\documentclass[reqno]{amsart} \usepackage{hyperref} \usepackage{mathrsfs} \AtBeginDocument{{\noindent\small \emph{Electronic Journal of Differential Equations}, Vol. 2018 (2018), No. 61, pp. 1--15.\newline ISSN: 1072-6691. URL: http://ejde.math.txstate.edu or http://ejde.math.unt.edu} \thanks{\copyright 2018 Texas State University.} \vspace{8mm}} \begin{document} \title[\hfilneg EJDE-2018/61\hfil Semiclassical ground states] {Semiclassical ground states for nonlinear Schr\"odinger-Poisson systems} \author[H. Zhang, F. Zhang \hfil EJDE-2018/61\hfilneg] {Hui Zhang, Fubao Zhang} \address{Hui Zhang (corresponding author) \newline Department of Mathematics, Jinling Institute of Technology, Nanjing 211169, China} \email{huihz0517@126.com} \address{Fubao Zhang \newline Department of Mathematics, Southeast University, Nanjing 210096, China} \email{zhangfubao@seu.edu.cn} \dedicatory{Communicated by Claudianor O. Alves} \thanks{Submitted September 29, 2017. Published March 5, 2018.} \subjclass[2010]{35J50, 35J60, 35A15} \keywords{Schr\"odinger-Poisson system; variational method; concentration; \hfill\break\indent Nehari manifold} \begin{abstract} In this article, we study the Schr\"odinger-Poisson system \begin{gather*} -\epsilon^2\Delta u+V(x)u+\phi(x) u=Q(x)u^3,\quad x\in \mathbb{R}^3,\\ -\epsilon^2\Delta\phi=u^2, \quad x\in \mathbb{R}^3, \end{gather*} where $\epsilon>0$ is a parameter, $V$ and $Q$ are positive bounded functions. We establish the existence of ground states for $\epsilon$ small, and describe the concentration phenomena of ground states as $\epsilon\to 0$. \end{abstract} \maketitle \numberwithin{equation}{section} \newtheorem{theorem}{Theorem}[section] \newtheorem{lemma}[theorem]{Lemma} \allowdisplaybreaks \section{Introduction and statement of main results} The Schr\"odinger-Poisson system \begin{equation}\label{1.1} \begin{gathered} -\epsilon^2\Delta u+V(x)u+\phi u=l(x,u),\quad x\in \mathbb{R}^3,\\ -\epsilon^2\Delta\phi=u^2, \quad x\in \mathbb{R}^3, \end{gathered} \end{equation} was first introduced in \cite{BF} as a physical model describing a charged wave interacting with its own electrostatic field in quantum mechanics. The unknowns $u$ and $\phi$ represent the wave functions associated to the particle and electric potential, the function $V$ is an external potential, and the nonlinearity $l(x,u)$ simulates the interaction between many particles or external nonlinear perturbations. For more information on the physical aspects, we refer the reader to \cite{BF}. There are many results on the existence and concentration of solutions for \eqref{1.1} and similar problems. Equation \eqref{1.1} is usually studied in two cases when $\epsilon$ is regarded as a small parameter, and when $\epsilon$ is fixed ($\epsilon=1$). For fixed $\epsilon$, see \cite{ALVES1,ALVES2,AM1,AA,AP,CV,JZ,HK,LPY,LW,RD,ZLZ,ZZ,ZT} and references therein. In this article, we study onlythe case $\epsilon$ is small. So we shall recall some results for this case. In \cite{IG} the authors considered the system \begin{equation}\label{1.2} \begin{gathered} -\epsilon^2\Delta u+V(x)u+\phi(x) u=f(u),\quad x\in \mathbb{R}^3,\\ -\Delta\phi=u^2, \quad x\in \mathbb{R}^3, \end{gathered} \end{equation} and proved that \eqref{1.2} has a single bump solution, which concentrates on the critical points of $V(x)$. Later, D'Aprile and Wei \cite{DW} constructed positive radially symmetric bound states of \eqref{1.2} with $f(u)=u^p$, $14$ such that $0<\mu F(s):=\mu\int_{0}^{s}f(t)dt\leq sf(s)$, $s>0$, and $$ f'(s)s^{2}-3f(s)s\geq Cs^{\sigma},\quad C>0,\; \sigma\in(4,6). $$} \end{equation} By using Ljusternik-Schnirelmann theory and minimax methods, he showed the multiplicity of positive solutions of \eqref{1.3} which concentrate on the minima of $V(x)$ as $\epsilon\to 0$. Later, Wang et al. \cite{WANGJUN} studied the system \begin{equation}\label{1.4} \begin{gathered} -\epsilon^2\Delta u+V(x)u+\phi(x) u=b(x)f(u),\quad x\in \mathbb{R}^3,\\ -\epsilon^2\Delta\phi=u^2, \quad x\in \mathbb{R}^3. \end{gathered} \end{equation} Suppose that $V(x)$ has at least one minimum, $b(x)$ has at least one maximum, and $f$ satisfies some weaker conditions than \eqref{aj3}, namely \begin{gather*} f\in C(\mathbb{R}^3),\quad f(s)=o(s^{3}) \text{ as } s\to 0,\quad \frac{f(s)}{s^{3}} \text{ is increasing on } (0,\infty),\\ \frac{F(s)}{s^4}\to \infty \quad \text{as } s\to \infty, \end{gather*} Wang et al.\ obtained the existence and concentration of positive ground states for \eqref{1.4} using the method of Nehari manifold and minimax methods. He and Zou \cite{He2} considered the existence and concentration behavior of ground states of \eqref{1.1} with critical growth, \begin{gather*} -\epsilon^2\Delta u+V(x)u+\phi(x) u=|u|^{4}u+f(u),\quad x\in \mathbb{R}^3,\\ -\epsilon^2\Delta\phi=u^2, \quad x\in \mathbb{R}^3, \end{gather*} where $f$ satisfies \eqref{aj3} and $f(t)\geq \lambda t^\sigma$ for all $t>0$, where $\sigma\in(3,5)$. Recently, He et al.