\documentclass[reqno]{amsart} \usepackage{hyperref} \AtBeginDocument{{\noindent\small \emph{Electronic Journal of Differential Equations}, Vol. 2015 (2015), No. 296, pp. 1--15.\newline ISSN: 1072-6691. URL: http://ejde.math.txstate.edu or http://ejde.math.unt.edu \newline ftp ejde.math.txstate.edu} \thanks{\copyright 2015 Texas State University.} \vspace{9mm}} \begin{document} \title[\hfilneg EJDE-2015/296\hfil Existence and concentration ] {Existence and concentration of positive bound states for Schr\"odinger-Poisson systems with potential functions} \author[P. L. Cunha \hfil EJDE-2015/296\hfilneg] {Patr\'icia L. Cunha} \address{Patr\'icia L. Cunha \newline Departamento de Inform\'atica e M\'etodos Quantitativos, Funda\c{c}\~ao Getulio Vargas, S\~ao Paulo, Brazil} \email{patcunha80@gmail.com} \thanks{Submitted July 8, 2015. Published November 30, 2015.} \subjclass[2010]{35B40, 35J60, 35Q55} \keywords{Schr\"odinger-Poisson system; variational methods; concentration} \begin{abstract} In this article we study the existence and concentration behavior of bound states for a nonlinear Schr\"odinger-Poisson system with a parameter $\varepsilon>0$. Under suitable conditions on the potential functions, we prove that for $\varepsilon$ small the system has a positive solution that concentrates at a point which is a global minimum of the minimax function associated to the related autonomous problem. \end{abstract} \maketitle \numberwithin{equation}{section} \newtheorem{theorem}{Theorem}[section] \newtheorem{lemma}[theorem]{Lemma} \newtheorem{proposition}[theorem]{Proposition} \newtheorem{remark}[theorem]{Remark} \allowdisplaybreaks \section{Introduction} In this article we study the Schr\"odinger-Poisson system \begin{equation} \begin{gathered} -\varepsilon^2\Delta v+ V(x)v+K(x)\phi(x) v= |v|^{q-2}v \quad \text{in } \mathbb{R}^3\\ -\Delta \phi=K(x)v^{2} \quad \text{in } \mathbb{R}^3 \end{gathered} \label{SPe} \end{equation} where $\varepsilon>0$ is a parameter, $q\in(4,6)$ and $V,K:\mathbb{R}^3\to\mathbb{R}$ are, respectively, an external potential and a charge density. The unknowns of the system are the field $u$ associated with the particles and the electric potential $\phi$. We are interested in the existence and concentration behavior of solutions of \eqref{SPe} in the semiclassical limit $\varepsilon\to 0$. The first equation of \eqref{SPe} is a nonlinear equation in which the potential $\phi$ satisfies a nonlinear Poisson equation. For this reason, \eqref{SPe} is called a Schr\"odinger-Poisson system, also known as Schr\"odinger-Maxwell system. For more information about physical aspects, we refer the reader to \cite{Benci-Fortunato-1998,D'Aprile-Mugnai} and references therein. We observe that when $\phi\equiv 0$, \eqref{SPe} reduces to the well known Schr\"odinger equation \begin{equation} -\varepsilon^2\Delta u+ V(x)u = f(x,u) \quad x\in\mathbb{R}^N \label{S}. \end{equation} In the previous years, the nonlinear stationary Schr\"odinger equation has been widely investigated, mainly in the semiclassical limit as $\varepsilon\to 0$ (see e.g. \cite{Rabinowitz,Wang,Wang-Zeng} and its references). Rabinowitz \cite{Rabinowitz} studied problem \eqref{S} using