}(x,w_{n})\varphi dx+o_{n}(1)\\ & =\langle I'_{p}(w),\varphi\rangle, \end{align*} that is, $w$ is a solution of periodic system \eqref{e1.7}. By Lemma \eqref{lem2.2}, Lemma \ref{lem2.8}, (A5) and Fatou lemma, we have \begin{align*} c & =I(u_{n})-\frac{1}{4}\langle I'(u_{n}),u_{n}\rangle+o_{n}(1) \\ & =\frac{1}{4}\|u_{n}\|^2+\int_{\mathbb{R}^3}(\frac{1}{4}f(x,u_{n})u_{n} -F(x,u_{n}))dx+o_{n}(1)\\ & \geq \frac{1}{4}\|u_{n}\|^2+\int_{\mathbb{R}^3}(\frac{1}{4}f_{p}(x,u_{n})u_{n} -F_{p}(x,u_{n}))dx+o_{n}(1)\\ & =\frac{1}{4}\|w_{n}\|^2+\int_{\mathbb{R}^3}(\frac{1}{4}f_{p}(x,w_{n})w_{n} -F_{p}(x,w_{n}))dx+o_{n}(1)\\ & \geq \frac{1}{4}\|w\|^2+\int_{\mathbb{R}^3}(\frac{1}{4}f_{p}(x,w)w -F_{p}(x,w))dx+o_{n}(1)\\ & =I_{p}(w)-\frac{1}{4}\langle I'_{p}(w),w\rangle\\ & =I_{p}(w) \geq c_{p}. \end{align*} Using Remark \eqref{00001}, $I_{p}(w)=c_{p}=c$. By the properties of $c$ and $\mathcal{N}$, there exits $t_{w}>0$ such that $t_{w}w\in \mathcal{N}$. Thus, we obtain $c\leq I(t_{w}w)\leq I_{p}(t_{w}w)\leq I_{p}(w)=c$. So $c$ is achieved by $t_{w}w$. By Lemma \ref{lem2.6}, we have $I'(t_{w}w)=0$. 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