\ \cite{HY} studied the system \begin{gather*} -\epsilon^2\Delta u+V(x)u+\phi(x) u=\lambda|u|^{p-2}u+ |u|^{4}u,\quad x\in \mathbb{R}^3,\\ -\epsilon^2\Delta\phi=u^2, \quad x\in \mathbb{R}^3, \end{gather*} where $30$ and $\inf_{x\in\mathbb{R}^3}Q(x)>0$; \\ \item[(A2)] $\nu_{\rm min}<\nu_\infty$, and there exist $R>0$ and $x_{\rm min}\in\mathcal{V}$ such that $Q(x_{\rm min})\geq Q(x)$ for all \indent\quad $|x|\geq R$. \item[(A3)] $q_{\rm max}>q_\infty$, and there exist $R>0$ and $x_{\rm max}\in\mathcal{Q}$ such that $V(x_{\rm max})\leq V(x)$ for all \indent\quad $|x|\geq R$. \end{itemize} Observe that, for case (A2), we can assume that $Q(x_{\rm min})=\max_{x\in\mathcal{V}} Q(x)$ and set $$ \mathcal{A}_{V}:=\{x\in \mathcal{V}: Q(x)=Q(x_{\rm min})\} \cup\{x\not\in\mathcal{V}: Q(x)>Q(x_{\rm min})\}; $$ while for case (A3), we can assume that $V(x_{\rm max})=\min_{x\in\mathcal{Q}} V(x)$ and set $$ \mathcal{A}_{Q}:=\{x\in \mathcal{Q}: V(x)=V(x_{\rm max})\} \cup\{x\not\in\mathcal{Q}: V(x)0$, considering the linear functional $L_u$ defined in $D^{1,2}(\mathbb{R}^3)$ by $$ L_u(v)=\int_{\mathbb{R}^3} u^2vdx. $$ By the H\"older inequality and the Sobolev inequality, we have \begin{equation}\label{2.5 }|L_u(v)|\leq|u|^2_{\frac{12}{5}}|v|_6\leq C|u|^2_{\frac{12}{5}}\|v\|_{D^{1,2}}. \end{equation} Hence the Lax-Milgram theorem implies that there exists a unique $\phi^\epsilon_u\in D^{1,2}(\mathbb{R}^3)$ such that \begin{equation}\label{2.6} \epsilon^2\int_{\mathbb{R}^3}\nabla\phi^\epsilon_u\nabla v dx =L_u(v)=\int_{\mathbb{R}^3} u^2vdx,\quad \forall v\in D^{1,2}(\mathbb{R}^3). \end{equation} Namely, $\phi^\epsilon_u$ is the unique solution of $-\epsilon^2\Delta\phi^\epsilon_u=u^2$. Moreover, $\phi^\epsilon_u$ can be expressed as $$ \phi^\epsilon_u(x)=\frac{1}{4 \pi \epsilon^2 }\int_{\mathbb{R}^3} \frac{u^2(y)}{|x-y|}dy. $$ Substituting $\phi^\epsilon_u$ into the first equation of \eqref{SPe}, we obtain \begin{equation} \label{SPe'} -\epsilon^2\Delta u+V(x)u+\phi^\epsilon_u(x)u=Q(x)u^3. \end{equation} Let $\mathscr{L}_\epsilon$ denote the set of all positive ground states of \eqref{SPe'}. Now we state our main results. \begin{theorem} \label{thm1.1} Let {\rm (A1)} and {\rm (A2)} hold. Then for any $\epsilon>0$ small we have: \\ (1) Equation \eqref{SPe} has a positive ground state $\psi_\epsilon=(w_\epsilon,\phi^\epsilon_{w_\epsilon})$ in $H^1(\mathbb{R}^3)\times D^{1,2}(\mathbb{R}^3)$; \\ (2) $\mathscr{L}_\epsilon$ is compact in $H^1(\mathbb{R}^3)$; \\ (3) If additionally $V$ and $Q$ are uniformly continuous functions, then $w_\epsilon$ satisfies: \begin{itemize} \item[(i)] there exists a maximum point $x_\epsilon\in\mathbb{R}^3$ of $w_\epsilon$, such that $\lim_{\epsilon\to 0}\operatorname{dist}(x_\epsilon, \mathcal{A}_{V})=0$. Setting $v_\epsilon(x):=w_\epsilon(\epsilon x+x_\epsilon)$, for any sequence $x_\epsilon\to x_0$, $\epsilon\to 0$, $v_\epsilon$ converges in $H^1(\mathbb{R}^3)$ to a ground state $v$ of \begin{equation*} -\Delta u+V(x_0)u+\phi_u(x)u=Q(x_0)u^3, \end{equation*} where $\phi_u(x)=\frac{1}{4\pi}\int_{\mathbb{R}^3}\frac{u^2(y)}{|x-y|}dy$. \\ In particular, if $\mathcal{V}\cap\mathcal{Q}\neq\emptyset$, then $\lim_{\epsilon\to 0}\operatorname{dist}(x_\epsilon,\mathcal{V}\cap\mathcal{Q})=0$ and up to a subsequence, $v_\epsilon$ converges in $H^1(\mathbb{R}^3)$ to a ground state $v$ of \begin{equation*} -\Delta u+\nu_{\rm min} u+\phi_u(x)u=q_{\rm max}u^3, \end{equation*} \item[(ii)] $|w_\epsilon(x)|\leq C exp\bigl(-\frac c\epsilon|x-x_{\epsilon}|\bigr)$, where $C, c>0$. \end{itemize} \end{theorem} \begin{theorem} \label{thm1.2} Suppose that {\rm (A1), (A3)} hold. Then, all the conclusions of Theorem \ref{thm1.1} remain true with $\mathcal{A}_{V}$ replaced by $\mathcal{A}_{Q}$. \end{theorem} \begin{proof}[Outline for the proof] Compared with the previous results \cite{He,He2,HY,WANGJUN}, the main difficulty is the lack of the higher-order term and the competing effect of the nonlocal term with three times growth term, which causes that the standard method of Nehari manifold is invalid. Inspired by \cite{FS}, by restricting the functional in a set, the functional has a unique maximum point along the nontrivial direction $u$ in $H^1(\mathbb{R}^3)$. Then we use the one-to-one correspondence of the functionals on the manifold and an open set of the unit sphere to establish the new method of Nehari manifold. We also would like to point out that, using the similar ideas, we \cite{ZXZ} showed the existence of classical ground states of system \eqref{1.1} with $\epsilon=1$ when the potentials are asymptotically periodic. However, in this paper, we prove the existence and concentration of semiclassical ground states for system \eqref{1.1} with small enough $\epsilon$. In addition, in the period of investigating the concentration behavior of ground