mountain pass arguments to find least energy solutions, for $\varepsilon>0$ sufficiently small. Then, Wang \cite{Wang} proved that the solution in \cite{Rabinowitz} concentrates around the global minimal of $V$ when $\varepsilon$ tends to 0. Wang and Zeng \cite{Wang-Zeng} considered the Schr\"odinger equation \begin{equation} -\varepsilon^2\Delta u+V(x)u =K(x) |u|^{p-1}u+Q(x) |u|^{q-1}u, \quad x\in\mathbb{R}^N \label{WZ} \end{equation} where $10$ sufficiently small, they proved the existence of a solution $u_\varepsilon$ for \eqref{WZ}, whose global maximum approaches to a point $y^*$ when $\varepsilon$ tends to 0. Moreover, under suitable hypothesis on the potentials $V$ and $W$, the function $\xi\mapsto C(\xi)$ assumes a minimum at $y^*$. Motivated by these results, Alves and Soares \cite{Alves-Soares} investigated the same phenomenon for the gradient system \begin{equation} \begin{gathered} -\varepsilon^2\Delta u+ V(x)u= Q_u(u,v) \quad \text{in } \mathbb{R}^{N}\\ -\varepsilon^2\Delta v+ W(x)v= Q_v(u,v) \quad \text{in } \mathbb{R}^{N}\\ u(x), v(x)\to 0, \quad \text{as } |x|\to\infty \\ u,v>0 \quad \mathbb{R}^{N} \end{gathered}\label{AS} \end{equation} In this system is natural to expect some competition between the potentials $V$ and $W$, each one trying to attract the local maximum points of the solutions to its minimum points. In fact, in \cite{Alves-Soares} the authors proved that functions $u_\varepsilon$ and $v_\varepsilon$ satisfies \eqref{AS} and concentrate around the same point which is the minimum of the respective function $C(s)$. Ianni and Vaira \cite{Ianni-Vaira} studied the Schr\"odinger-Poisson system \eqref{SPe} proving that if $V$ has a non-degenerated critical point $x_0$, then there exists a solution that concentrates around this point. Moreover, they also proved that if $x_0$ is degenerated for $V$ and a local minimum for $K$, then there exist a solution concentrating around $x_0$. The proof was based in the Lyapunov-Schmidt reduction. The double parameter perturbation was also considered for system \eqref{SPe} by \cite{He,He-Zou2012}. He and Zhou \cite{He-Zou2012} studied the existence and behavior of a ground state solution which concentrates around the global minimum of the potential $V$. They considered $K\equiv 1$ and the presence of the nonlinear term $f(x,u)$. Yang and Han \cite{Yang-Han} studied the Schr\"odinger-Poisson system \begin{equation} \begin{gathered} -\Delta v+ V(x)v+K(x)\phi(x) v= |v|^{q-2}v \quad \text{in } \mathbb{R}^3\\ -\Delta \phi=K(x)v^{2} \quad \text{in } \mathbb{R}^3 \end{gathered} \label{SP}. \end{equation} Under suitable assumptions on $V$, $K$ and $f$ they proved existence and multiplicity results by using the mountain pass theorem and the fountain theorem. Later, Zhao, Liu and Zhao \cite{Zhao-Liu-Zhao}, using variational methods, proved the existence and concentration of solutions for the system \begin{equation} \begin{gathered} -\Delta v+ \lambda V(x)v+K(x)\phi(x) v= |v|^{q-2}v \quad \text{in } \mathbb{R}^3\\ -\Delta \phi=K(x)v^{2} \quad \text{in } \mathbb{R}^3 \end{gathered} \end{equation} when $\lambda>0$ is a parameter and $20$ such that $V(x), K(x)\geq \alpha>0$ for all $x\in\mathbb{R}^3$, \item[(H1)] $V(x)$ and $K(x)$ are continuous functions and $V_\infty, K_\infty$ are defined by \begin{gather*} V_\infty=\liminf _{|x|\to\infty}V(x)>\inf_{x\in\mathbb{R}^3}V(x)\\ K_\infty=\liminf _{|x|\to\infty}K(x)>\inf_{x\in\mathbb{R}^3}K(x). \end{gather*} \end{itemize} We prove that if \[ C_\infty>\inf_{\xi\in\mathbb{R}^3} C(\xi), \] then \eqref{SPe} has a positive solution $v_\varepsilon$ as $\varepsilon$ tends to zero. After passing to a subsequence, $v_\varepsilon$ concentrates at a global minimum point of $C(\xi)$ for $\xi\in\mathbb{R}^3$, where the energy function $C(\xi)$ is defined to be the minimax function associated with the problem \begin{equation} \begin{gathered} -\Delta u+ V(\xi)u+K(\xi)\phi(\xi) u= |u|^{q-2}u \quad \text{in } \mathbb{R}^3\\ -\Delta \phi=K(\xi)u^{2} \quad \text{in } \mathbb{R}^3 \end{gathered} \label{SPx} \end{equation} Therefore, $C(\xi)$ plays a central role in our study. The main result for \eqref{SPe}) reads as follows. \begin{theorem}\label{principal} Suppose {\rm (H0)--(H1)} hold. If \begin{equation}\label{Cinfty} C_\infty>\inf_{\xi\in\mathbb{R}^3} C(\xi), \end{equation} then there exists $\varepsilon^*>0$ such that system \eqref{SPe}) has a positive solution $v_\varepsilon$ for $\varepsilon\in(0,\varepsilon^*)$. Moreover, $v_{\varepsilon}$ concentrates at a local (hence global) maximum point $y^*\in\mathbb{R}^3$ such that $$ C(y^*)=\min_{\xi\in\mathbb{R}^3}C(\xi). $$ \end{theorem} Theorem \ref{principal} complements the study made in \cite{Fang-Zhang,Ianni-Vaira,Yang-Han,Zhao-Liu-Zhao} in the following sense: we deal with the perturbation problem \eqref{SPe} and study the concentration behavior of positive bound states. To the best of our knowledge, the only previous article regarding the concentration of solutions for the perturbed Schr\"odinger-Poisson system with potentials $V$ and $K$ is \cite{Ianni-Vaira}, where the smoothness of such potentials is considered. We only need the boundedness of $V$ and $K$. Moreover, we do not assume that the concentration point of solutions $v_\varepsilon$ for the system \eqref{SPe} is a local minimum (or maximum) of such potentials, as in the previous paper. In our research we shall consider a different variational approach. The outline of this paper is as follows: in Section 2 we set the variational framework. In Section 3 we study the autonomous system related to \eqref{SPe}. In section 4 we establish an existence result for system \eqref{SPe} with $\varepsilon=1$. In section 5, we prove Theorem \ref{principal}. \section{Variational framework and preliminary results} Throughout this article we use the following notation: $\bullet$ $H^1(\mathbb{R}^3)$ is the usual Sobolev space endowed with the standard scalar product and norm \[ (u,v)=\int_{\mathbb{R}^3}(\nabla u\nabla v+uv)\,dx, \quad \|u\|^2=\int_{\mathbb{R}^3}(|\nabla u|^2+u^2)\,dx. \] $\bullet$ $\mathcal{D}^{1,2}=\mathcal{D}^{1,2}(\mathbb{R}^3)$ represents the completion of $C_0^\infty(\mathbb{R}^3)$ with respect to the norm \[ \|u\|_{\mathcal{D}^{1,2}}^2=\int_{\mathbb{R}^3}|\nabla u|^2\,dx. \] $\bullet$ $L^p(\Omega)$, $1\leq p\leq \infty$, $\Omega\subset\mathbb{R}^3$, denotes a Lebesgue space; the norm in $L^p(\Omega)$ is denoted by $\|u\|_{L^p(\Omega)}$, where $\Omega $ is a proper subset of $\mathbb{R}^3$; $\|u\|_p$ is the norm in $L^p(\mathbb{R}^3)$. We recall that by the Lax-Milgram theorem, for every $v\in H^1(\mathbb{R}^3)$, the Poisson equation $-\Delta \phi=v^{2}$ has a unique positive solution $\phi=\phi_{v}\in \mathcal{D}^{1,2}(\mathbb{R}^3)$ given by \begin{equation}\label{1} \phi_{v}(x)=\int_{\mathbb{R}^3}\frac{v^{2}(y)}{|x-y|}\,dy. \end{equation} The function $\phi: H^{1}(\mathbb{R}^3)\to \mathcal{D}^{1,2}(\mathbb{R}^3)$, $\phi[v]=\phi_v$ has the following properties (see for instance Cerami and Vaira \cite{Cerami-Vaira}). \begin{lemma} \label{propriedade phi} For any $v\in H^{1}(\mathbb{R}^3)$, we have \begin{itemize}\item[(i)] $\phi$ is continuous and maps bounded sets into bounded sets; \item [(ii)] $\phi_v\geq 0$; \item [(iii)] there exists $C>0$ such that $\|\phi\|_{D^{1,2}}\leq C\|v\|^2$ and $$ \int_{\mathbb{R}^3}|\nabla v |^2\, dx =\int_{\mathbb{R}^3}\phi_{v} v^2\, dx\leq C\|v\|^4; $$ \item [(iv)] $\phi_{tv}=t^2\phi_{v}$, $\forall\,t>0$; \item [(v)] if $v_n\rightharpoonup v$ in $H^1(\mathbb{R}^3)$, then $\phi_{v_n}\rightharpoonup\phi_v$ in $\mathcal{D}^{1,2}(\mathbb{R}^3)$. \end{itemize} \end{lemma} As in \cite{Ambrosetti}, for every $v\in H^1(\mathbb{R}^3)$, there exist a unique solution $\phi=\phi_{K,v}\in \mathcal{D}^{1,2}(\mathbb{R}^3)$ of $-\Delta \phi=K(x)v^{2}$ where \begin{equation}\label{1b} \phi_{K,v}(x)=\int_{\mathbb{R}^3}\frac{K(y)v^{2}(y)}{|x-y|}dy. \end{equation} and it is easy to see that $\phi_{K,v}$ satisfies Lemma \ref{propriedade phi} if $K$ satisfies conditions (H0)--(H1). Substituting \eqref{1b} into the first equation of \eqref{SPe}, we obtain \begin{equation}\label{2} -\varepsilon^2\Delta v+ V(x)v+K(x)\phi_{K,v}(x) v= |v|^{q-2}v. \end{equation} Making the changing of variables $x\mapsto \varepsilon x$ and setting $u(x)=v(\varepsilon x)$, \eqref{2} becomes \begin{equation}\label{3} -\Delta u+ V(\varepsilon x)u+K(\varepsilon x)\phi_{K,v}(\varepsilon x) u = |u|^{q-2}u. \end{equation} A simple computation shows that $$ \phi_{K,v}(\varepsilon x)=\varepsilon^2 \phi_{\varepsilon,u}(x), $$ where $$ \phi_{\varepsilon,u}(x)=\int_{\mathbb{R}^3}\frac{K(\varepsilon y)u^2(y)}{|x-y|}dy. $$ Substituting this into \eqref{3}, Equation \eqref{SPe} can be rewritten in the equivalent equation \begin{equation} \label{4} %\label{Se} -\Delta u+ V(\varepsilon x)u+\varepsilon^2 K(\varepsilon x)\phi_{\varepsilon,u} u = |u|^{q-2}u. \end{equation} Note that if $u_\varepsilon$ is a solution of \eqref{4}, then $v_\varepsilon(x)=u_\varepsilon(x/\varepsilon)$ is a solution of \eqref{2}. We denote by $H_\varepsilon=\{u\in H^{1}(\mathbb{R}^3): \int_{\mathbb{R}^3}V(\varepsilon x)u^2<\infty\}$ is a Sobolev space endowed with the norm \[ \|u\|_{\varepsilon}^{2}=\int_{\mathbb{R}^3}(|\nabla u|^2+V(\varepsilon x)u^2)\,dx. \] At this step, we see that \eqref{Se} is variational and its solutions are critical points of the functional \[ \mathcal{I}_\varepsilon(u)=\frac{1}{2} \int_{\mathbb{R}^3}(|\nabla u|^2+V(\varepsilon x)u^2)\,dx+ \frac{\varepsilon^2}{4}\int_{\mathbb{R}^3}K(\varepsilon x) \phi_{\varepsilon,u}(x) u^2\,dx-\frac{1}{q}\int_{\mathbb{R}^3}|u|^q\,dx. \] \section{Autonomous