states, the competing effect of the nonlocal term $\phi u$ and three times growth term $Q(x)u^3$ makes that some estimations and verifications become complex. \end{proof} In this paper we use the following notation. For $1\leq p\leq\infty$, the norm in $L^p(\mathbb{R}^3)$ is denoted by $|\cdot|_{p}$. $\int_{\mathbb{R}^3} f(x)dx$ will be represented by $\int_{\mathbb{R}^3} f(x)$. For any $r>0$ and $x\in\mathbb{R}^3$, $B_r(x)$ denotes the ball centered at $x$ with the radius $r$. This article is organized as follows. In Section 2 we introduce the variational framework. In Section 3 we study the autonomous problem. In Section 4 we are devoted to investigating an auxiliary problem. In Section 5, we give the proof of Theorems \ref{thm1.1} and \ref{thm1.2}. \section{The new method of Nehari manifold} For the proof of our theorems, we shall consider an equivalent equation to \eqref{SPe'}. By making the change of variable $x\to \epsilon x$, the problem \eqref{SPe'} turns out to be \begin{equation} -\Delta u+V(\epsilon x)u+\phi_u(x)u=Q(\epsilon x)u^3,\quad u\in H^{1}(\mathbb{R}^{3}),\label{SPe*} \end{equation} where $H^1(\mathbb{R}^3)$ is the Sobolev space with standard norm $$ \|u\|^2=\int_{\mathbb{R}^{3}}(|\nabla u|^2+u^2). $$ Let $S_1=\{u\in H^1(\mathbb{R}^3):\|u\|^2=1\}$. From assumption (A1), it follows that $$ \|u\|^2_\epsilon=\int_{\mathbb{R}^{3}}(|\nabla u|^2+V(\epsilon x)u^2) $$ is an equivalent norm on $H^1(\mathbb{R}^3)$. The functional associated with the equation \eqref{SPe*} is $$ I_\epsilon(u)=\frac{1}{2}\|u\|^2_\epsilon+\frac{1}{4}\int_{\mathbb{R}^3} {\phi}_uu^2-\frac14\int_{\mathbb{R}^3} Q(\epsilon x)u^4. $$ Now we recall some standard properties of $\phi_u$, see \cite{ZXZ}. \begin{lemma}\label{l2.1.1} Let {\rm (A1)} hold. For any $\epsilon>0$, we have: \begin{itemize} \item[(i)] If $u_n\rightharpoonup u$ in $H^1(\mathbb{R}^3)$, then $\phi_{u_n}\rightharpoonup\phi_{u}$ in $D^{1,2}(\mathbb{R}^3)$. \item[(ii)] If $u_n\rightharpoonup u$ in $H^1(\mathbb{R}^3)$, then $\int_{\mathbb{R}^3}\phi_{u_n}u_nv\to \int_{\mathbb{R}^3}\phi_{u}uv$, for any $v\in C^\infty_0(\mathbb{R}^3)$. \end{itemize} \end{lemma} Below we describe the variational framework for our problem. Firstly we give the Nehari manifold $N_\epsilon$ corresponding to $I_\epsilon$: $$ N_\epsilon=\{u\in H^{1}(\mathbb{R}^3)\backslash\{0\}:\langle I'_\epsilon(u),u\rangle=0\}, $$ where $$ \langle I'_\epsilon(u),u\rangle=|\nabla u|^2_2+\int_{\mathbb{R}^3}V(\epsilon x)u^2 +\int_{\mathbb{R}^3}\phi_u u^2-\int_{\mathbb{R}^3} Q(\epsilon x)u^4, $$ and the least energy on $N_\epsilon$ is defined by $c_\epsilon:=\inf_{N_\epsilon}I_\epsilon$. \begin{lemma}\label{l2.1} Let {\rm (A1)} hold. Then $I_\epsilon$ is coercive on $N_\epsilon$. \end{lemma} \begin{proof} For all $u\in N_\epsilon$, we have \begin{equation}\label{3.2} I_\epsilon(u)=I_\epsilon(u)-\frac{1}{4}\langle I'_\epsilon(u),u\rangle =\frac{1}{4}\|u\|^2_\epsilon.\end{equation} Then ${I_\epsilon}|_{N_\epsilon}$ is coercive. \end{proof} Next we introduce a set to construct the new method of Nehari manifold as in \cite{ZXZ}. Define $$ \Theta_\epsilon:=\{u\in H^1(\mathbb{R}^3):\int_{\mathbb{R}^3} \phi_uu^2<\int_{\mathbb{R}^3} Q(\epsilon x)u^4\}. $$ As in \cite{ZXZ}, we can show that $\Theta_\epsilon\neq\emptyset$ since $\inf_{\mathbb{R}^3}Q>0$ by (A1). Set $$ h_\epsilon(t):=I_\epsilon(tu) =\frac {t^2}2\|u\|^2_\epsilon+\frac {t^4}4\bigl[\int_{\mathbb{R}^3}\phi_u u^2 -\int_{\mathbb{R}^3} Q(\epsilon x)u^4\bigr], $$ using the similar argument in \cite{ZXZ}, we obtain the following two lemmas. \begin{lemma} \label{l3.2} Let {\rm (A1)} hold. Then for any $\epsilon>0$, we have: \begin{itemize} \item[(i)] For all $u\in \Theta_\epsilon$, there exists a unique $t_\epsilon:=t_\epsilon(u)>0$ such that $h'_\epsilon(t)>0$ for $0t_\epsilon$. Moreover, $t_{\epsilon}u\in N_\epsilon$ and $I_\epsilon(t_{\epsilon} u)=\max_{t>0}I_\epsilon(tu)$. \item[(ii)] If $u\not\in \Theta_\epsilon$, then $tu\not\in N_\epsilon$ for any $t>0$. \item[(iii)] For each compact subset $W$ of $\Theta_\epsilon\cap {S_1}$, there exists $C_W>0$ such that $t_w\leq C_W$ for all $w\in W$. \end{itemize} \end{lemma} \begin{lemma} \label{l3.11} Under assumption {\rm (A1)}, for $\epsilon>0$ we have: \begin{itemize} \item[(1)] $c_\epsilon>0$; \item[(2)] $\|u\|^2_\epsilon\geq4c_\epsilon$ for all $u\in N_\epsilon$. \end{itemize} \end{lemma} From Lemma \ref{l3.2} (i), we define the mapping $\hat{m}_\epsilon:\Theta_\epsilon\to N_\epsilon$ by $\hat{m}_\epsilon(u)=t_\epsilon u$. In addition, $\forall v\in\mathbb{R}^+u$ we have $\hat{m}_\epsilon(v)=\hat m_\epsilon(u)$. Let $U_\epsilon:=\Theta_\epsilon\cap S_1$, we easily infer that $U_\epsilon$ is an open subset of $S_1$. Define $ m_\epsilon:=\hat{m}_\epsilon|_{U_\epsilon}$. Then $m_\epsilon$ is a bijection from $U_\epsilon$ to $N_\epsilon$. Moreover, by Lemmas \ref{l3.2} and \ref{l3.11}, as in the proof of \cite[Proposition 3.1]{SW}, we have the following result. \begin{lemma} \label{l3.6} If {\rm (A1)} is satisfied, then the mapping $m_\epsilon$ is a homeomorphism between $U_\epsilon$ and $N_\epsilon$, and the inverse of $m_\epsilon$ is given by $m^{-1}_\epsilon(u)=\frac{u}{\|u\|}$. \end{lemma} By Lemma \ref{l3.6}, the least energy $c_\epsilon$ has the following minimax characterization: \begin{equation}\label{2.1} c_\epsilon:=\inf_{u\in N_\epsilon}I_\epsilon(u) =\inf_{u\in U_\epsilon}\max_{t\geq0}I_\epsilon(tu). \end{equation} Considering the functional $\Psi_\epsilon:U_\epsilon\to \mathbb{R}$ given by $\Psi_\epsilon(w):=I_\epsilon(m_\epsilon(w))$, as in \cite[Corollary 3.3]{SW} we easily deduce the following statement. \begin{lemma} \label{l3.7} Let {\rm (A1)} hold. Then the following results hold: \begin{itemize} \item[(1)] If $\{w_n\}$ is a PS sequence for $\Psi_\epsilon$, then $\{m_\epsilon(w_n)\}$ is a PS sequence for $I_\epsilon$. If $\{u_n\}\subset N_\epsilon$ is a bounded PS sequence for $I_\epsilon$, then $\{m^{-1}_\epsilon(u_n)\}$ is a PS sequence for $\Psi_\epsilon$. \item[(2)] $w$ is a critical point of $\Psi_\epsilon$ if and only if $m_\epsilon(w)$ is a nontrivial critical point of $I_\epsilon$. Moreover, $\inf_{N_\epsilon}I_\epsilon=\inf_{U_\epsilon}\Psi_\epsilon$. \end{itemize} \end{lemma} \section{Autonomous problem} This section concerns the autonomous equation. Precisely, for any positive constants $\nu$ and $q$, we consider \begin{equation} -\Delta u+\nu u+\phi_u(x)u=qu^3,\ u\in H^{1}(\mathbb{R}^{3}). \label{SPnq} \end{equation} The functional of \eqref{SPnq} is denoted by $$ J_{\nu, q}(u)=\frac{1}{2}\int_{\mathbb{R}^{3}}(|\nabla u|^{2}+\nu|u|^{2})+\frac{1}{4}\int_{\mathbb{R}^{3}}\phi_uu^{2} -\frac{q}{4}\int_{\mathbb{R}^{3}}|u|^{4}. $$ The Nehari manifold corresponding to \eqref{SPnq} is defined by $$ M_{\nu,q}=\{u\in H^{1}(\mathbb{R}^3)\backslash\{0\}: \langle J'_{\nu, q}(u),u\rangle=0\}, $$ and the least energy on $M_{\nu, q}$ is defined by $m_{\nu, q}:=\inf_{M_{\nu, q}}J_{\nu, q}$. Denote $$ \Theta^q=\{u\in H^1(\mathbb{R}^3):\int_{\mathbb{R}^3} \phi_uu^20}J_{\nu, q}(tw). \end{equation} From \cite{ZXZ} we have the following result. \begin{lemma}\label{l2.2} For any $\nu,q>0$, problem \eqref{SPnq} has a positive ground state $u_{\nu,q}$ with $J_{\nu,q}(u_{\nu,q})=m_{\nu,q}$. \end{lemma} The following lemma describes the behavior of the least energy for different parameters $\nu,q>0$, which will play an important role in proving the existence results for \eqref{SPe*}. \begin{lemma}\label{l2.3} Let $\nu_i,q_i>0$, $i=1,2$, with $\min\{\nu_2-\nu_1, q_1-q_2\}\geq0$. Then $m_{\nu_1,q_1}\leq m_{\nu_2,q_2}$. If additionally $\max\{\nu_2-\nu_1, q_1-q_2\}>0$, then $m_{\nu_1,q_1}q_1$. \end{lemma} \begin{proof} We prove that $m_{\nu_1,q_1}\leq m_{\nu_2,q_2}$ for example. From Lemma \ref{l2.2}, choose $u$ to be a positive ground state of problem \eqref{SPnq} with $\nu=\nu_2$ and $q=q_2$. Then $u\in \Theta^{q_2}$ and $$ m_{\nu_2,q_2}=J_{\nu_2,q_2}(u)=\max_{t\geq0}J_{\nu_2,q_2}(tu). $$ With the use of $q_1\geq q_2$, $u\in \Theta^{q_1}$. Then there exists $t_0>0$ such that $J_{\nu_1,q_1}(t_0u)=\max_{t\geq0}J_{\nu_1,q_1}(tu)$. By $\min\{\nu_2-\nu_1, q_1-q_2\}\geq0$, we obtain $J_{\nu_1,q_1}(t_0u)\leq J_{\nu_2,q_2}(t_0u)$. Then by \eqref{3.2.2} we have \begin{align*} m_{\nu_1,q_1} &=\inf_{w\in{\Theta^{q_1} \cap {S_1}}}\max_{t\geq0}J_{\nu_1,q_1}(tw) \leq \max_{t\geq0}J_{\nu_1,q_1}(tu)=J_{\nu_1,q_1}(t_0u)\\ &\leq J_{\nu_2,q_2}(t_0u)\leq J_{\nu_2,q_2}(u)=m_{\nu_2,q_2}. \end{align*} \end{proof} \section{An auxiliary problem} Now, we are ready to introduce the auxiliary problem for \eqref{SPe*}. For any $\nu_{\rm min}\leq \tilde{a}\leq\nu_{\infty}$, $q_{\infty}\leq \tilde{b}\leq q_{\rm max}$. Set $$ V^{\tilde{a}}_{\epsilon}(x):=\max\{\tilde{a}, V(\epsilon x)\},\quad Q^{\tilde{b}}_{\epsilon}(x):=\min\{\tilde{b}, Q(\epsilon x)\}, $$ and consider the auxiliary equation \begin{equation} -\Delta u+V^{\tilde{a}}_{\epsilon}(x)u+\phi_u(x)u=Q^{\tilde{b}}_{\epsilon}(x) u^3. \label{AP*} \end{equation} The functional is $$ I^{\tilde{a},\tilde{b}}_\epsilon(u) =\frac{1}{2}\int_{\mathbb{R}^{3}}(|\nabla u|^2 +V^{\tilde{a}}_{\epsilon}(x)u^2)+\frac{1}{4}\int_{\mathbb{R}^{3}} \phi_uu^2-\frac14\int_{\mathbb{R}^{3}} Q^{\tilde{b}}_{\epsilon}(x)u^4, $$ and the Nehari manifold is $$ N^{\tilde{a},\tilde{b}}_\epsilon=\{u\in H^{1}(\mathbb{R}^3)\backslash\{0\}: \langle (I^{\tilde{a},\tilde{b}}_\epsilon)'(u),u\rangle=0\}, $$ and the least energy on $N^{\tilde{a},\tilde{b}}_\epsilon$ is denoted by $c^{\tilde{a},\tilde{b}}_\epsilon$. Moreover, as in Section 2, denote $$ \Theta^{\tilde{b}}_{\epsilon}:=\{u\in H^1(\mathbb{R}^3) :\int_{\mathbb{R}^3} \phi_uu^2<\int_{\mathbb{R}^3}Q^{\tilde{b}}_{\epsilon}(x)u^4\}. $$ and $c^{\tilde{a},\tilde{b}}_\epsilon$ can be characterized by \begin{equation}\label{4.1.1} c^{\tilde{a},\tilde{b}}_\epsilon=\inf_{u\in