Case} In this section we study the autonomous system \begin{equation} \begin{gathered} -\Delta u+ V(\xi)u+K(\xi)\phi(x) u= |u|^{q-2}u \quad \text{in } \mathbb{R}^3\\ -\Delta \phi=K(\xi)u^{2} \quad \text{in } \mathbb{R}^3 \end{gathered} \label{SPx2} \end{equation} where $\xi\in\mathbb{R}^3$. To this system we associate the functional $I_\xi: H_\xi\mapsto\mathbb{R}$, \begin{equation}\label{Ixi} I_\xi(u)=\frac{1}{2}\int_{\mathbb{R}^3}(|\nabla u|^2+V(\xi) u^2)\,dx +\frac{1}{4}\int_{\mathbb{R}^3}K(\xi)\phi_u(x) u^2\,dx-\frac{1}{q}\int_{\mathbb{R}^3}|u|^q\,dx. \end{equation} Hereafter, the Sobolev space $H_\xi=H^1(\mathbb{R}^3)$ is endowed with the norm \[ \|u\|_\xi=\int_{\mathbb{R}^3}(|\nabla u|^2 +V(\xi) u^2)\,dx. \] By standard arguments, the functional $I_\xi$ verifies the Mountain-Pass Geometry, more exactly it satisfies the following lemma. \begin{lemma}\label{mountain pass geometry} The functional $I_\xi$ satisfies \begin{itemize} \item[(i)] There exist positive constants $\beta,\rho$ such that $I_\xi(u)\geq\beta$ for $\|u\|_\xi=\rho$, \item[(ii)] There exists $u_{1}\in H^1(\mathbb{R}^3)$ with $\|u_{1}\|_\xi>\rho$ such that $I_\xi(u_{1})< 0$. \end{itemize} \end{lemma} Applying a variant of the Mountain Pass Theorem (see \cite{Willem}), we obtain a sequence $(u_n)\subset H^1(\mathbb{R}^3)$ such that \[ I_\xi(u_{n})\to C(\xi) \quad\text{and}\quad I_\xi'(u_{n})\to 0, \] where \begin{gather}\label{valorC} C(\xi)=\inf_{\gamma\in\Gamma}\max_{0\leq t\leq 1}I_\xi(\gamma(t)), \quad C(\xi)\geq\alpha, \\ \Gamma=\{\gamma\in\mathcal{C}([0,1],H^1(\mathbb{R}^3) )|\gamma(0)=0, \gamma(1)=u_{1} \}. \end{gather} We observe that $C(\xi)$ can be also characterized as \[ C(\xi)=\inf_{u\neq 0}\max_{t>0}I_\xi(tu). \] \begin{proposition}\label{prop1} Let $\xi\in\mathbb{R}^3$. Then system \eqref{SPx2} has a positive solution $u\in H^1(\mathbb{R}^3)$ such that $I'_\xi(u)=0$ and $I_\xi(u)=C(\xi)$, for any $q\in(4,6)$. \end{proposition} The proof of the above propostion is an easy adaptation of Azzollini and Pomponio \cite[Theorem 1.1]{Azzollini-Pomponio-SM} and we omit it. \begin{lemma} \label{lem3.2} The function $\xi\mapsto C(\xi)$ is continuous. \end{lemma} \begin{proof} The proof consists in proving that there exist sequences $(\zeta_n)$ and $(\lambda_n)$ in $\mathbb{R}^3$ such that $C(\zeta_n), C(\lambda_n)\to C(\xi)$ as $n\to 0$, where \begin{itemize} \item $\zeta_n\to\xi $ and $C(\zeta_n)\geq C(\xi)$ for all $n$, \item $\lambda_n\to \xi$ and $C(\lambda_n)\geq C(\xi)$ for all $n$, \end{itemize} as we know by Alves and Soares \cite{Alves-Soares}. \end{proof} \section{System \eqref{SPe} with $\varepsilon=1$}\label{S-1} Setting $\varepsilon=1$, in this section we consider the system \begin{equation} \begin{gathered} -\Delta u+ V(x)u+K(x)\phi(x) u= |u|^{q-2}u \quad \text{in } \mathbb{R}^3\\ -\Delta \phi=K(x)u^{2} \quad \text{in } \mathbb{R}^3 \end{gathered}\label{SP1} \end{equation} whose solutions are critical points of the corresponding functional \[ I(u)=\frac{1}{2}\int_{\mathbb{R}^3}(|\nabla u|^2+V( x)u^2)\,dx + \frac{1}{4}\int_{\mathbb{R}^3}K(x)\phi_{u}(x) u^2\,dx -\frac{1}{p}\int_{\mathbb{R}^3}|u|^q\,dx \] which is well defined for $u\in H_1$, where \[ H_1=\{u\in H^{1}(\mathbb{R}^3): \int_{\mathbb{R}^3}V(x)u^2\,dx<\infty\} \] with the same norm notation of the Sobolev space $H^1(\mathbb{R}^3)$. Similar to the autonomous case, the functional $I$ satisfies the mountain pass geometry, then there exists a sequence $(u_n)\subset H_1$ such that \begin{equation}\label{5} I(u_n)\to c \quad \text{and} \quad I'(u_n)\to 0\,, \end{equation} where \begin{gather*} c=\inf_{\gamma\in\Gamma}\max_{0\leq t\leq 1}I(\gamma(t)), \\ \Gamma=\{\gamma\in\mathcal{C}([0,1],H_{1}(\mathbb{R}^3)) |\gamma(0)=0, I(\gamma(1))<0 \}. \end{gather*} \begin{remark} \label{rmk4.1} \rm The function $(\mu,\nu)\mapsto c_{\mu,\nu}$ is continuous, where $c_{\mu,\nu}$ is the minimax level of \begin{equation}\label{Imu} I_{\mu,\nu}(u)=\frac{1}{2}\int_{\mathbb{R}^3}(|\nabla u|^2 +\mu u^2)\,dx+\frac{1}{4}\int_{\mathbb{R}^3}\nu\phi_u(x) u^2\,dx-\frac{1}{q}\int_{\mathbb{R}^3}|u|^q\,dx. \end{equation} \end{remark} \begin{remark} \label{rmk4.2} \rm We denote by $C_\infty$ the minimax value related to the functional \[ I_\infty(u)=\frac{1}{2}\int_{\mathbb{R}^3}(|\nabla u|^2+V_\infty u^2)\,dx +\frac{1}{4}\int_{\mathbb{R}^3}K_\infty\phi_u u^2\,dx -\frac{1}{q}\int_{\mathbb{R}^3}|u|^q\,dx\,, \] where $V_\infty$ and $K_\infty$, given by condition $(H_1)$, belong to $(0,\infty)$. Otherwise, define $C_\infty=\infty$. $I_\infty(u)$ is well defined for $u\in E_\infty$, where $E_\infty$ is a Sobolev space endowed with the norm $$ \|u\|_\infty=\int_{\mathbb{R}^3}(|\nabla u|^2 +V_\infty u^2)\,dx $$ equivalent to the usual Sobolev norm on $H^1(\mathbb{R}^3)$. \end{remark} An important tool in our analysis is the following theorem. \begin{theorem}\label{critical value} If $c0, \] then $u\neq 0$. By contradiction, consider $u\equiv 0$. Hence, there exists a subsequence of $(u_n)$, still denoted by $(u_n)$, such that \[ \lim_{n\to+\infty}\int_{B_R(0)}u_n^2\,dx=0. \] Let $\mu$ and $\nu$ be such that \[ \inf_{x\in\mathbb{R}^3}V(x)<\mu<\liminf_{|x|\to\infty}V(x)=V_{\infty}\\ \inf_{x\in\mathbb{R}^3}K(x)<\nu<\liminf_{|x|\to\infty}K(x)=K_{\infty} \] and take $R>0$ such that \[ V(x)>\mu, \quad \forall\,x\in \mathbb{R}^3\backslash B_R(0)\\ K(x)>\nu, \quad \forall\,x\in \mathbb{R}^3\backslash B_R(0). \] For each $n\in \mathbb{N}$, there exist $t_n>0$, $t_n\to 1$ such that $ I(t_n u_n)=\max_{t\geq 0}I(t u_n)$. The convergence of $(t_n)$ follows from \eqref{5}. In fact, since $I'(u_n)u_n=o_n(1)$ and $I'(t_n u_n)t_n u_n = o_n(1)$, we have \[ \|u_n\|^2+\int_{\mathbb{R}^3}K(x)\phi_{u_n} u_n^2\,dx =\int_{\mathbb{R}^3}|u_n|^q\,dx+o_n(1) \] we have \[ t_n^2\|u_n\|^2+t_n^4\int_{\mathbb{R}^3}K(x)\phi_{u_n} u_n^2\,dx =t_n^q\int_{\mathbb{R}^3}|u_n|^q\,dx+o_n(1). \] Then \[ (1-t_n^2)\|u_n\|^2=(t_n^{q-2}-t_n^2)\int_{\mathbb{R}^3}|u_n|^q\,dx + o_n(1) \] Observe that $t_n$ neither converge to 0 nor to $\infty$, otherwise we would have $\|u_n\|\to\infty$ as $n\to\infty$, which is impossible since $c>0$. See e.g. \cite{Alves-Carriao-Miyagaki}. Suppose $t_n\to t_0$. Letting $n\to+\infty$, \[ 0=(t_0^2-1)\ell_1+t_0^2(t_0^{q-4}-1)\ell_2 \] where $\ell_1,\ell_2>0$. Hence, $t_0=1$. Consequently, we have \begin{align*} &I(u_n)-I(t_n u_n)\\ &=\frac{1-t_n^2}{2}\|u_n\|^2+\frac{1}{4}(1-t_n^4) \int_{\mathbb{R}^3}K(x)\phi_{u_n} u_n^2\,dx +\frac{t_n^q-1}{q}\int_{\mathbb{R}^3}|u_n|^q\,dx = o_n(1) \end{align*} which implies, for every $t\geq 0$, \begin{equation}\label{bla} \begin{aligned} I(u_n) &\geq I(t u_n)+o_n(1) \\ &= \frac{t^2}{2}\int_{\mathbb{R}^3}|\nabla u_n|^2+V(x)u_n^2\,dx + \frac{t^4}{4}\int_{\mathbb{R}^3}K(x)\phi_{u_n} u_n^2\,dx \\ &\quad -\frac{t^q}{q}\int_{\mathbb{R}^3}|u_n|^q\,dx+I_{\mu,\nu}(t u_n) -I_{\mu,\nu}(t u_n)+o_n(1) \\ &\geq \frac{t^2}{2}\int_{B_R(0)}(V(x)-\mu)u_n^2\,dx +\frac{t^4}{4}\int_{B_R(0)}(K(x)-\nu)\phi_{u_n} u_n^2\,dx \\ &\quad + I_{\mu,\nu}(t u_n) + o_n(1), \end{aligned} \end{equation} where $I_{\mu,\nu}(u)$ is given by \eqref{Imu}. Consider $\tau_n$ such that $ I_{\mu,\nu}(\tau_n u_n)=\max_{t\geq 0}I_{\mu,\nu}(tu_n)$. As in the above arguments, $\tau_n\to 1$. Letting $t=\tau_n$ in \eqref{bla}, we have \begin{align*} I(u_n) &\geq \frac{\tau_n^2}{2}\int_{B_R(0)}(V(x)-\mu)u_n^2\,dx +\frac{\tau_n^4}{4}\int_{B_R(0)}(K(x)-\nu)\phi_{u_n} u_n^2\,dx \\ &\quad + c_{\mu,\nu} + o_n(1). \end{align*} Taking the limit $n\to +\infty$, we have $c\geq c_{\mu,\nu}$. Next, taking $\mu\to V_\infty$ and $\nu\to K_\infty$, we obtain $c\geq C_\infty$, proving Theorem \ref{critical value}. \end{proof} \section{Proof of Theorem \ref{principal}} This section is devoted to study the existence, regularity and the asymptotic behavior of solutions for the system \eqref{SPe}), which is equivalent to \begin{equation}\label{Se} -\Delta u+ V(\varepsilon x)u+\varepsilon^2 K(\varepsilon x) \phi_{\varepsilon,u} u= |u|^{q-2}u. \end{equation} where \[ \mathcal{I}_\varepsilon(u) =\frac{1}{2}\int_{\mathbb{R}^3}(|\nabla u|^2+V(\varepsilon x)u^2)\,dx+ \frac{\varepsilon^2}{4}\int_{\mathbb{R}^3}K(\varepsilon x) \phi_{\varepsilon,u}(x) u^2\,dx-\frac{1}{q}\int_{\mathbb{R}^3}|u|^q\,dx. \] is the Euler-Lagrange functional related to \eqref{Se}. The proof of Theorem \ref{principal} is divided into three subsections as follows: \subsection{Existence of a solution} \begin{theorem} Suppose {\rm (H0)--(H1)} hold and consider \begin{equation} \label{Cinfty2} C_\infty>\inf_{\xi\in\mathbb{R}^3} C(\xi)\,. \end{equation} Then, there exists $\varepsilon^*>0$ such that system \eqref{Se} has a positive solution for every $0<\varepsilon<\varepsilon^*$. \end{theorem} \begin{proof} By hypothesis \eqref{Cinfty2}, there exists $b\in\mathbb{R}^3$ and $\delta>0$ such that \begin{equation}\label{8} C(b)+\delta0$ such that $c_\varepsilon0,\, x\in\mathbb{R}^3. \] Then, $u\in L^t(\mathbb{R}^3)$ for every $t\geq 2$. Moreover, there exists a positive constant $C=C(t,C_f)$ such that \[ \|u\|_{L^t(\mathbb{R}^3)}\leq C\|u\|_{H^1(\mathbb{R}^3)}. \] \end{proposition} \begin{proposition}\label{R2} Consider $t>3$ and $g\in L^{1/2}(\Omega)$, where $\Omega$ is an open subset of $\mathbb{R}^3$. Then, if $u\in H^1(\Omega)$ is a subsolution of \[ \Delta u=g \quad \text{in } \Omega, \] we have that for any $y\in\mathbb{R}^3$ and $B_{2R}(y)\subset \Omega$, $R>0$ and \[ \sup_{B_{R}(y)}u \leq C \Big(\|u^+\|_{L^2(B_{2R}(y))} +\|g\|_{L^{1/2}(B_{2R}(y)) }\Big) \] where $C=C(t,R)$. \end{proposition} In view of Propositions \ref{R1} and \ref{R2}, the positive solutions of \eqref{SPe} are in $C^2(\mathbb{R}^3)\cap