N^{\tilde{a},\tilde{b}}_\epsilon} I^{\tilde{a},\tilde{b}}_\epsilon(u) =\inf_{u\in\Theta^{\tilde{b}}_{\epsilon}\cap S_1} \max_{t>0}I^{\tilde{a},\tilde{b}}_\epsilon(tu). \end{equation} \begin{lemma}\label{l3.1} $m_{\tilde{a},\tilde{b}}\leq c^{\tilde{a},\tilde{b}}_\epsilon$. \end{lemma} \begin{proof} For any $u\in {\Theta^{\tilde{b}}_{\epsilon}\cap S_1}$, we have $u\in {\Theta^{\tilde{b}}}\cap S_1$ since $\tilde{b}\geq Q^{\tilde{b}}_{\epsilon}(x)$. Then $$ m_{\tilde{a},\tilde{b}}\leq\max_{t>0}J_{\tilde{a},\tilde{b}}(tu) \leq\max_{t>0}I^{\tilde{a},\tilde{b}}_\epsilon(tu). $$ By the arbitrary of $u$, from \eqref{4.1.1} we obtain that $m_{\tilde{a},\tilde{b}}\leq c^{\tilde{a},\tilde{b}}_\epsilon$. \end{proof} \section{Proof of the main results} In this part, we only prove Theorem \ref{thm1.1}, since the arguments for Theorem \ref{thm1.2} are quite similar. Without loss of generality, we may assume that $x_{\rm min}:=0\in \mathcal{V}$ in (A2) or $x_{\rm min}:=0\in \mathcal{V}\cap \mathcal{Q}$ if $\mathcal{V}\cap \mathcal{Q}\neq\emptyset$. Denote $$ d:=Q(0)=\max_{x\in \mathcal{V}}Q(x)\geq Q(x),\quad \forall |x|\geq R $$ and consider the functional $I_\epsilon$. \begin{lemma}\label{l4.2} $\limsup_{\epsilon\to 0}c_{\epsilon}\leq m_{V(y_{1}),Q(y_{1})}$, where $y_{1}\in\mathbb{R}^3$. In particular, $\limsup_{\epsilon\to 0}c_{\epsilon}\leq m_{V(0),Q(0)}:=m_{\nu_{\rm min},d}$. \end{lemma} \begin{proof} Since \eqref{SPnq} has a positive ground state for each $\nu, q>0$, we can take $u\in M_{V(y_{1}),Q(y_{1})}$ such that $J_{V(y_{1}),Q(y_{1})}(u)=m_{V(y_{1}),Q(y_{1})}$. Then \begin{equation}\label{4.1} \int_{\mathbb{R}^{3}}(|\nabla u|^{2}+V(y_{1})|u|^{2})+\int_{\mathbb{R}^{3}}\phi_uu^2= \int_{\mathbb{R}^{3}}Q(y_{1})|u|^{4}. \end{equation} Then for small $\epsilon>0$, it holds \begin{equation*} \int_{\mathbb{R}^{3}}(|\nabla u|^{2}+V(\epsilon x+y_{1})|u|^{2})+\int_{\mathbb{R}^{3}}\phi_uu^{2} =\int_{\mathbb{R}^{3}}Q(\epsilon x+y_{1})u^4+o_\epsilon(1). \end{equation*} Set $w_{\epsilon}(x)=u(x-\frac{y_{1}}{\epsilon})$. Then \begin{equation}\label{5.4.4} \int_{\mathbb{R}^{3}}(|\nabla w_{\epsilon}|^{2}+V(\epsilon x)|w_{\epsilon}|^{2})+ \int_{\mathbb{R}^{3}}\phi_{w_{\epsilon}}w^{2}_\epsilon =\int_{\mathbb{R}^{3}}Q(\epsilon x)w_{\epsilon}^{4}+o_\epsilon(1). \end{equation} Since for small $\epsilon>0$, \begin{equation}\label{5.4.5} \int_{\mathbb{R}^{3}}Q(\epsilon x)w_{\epsilon}^{4} -\int_{\mathbb{R}^{3}}\phi_{w_{\epsilon}}w^{2}_\epsilon\geq C>0, \end{equation} we have that $w_{\epsilon}\in {\Theta_\epsilon}$. So there exists $t_{\epsilon}>0$ such that $t_{\epsilon}w_{\epsilon}\in N_{\epsilon}$. Then \begin{equation}\label{4.1.2} t_{\epsilon}^{2}\int_{\mathbb{R}^{3}}(|\nabla w_{\epsilon}|^{2}+V(\epsilon x)|w_{\epsilon}|^{2})+t_{\epsilon}^{4} \int_{\mathbb{R}^{3}}\phi_{w_{\epsilon}}w^{2}_\epsilon =t_{\epsilon}^{4}\int_{\mathbb{R}^{3}}Q(\epsilon x)w_{\epsilon}^{4}. \end{equation} By \eqref{5.4.4} and \eqref{4.1.2} we obtain $$ (t^2_{\epsilon}-1)\bigl[\int_{\mathbb{R}^{3}}Q(\epsilon x)w_{\epsilon}^{4}- \int_{\mathbb{R}^{3}}\phi_{w_{\epsilon}}w^{2}_\epsilon\bigr] =o_\epsilon(1). $$ Using \eqref{5.4.5} we have that $t_{\epsilon}\to 1$ as $\epsilon\to 0$. Since $t_{\epsilon}w_{\epsilon}\in N_{\epsilon}$, one has \begin{align*} c_{\epsilon}\leq I_{\epsilon}(t_{\epsilon}w_{\epsilon}) =&J_{V(y_{1}),Q(y_{1})}(t_{\epsilon}w_{\epsilon}) +\frac{t_{\epsilon}^{2}}{2}\int_{\mathbb{R}^{3}}(V(\epsilon x)-V(y_{1}))w_{\epsilon}^{2}\\ &+\frac{t_{\epsilon}^{4}}{4}\int_{\mathbb{R}^{3}}(Q(y_{1})-Q(\epsilon x))|w_{\epsilon}|^{4}\\ =&J_{V(y_{1}),Q(y_{1})}(t_{\epsilon}w_{\epsilon}) +\frac{t_{\epsilon}^{2}}{2}\int_{\mathbb{R}^{3}}(V(\epsilon x+y_{1})-V(y_{1}))u^{2}\\ &+\frac{t_{\epsilon}^{4}}{4}\int_{\mathbb{R}^{3}}(Q(y_{1})-Q(\epsilon x+y_{1}))|u|^{4}. \end{align*} Therefore, \begin{equation*} c_{\epsilon} \leq J_{V(y_{1}),Q(y_{1})}(t_{\epsilon}w_{\epsilon}) +o_{\epsilon}(1) =J_{V(y_{1}),Q(y_{1})}(t_{\epsilon}u)+o_{\epsilon}(1) = J_{V(y_{1}),Q(y_{1})}(u)+o_{\epsilon}(1). \end{equation*} Then \begin{equation*} \limsup_{\epsilon\to 0}c_{\epsilon}\leq J_{V(y_{1}),Q(y_{1})}(u)=m_{V(y_{1}),Q(y_{1})}. \end{equation*} In particular, we take $y_{1}=0$, it follows that \begin{equation*} \limsup_{\epsilon\to 0}c_{\epsilon}\leq m_{V(0),Q(0)}=m_{\nu_{\rm min},d}. \end{equation*} \end{proof} \begin{lemma}\label{l4.6} The minimax value $c_\epsilon$ is achieved if $\epsilon$ is small enough. Hence, problem \eqref{SPe*} has a positive ground state if $\epsilon$ is small enough. \end{lemma} \begin{proof} Assume that $w_n\in U_\epsilon$ satisfies that $\Psi_\epsilon(w_n)\to \inf_{U_\epsilon}\Psi_\epsilon$. By the Ekeland variational principle, we may suppose that $\Psi'_\epsilon(w_n)\to 0$. Then from Lemma \ref{l3.7} (1) it follows that $I'_\epsilon(u_n)\to 0$, where $u_n=m_\epsilon(w_n)\in N_\epsilon$. By Lemma \ref{l3.7} (2), we have $I_\epsilon(u_n)=\Psi_\epsilon(w_n)\to c_\epsilon$. By Lemma \ref{l2.1}, we obtain that $\{u_n\}$ is bounded in $H^1(\mathbb{R}^3)$. Up to a subsequence, we assume that $u_n\rightharpoonup \tilde{u}_\epsilon$ in $H^1(\mathbb{R}^{3})$, $u_n\to \tilde{u}_\epsilon$ in $L^{2}_{loc}(\mathbb{R}^3)$ and $u_n\to \tilde{u}_\epsilon$ a.e. on $\mathbb{R}^{3}$. Then $I'_\epsilon(\tilde{u}_\epsilon)=0$. Below we discuss for two cases. (i) $\tilde{u}_\epsilon\neq0$ if $\epsilon$ is small enough. Then $\tilde{u}_{\epsilon}\in N_\epsilon$. So $I_\epsilon(\tilde{u}_\epsilon)\geq c_\epsilon$. On the other hand, \begin{equation}\label{4.2.2} \begin{aligned} I_\epsilon(\tilde{u}_\epsilon) -\frac14\langle I'_\epsilon(\tilde{u}_\epsilon),\tilde{u}_\epsilon\rangle &=\frac {1}{4}\|\tilde{u}_\epsilon\|^2_\epsilon \leq \frac {1}{4}\|{u}_n\|^2_\epsilon+o_n(1)\\ &=I_\epsilon({u}_n)-\frac14\langle I'_\epsilon({u}_n),{u}_n\rangle+o_n(1) =c_\epsilon+o_n(1). \end{aligned} \end{equation} Then $I_\epsilon(\tilde{u}_\epsilon)\leq c_\epsilon$. Therefore, $I_\epsilon(\tilde{u}_\epsilon)=c_\epsilon$, and then $u_n\to \tilde{u}_\epsilon$ in $H^1(\mathbb{R}^3)$ by \eqref{4.2.2}. (ii) There exists a sequence $\epsilon_j$ with $\tilde{u}_{\epsilon_j}=0$. For each fixed $j$, there exists a sequence $u_n\in N_{\epsilon_j}$ such that $u_n\rightharpoonup \tilde{u}_{\epsilon_j}=0$ in $H^1(\mathbb{R}^3)$. By $c_{\epsilon_j}>0$ in Lemma \ref{l3.11}, it is easy to see that $\{u_n\}$ is non-vanishing. Then there exists $x_n\in \mathbb{R}^3$ and $\delta_0>0$ such that \begin{equation}\label{4.2.6} \int_{B_1(x_n)}u^2_n(x)dx>\delta_0. \end{equation} Select $\tilde{a}\in (\nu_{\rm min},\nu_\infty)$. Since $u_n\in N_{\epsilon_j}$, we know that $$ |\nabla u_n|^2_2+\int_{\mathbb{R}^3} V(\epsilon_j x)u^2_n +\int_{\mathbb{R}^3}\phi_{ u_n}u^2_n=\int_{\mathbb{R}^3} Q(\epsilon_j x)u^4_n. $$ It is easy to see that \begin{equation}\label{4.3.1} \int_{\mathbb{R}^3}(V^{\tilde{a}}_{\epsilon_j}(x)-V(\epsilon_j x))u^2_n =\int_{\{x| V(\epsilon_jx)\leq \tilde{a}\}}(\tilde{a}-V(\epsilon_jx))u^2_n=o_n(1). \end{equation} Similarly, \begin{equation}\label{4.3.2} \int_{\mathbb{R}^3}(Q^{d}_{\epsilon_j}(x)-Q(\epsilon_j x))u^4_n =\int_{\{x| Q(\epsilon_jx)\geq d\}}(Q(\epsilon_jx)-d)u^4_n=o_n(1). \end{equation} Then $$ |\nabla u_n|^2_2+\int_{\mathbb{R}^3} V^{\tilde{a}}_{\epsilon_j}(x)u^2_n +\int_{\mathbb{R}^3}\phi_{ u_n}u^2_n =\int_{\mathbb{R}^3} Q^{d}_{\epsilon_j}(x)u^4_n+o_n(1). $$ By \eqref{4.2.6}, we obtain that \begin{equation}\label{5.5} \int_{\mathbb{R}^3} Q^{d}_{\epsilon_j}(x)u^4_n -\int_{\mathbb{R}^3}\phi_{ u_n}u^2_n\geq\int_{\mathbb{R}^3} V^{\tilde{a}}_{\epsilon_j}(x)u^2_n+o_n(1) \geq C\int_{\mathbb{R}^3} u^2_n+o_n(1)>\delta>0. \end{equation} Hence, $u_n\in \Theta^{d}_{\epsilon_j}$. Then there exists $t_n>0$ such that $t_nu_n\in N^{\tilde{a},d}_{\epsilon_j}$. We claim that $t_n$ is bounded. Otherwise, assume that $t_n\to \infty$. By $t_nu_n\in N^{\tilde{a},d}_{\epsilon_j}$, we know that \begin{equation}\label{4.2.7} |\nabla u_n|^2_2+\int_{\mathbb{R}^3} V^{\tilde{a}}_{\epsilon_j}(x)u^2_n +t^2_n \int_{\mathbb{R}^3}\phi_{ u_n}u^2_n=t^2_n\int_{\mathbb{R}^3} Q^{d}_{\epsilon_j}(x)u^4_n. \end{equation} By \eqref{5.5}, there is a contradiction for \eqref{4.2.7} if $t_n\to \infty$. Therefore, $t_n$ is bounded. Then \begin{align*} c^{\tilde{a},d}_{\epsilon_j} &\leq I^{\tilde{a},d}_{\epsilon_j}(t_nu_n) \\ &=I_{\epsilon_j}(t_nu_n)+\frac12\int_{\mathbb{R}^3}[V^{\tilde{a}}_{\epsilon_j}(x) -V(\epsilon_j x)]t^2_nu^2_n -\frac14\int_{\mathbb{R}^3}[Q^{d}_{\epsilon_j}(x)-Q(\epsilon_j x)]t^4_nu^4_n. \end{align*} By the boundedness of $t_n$, \eqref{4.3.1} and \eqref{4.3.2}, we obtain $$ c^{\tilde{a},d}_{\epsilon_j}\leq I_{\epsilon_j}(t_nu_n)+o_n(1) \leq I_{\epsilon_j}(u_n)+o_n(1). $$ Hence $c^{\tilde{a},d}_{\epsilon_j}\leq c_{\epsilon_j}$ as $n\to \infty$. However, from Lemma \ref{l3.1}, it holds $$ m_{\tilde{a},d}\leq c^{\tilde{a},d}_{\epsilon_j}, $$ leading to $m_{\tilde{a},d}\leq c_{\epsilon_j}$. Taking the limit $j\to \infty$ and using Lemma \ref{l4.2}, we obtain $$ m_{\tilde{a},d}\leq m_{\nu_{\rm min},d}, $$ which is a contradiction to Lemma \ref{l2.3} since $\tilde{a}>\nu_{\rm min}$. Now we find the ground state $\tilde{u}_\epsilon$ for \eqref{SPe*}. Using the standard argument, we can further find a positive ground state for \eqref{SPe*}. This completes the proof. \end{proof} Denote $\mathscr{J}_{\epsilon}$ as the set of all positive ground states of \eqref{SPe*}. \begin{lemma}\label{l4.3} Let {\rm (A1), (A2)} hold. Then $\mathscr{J}_{\epsilon}$ is compact in $H^{1}(\mathbb{R}^{3})$ for all small $\epsilon>0$. \end{lemma} \begin{proof} Let the bounded sequence $\{u_{n}\}\subset\mathscr{J}_{\epsilon}\cap N_{\epsilon}$ be such that $I_{\epsilon}(u_{n})=c_{\epsilon}$ and $I'_{\epsilon}(u_{n})=0$. Without loss of generality we assume that $u_{n}\rightharpoonup u\in H^1(\mathbb{R}^3)$. As done in the proof of Lemma \ref{l4.6}, we obtain that $u_{n}\to u>0$, $u\in N_\epsilon$ and $I_\epsilon(u)=c_\epsilon$. Then $u\in \mathscr{J}_{\epsilon}$. \end{proof} \begin{lemma} \label{l4.5} Suppose that {\rm (A1), (A2)} are satisfied, and $V, Q$ are uniformly continuous. Let $u_{\epsilon}$ be the positive ground state obtained in Lemma \ref{l4.6}. Then there is $y_{\epsilon}\in\mathbb{R}^3$ such that $\lim_{{\epsilon}\to 0} \operatorname{dist}(\epsilon y_{\epsilon},\mathcal{A}_{V})=0$, and for each sequence $\epsilon y_{\epsilon}\to y_0$, $v_{\epsilon}(x):=u_{\epsilon}(x+y_\epsilon)$ converges in $H^1(\mathbb{R}^3)$ to a ground state $v$ of $$ -\Delta u+V(y_0)u+\phi_u(x)u=Q(y_0)u^3,\quad u>0. $$ In particular, if $\mathcal{V}\cap\mathcal{Q}\neq\emptyset$, it follows that $\lim_{{\epsilon}\to 0} \operatorname{dist}(\epsilon y_{\epsilon}, \mathcal{V}\cap\mathcal{Q})=0$, and up to subsequences, $v_{\epsilon}$ converges in $H^1(\mathbb{R}^3)$ to a ground state $v$ of $$ -\Delta u+\nu_{\rm min}u+\phi_u(x)u=q_{\max}u^3,\quad u>0. $$ \end{lemma} \begin{proof} Let $u_n$ be the positive ground states of problem \eqref{SPe*} with parameter $\epsilon_n\to 0$. It is easy to see that $u_n$ is bounded and non-vanishing. Then there exists $\delta>0$ such that \begin{equation}\label{4.3.4} \int_{B_1(y_n)}|u_n|^2dx\geq\delta. \end{equation} Setting $v_n(x)=u_n(x+y_n)$, $\tilde{V}_{\epsilon_n}(x)=V(\epsilon_n(x+y_n))$ and $\tilde{Q}_{\epsilon_n}(x)=Q(\epsilon_n(x+y_n))$, we see that $v_n$ solves the below problem $$ -\Delta u+\tilde{V}_{\epsilon_n}(x)u+\phi_u(x)u=\tilde{Q}_{\epsilon_n}(x)u^3, $$ with energy functional $$ \tilde{I}_{\epsilon_n}(v_n) =\frac12\bigl(\int_{\mathbb{R}^3}{|\nabla {v_n}|}^2+\tilde{V}_{\epsilon_n}(x)v^2_n \bigr)+\frac 14\int_{\mathbb{R}^3}\phi_{v_n} v^2_n -\frac14\int_{\mathbb{R}^3}\tilde{Q}_{\epsilon_n}(x)v^4_n. $$ Since $v_n$ is also bounded in $H^1(\mathbb{R}^3)$, from \eqref{4.3.4}, we may assume that $v_n\rightharpoonup v\neq0$ in $H^1(\mathbb{R}^3)$. \smallskip \noindent\textbf{Claim 1:} The sequence $\epsilon_ny_n$ must be bounded. Otherwise if $\epsilon_ny_n\to \infty$, then we may suppose that $ V(\epsilon_ny_n)\to V_0\geq \nu_\infty>\nu_{\rm min}$ and $Q(\epsilon_ny_n)\to Q_0\leq d:= Q(0)$. Since $V$ and $Q$ are uniformly continuous functions, it follows that for $R>0$ and $|x|\leq R$, $$ |\tilde{V}_{\epsilon_n}(x)-V_0| \leq|V(\epsilon_n(x+y_n))-V(\epsilon_ny_n)|+|V(\epsilon_ny_n)-V_0|\to 0. $$ Similarly, $$ |\tilde{Q}_{\epsilon_n}(x)-Q_0|\to 0,\quad \forall |x|\leq R. $$ Then for each $\eta\in C^\infty_0(\mathbb{R}^3)$, we infer that \begin{equation}\label{d22} \lim_{n\to \infty}\int_{\mathbb{R}^3}\tilde{V}_{\epsilon_n}(x)v_n\eta =\int_{\mathbb{R}^3}{V}_{0}v\eta ,\quad \lim_{n\to \infty}\int_{\mathbb{R}^3}\tilde{Q}_{\epsilon_n}(x)v^3_n\eta =\int_{\mathbb{R}^3}{Q}_{0}v^3\eta. \end{equation} Moreover, by Lemma \ref{l2.1.1} (ii), we have $$ \lim_{n\to \infty}\int_{\mathbb{R}^3}\phi_{v_n}v_n\eta =\int_{\mathbb{R}^3}\phi_{v}v\eta. $$ Thus, $v$ solves \begin{equation} -\Delta v+V_0v+\phi_v(x)v=Q_0v^{3}\quad \text{in } \mathbb{R}^{3}.\label{SPvq} \end{equation} From Fatou's lemma and Lemma \ref{l4.2} it follows that \begin{equation}\label{4.3.5} \begin{split} m_{\nu_{\rm min},d} &\geq\liminf_{n\to \infty}c_{\epsilon_{n}} =\liminf_{n\to \infty}[I_{\epsilon_{n}}(u_{n}) -\frac{1}{4}\langle I'_{\epsilon_{n}}(u_{n}),(u_{n})\rangle]\\ &=\liminf_{n\to \infty}[\tilde{I}_{\epsilon_{n}}(v_{n}) -\frac{1}{4}\langle\tilde{I}'_{\epsilon_{n}}(v_{n}),(v_{n})\rangle]\\ &=\liminf_{n\to \infty}\frac{1}{4}\int_{\mathbb{R}^{3}}(|\nabla v_{n}|^{2}+\tilde{V}_{\epsilon_{n}}(x)|v_{n}|^{2})\\ &\geq\frac{1}{4}\int_{\mathbb{R}^{3}}(|\nabla v|^{2}+V_0|v|^{2})\\&={J}_{V_0,Q_0}(v)-\frac14\langle{J}'_{V_0,Q_0}(v), v\rangle\geq m_{V_0,Q_0}. \end{split} \end{equation} However, from the fact that $\nu_{\rm min}Q(x_{\rm min})\}. \] If $y_0\not\in \mathcal{A}_{V}$, then it is easy to see that \begin{equation}\label{4.3.9} m_{\nu_{\rm min},d}0. $$ \end{proof} \begin{lemma}\label{l5.1} Suppose that {\rm (A1), (A2)} are satisfied, and $V,Q$ are uniformly continuous. Set ${v}_{n}:=u_n(x+y_n)$, where $u_n$ is the positive ground state obtained in Lemma \ref{l4.6} and $y_n$ is given in \eqref{4.3.4}. Then: \begin{itemize} \item[(i)] there exist $\delta'$ and $M>0$ such that $\delta'\leq|{v}_n|_\infty\leq M$ for all $n\in\mathbb{N}$. \item[(ii)] $$ \lim_{|x|\to \infty}{v}_n(x)=0\quad \text{uniformly in }n\in\mathbb{N}. $$ Moreover, there exist $C,c>0$ such that $$ {v}_{n}(x)\leq C e^{-c|x|},\quad \forall x\in\mathbb{R}^3. $$ \end{itemize} \end{lemma} \begin{proof} As in the proof of Lemma \ref{l4.5}, we have that ${v}_n$ is the solution of $$ -\Delta v_n+\tilde{V}_{\epsilon_n}(x)v_n+\phi_{v_n}(x)v_n =\tilde{Q}_{\epsilon_n}(x)v^3_n, $$ and $v_n\to v\neq0$ in $H^1(\mathbb{R}^3)$. Then \begin{equation}\label{5.2} \lim_{R\to \infty}\bigl(\int_{|x|\geq R}(v^2_n+v^6_n)\bigr)=0, \quad \text{uniformly for } n\in\mathbb{N}. \end{equation} Using \cite[Proposition 3.3]{He2}, we obtain that $v_n\in L^t(\mathbb{R}^3)$ for all $t\geq2$ and \begin{equation*} |v_n|_t\leq C_{t}\|v_n\|_{H^1(\mathbb{R}^3)}, \end{equation*} where $C_t$ does not depend on $n$. Then for $t>3$, $|v^3_n|_{\frac t2}\leq C$ for all $n$. Thus by \cite[Theorem 8.17]{Gilbarg}, we infer that for all $y\in\mathbb{R}^3$, \begin{equation}\label{5.4} \sup_{B_1(y)}v_n(x) \leq C\bigl(|v_n|_{L^2(B_2(y))}+|v^3_n|_{L^{\frac t2}(B_2(y))}\bigr). \end{equation} This implies that $|v_n|_\infty$ is uniformly bounded. Recall that by \eqref{4.3.4}, $$ \delta\leq\int_{B_1(y_n)}|u_n|^2dx\leq|B_1||v_n|^2_\infty. $$ Then $|v_n|_\infty\geq\delta'$, for all $n$. Combining \eqref{5.4} with \eqref{5.2}, we obtain $$ \lim_{|x|\to \infty}v_n(x)=0\ \text{uniformly for all}\ n\in\mathbb{N}. $$ Then we can take $\rho_0>0$ such that $$ {\tilde{Q}_{\epsilon_n}(x)}v^3_n\leq \frac{\nu_{\rm min}}{2}v_n, $$ for all $|x|>\rho_0$. Consequently, $$ -\Delta v_n+\frac{\tilde{V}_{\epsilon_n}(x)}{2}v_n\ leq{\tilde{Q}_{\epsilon_n}(x)}v^3_n-\frac{\tilde{V}_{\epsilon_n}(x)}{2}v_n\leq0, $$ for all $|x|\geq \rho_0$. Let $s$ and $T$ be positive constants such that $$ s^2<\frac{\nu_{\rm min}}{2}, \quad v_n(x)\leq Te^{-s{\rho_0}}, $$ for all $|x|=\rho_0$. Hence, the function $\psi(x)=Texp(-s|x|)$ satisfies $$ -\Delta \psi+\frac{\tilde{V}_{\epsilon_n}(x)}{2}\psi\geq(\frac{\nu_{\rm min}}{2}-s^2) \psi>0, $$ for all $x\neq0$. Thereby, taking $\eta=\max\{v_n-\psi,0\}\in H^1_0(|x|>\rho_0)$ as a test function, we obtain \begin{align*} 0&\geq\int_{\mathbb{R}^3}\bigl(\nabla v_n\nabla \eta +\frac{\tilde{V}_{\epsilon_n}(x)}{2}v_n\eta\bigr)\\ &\geq \int_{\mathbb{R}^3}\bigl((\nabla v_n-\nabla\psi)\nabla \eta +\frac{\tilde{V}_{\epsilon_n}(x)}{2}(v_n-\psi)\eta\big) \\ &\geq \frac{\nu_{\rm min}}{2}\int_{\{x\in\mathbb{R}^3:v_n>\psi\}}(v_n-\psi)^2 \geq0, \end{align*} for all $|x|>\rho_0$. Therefore, the set $\Omega_n:=\{x\in\mathbb{R}^3:|x|>\rho_0 \text{ and }v_n>\psi(x)\}$ is empty. Then we know that there exists $C,c>0$ such that $$ v_n(x)\leq C e^{-c|x|}, \forall x\in\mathbb{R}^3. $$ This completes the proof. \end{proof} \begin{proof}[Proof of Theorem \ref{thm1.1}] Going back to \eqref{SPe} with the substitution of variables $x\mapsto\frac{x}{\epsilon}$, Lemma \ref{l4.6} implies that \eqref{SPe} has a positive ground state $w_\epsilon=u_\epsilon(\frac{x}{\epsilon})$ for $\epsilon>0$ small. Lemma \ref{l4.3} implies that $\mathscr{L}_{\epsilon}$ is compact in $H^1(\mathbb{R}^3)$. Set $\epsilon_n\to 0$ as $n\to \infty$. If $b_n$ denotes a maximum point of $v_n$, then from Lemma \ref{l5.1} (i), it follows that it is bounded. Then we assume that $b_n\in B_R(0)$. Thereby, the global maximum point of $u_n$ is $z_n:=b_n+y_n$ and then $x_n:=\epsilon_nz_n$ is the maximum point of $w_n$. From the boundedness of $b_n$, by Lemma \ref{l4.5} we obtain that $\lim_{n\to \infty}x_n=y_0$, which together with the continuity of $V$ gives $$ \lim_{n\to \infty}V(x_n)=V(y_0),\quad \lim_{n\to \infty}Q(x_n)=Q(y_0). $$ Then from Lemma \ref{l4.5}, the proof of the conclusion (3)(i) in Theorem \ref{thm1.1} is complete. Moreover, from Lemma \ref{l5.1}, by the boundedness of $b_n$ we obtain $$ w_n(x)=u_n(\frac{x}{\epsilon_n})={v}_n(\frac{x}{\epsilon_n}-y_n) ={v}_n(\frac{x}{\epsilon_n}-\frac{x_n}{\epsilon_n}-b_n) \leq Ce^{-c|\frac{x}{\epsilon_n}-\frac{x_n}{\epsilon_n}-b_n|} \leq Ce^{-\frac{c}{\epsilon_n}|{x-x_n}|}. $$ Thus, for small $\epsilon>0$, we have that $$ w_\epsilon(x)\leq Ce^{-\frac{c}{\epsilon}|{x-x_\epsilon}|}. $$ \end{proof} \begin{proof}[Proof of Theorem \ref{thm1.2}] If the potential functions $V$ and $Q$ satisfy condition (A3), we can assume that $x_{\rm max}:=0\in \mathcal{Q}$ in (A3) or $x_{\rm max}:=0\in \mathcal{V}\cap \mathcal{Q}$ if $\mathcal{V}\cap \mathcal{Q}\neq\emptyset$. Denote $$ e:=V(0)=\min_{x\in \mathcal{Q}}V(x)\leq V(x),\quad \forall |x|\geq R. $$ The rest is similar to the proof of Theorem \ref{thm1.1}. \end{proof} \subsection*{Acknowledgements} This work was supported by the National Natural Science Foundation of China (Nos. 11601204,11671077,11571140) and by the Natural Science Foundation of Outstanding Young Scholars of Jiangsu Province (No: BK20160063). The authors would like to express their sincere gratitude to the anonymous referees for their helpful and insightful comments. \begin{thebibliography}{99} \bibitem{ALVES1} Claudianor O. Alves, Marco A. S. Souto, S\'{e}rgio H. M. Soares; \emph{A sign-changing solution for the Schr\"odinger-Poisson equation in $\mathbb{R}^3$}, Rocky Mountain J. Math., 47 (2017), 1-25. \bibitem{ALVES2} Claudianor O. Alves, Marco A. S. Souto, S\'{e}rgio H. M. Soares; \emph{Schr\"odinger-Poisson equations without Ambrosetti-Rabinowitz condition}, J. Math. Anal. Appl., 377 (2011), 584-592. \bibitem{AM1} A. 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