L^{\infty}(\mathbb{R}^3)$ for all $\varepsilon >0$. Similar arguments was employed by He and Zou \cite{He-Zou2012}. \subsection{Concentration of solutions} \begin{lemma}\label{beta0} Suppose {\rm (H0)--(H1)} hold. Then, there exists $\beta_0>0$ such that \[ c_\varepsilon\geq \beta_0, \] for every $\varepsilon>0$. Moreover, \[ \limsup_{\varepsilon\to 0}c_\varepsilon\leq\inf_{\xi\in\mathbb{R}^3}C(\xi). \] \end{lemma} \begin{proof} Let $w_\varepsilon\in H_\varepsilon$ be such that $c_\varepsilon=\mathcal{I}_\varepsilon(w_\varepsilon)$. Then, from condition $(H_0)$ \[ c_\varepsilon=\mathcal{I}_\varepsilon(w_\varepsilon) \geq \inf_{\genfrac{}{}{0pt}{}{u\in H^1}{u\neq 0} } \sup_{t\geq 0}J(tu)=\beta_0,\quad \forall \varepsilon>0, \] where \[ J(u)=\frac{1}{2}\int_{\mathbb{R}^3}(|\nabla u|^2+\alpha u^2)\,dx +\frac{1}{4}\int_{\mathbb{R}^3}\alpha\phi_u u^2\,dx -\frac{1}{q}\int_{\mathbb{R}^3}|u|^q\,dx. \] Let $\xi\in\mathbb{R}^3$ and consider $w\in H^1(\mathbb{R}^3)$ a least energy solution for system \eqref{SPx}, that is, $I_\xi(w)=C(\xi)$ and $I'_{\xi}(w)=0$. Let $w_\varepsilon(x)=w(x-\frac{\xi}{\varepsilon})$ and take $t_\varepsilon>0$ such that \[ c_\varepsilon\leq \mathcal{I}_\varepsilon(t_\varepsilon w_\varepsilon) =\max_{t\geq 0}\mathcal{I}_\varepsilon(t w_\varepsilon). \] Similar to the proof of Theorem \ref{critical value}, $t_\varepsilon\to 1$ as $\varepsilon\to 0$, then \[ c_\varepsilon\leq \mathcal{I}_\varepsilon(t_\varepsilon w_\varepsilon) \to I_\xi(w)=C(\xi), \quad \text{as } \varepsilon\to 0 \] which implies that $\limsup_{\varepsilon\to 0}c_{\varepsilon}\leq C(\xi)$ for all $\xi\in\mathbb{R}^3$. Therefore, \[ \limsup_{\varepsilon\to 0}c_{\varepsilon}\leq \inf_{\xi\in\mathbb{R}^3} C(\xi). \] \end{proof} \begin{lemma}\label{awayfromzero} There exist a family $(y_\varepsilon)\subset\mathbb{R}^3$ and constants $R,\beta>0$ such that \[ \liminf_{\varepsilon\to 0} \int_{B_R(y_\varepsilon)}u_\varepsilon^2\, dx \geq \beta, \quad \text{for each }\varepsilon>0. \] \end{lemma} \begin{proof} By contradiction, suppose that there exists a sequence $\varepsilon_n\to 0$ such that \[ \lim_{n\to\infty} \sup_{y\in\mathbb{R}^3} \int_{B_R(y)}u_n^2\, dx = 0, \quad \text{for all }R>0, \] where, for the sake of simplicity, we denote $u_n(x)=u_{\varepsilon_n}(x)$. Hereafter, denote $\phi_{\varepsilon_n,u_n}(x)=\phi_{u_n}(x)$. From \cite[Lemma I.1]{Lions2}, we have \[ \int_{\mathbb{R}^3}|u_n|^q\,dx\to 0, \quad \text{as}\,\, n\to\infty. \] But, since \[ \int_{\mathbb{R}^3}(|\nabla u_n|^2+V(\varepsilon_n x)u_n^2)\,dx + \int_{\mathbb{R}^3}\varepsilon_n^2 K(\varepsilon_n x)\phi_{u_n} u_n^2\,dx=\int_{\mathbb{R}^3}|u_n|^q\,dx\,, \] we have \[ \int_{\mathbb{R}^3}(|\nabla u_n|^2+V(\varepsilon_n x)u_n^2)\,dx\to 0, \quad \text{as } n\to\infty. \] Therefore, \[ \lim_{n\to\infty}c_{\varepsilon_n}=\lim_{n\to\infty}I_{\varepsilon_n}(u_n)=0 \] which is an absurd, since for some $\beta_0>0$, $c_\varepsilon\geq\beta_0$, from Lemma \ref{beta0}. \end{proof} \begin{lemma}\label{lema1} The family $(\varepsilon y_\varepsilon)$ is bounded. Moreover, if $y^*$ is the limit of the sequence $(\varepsilon_n y_{\varepsilon_n})$ in the family $(\varepsilon y_\varepsilon)$, then we have \[ C(y^*)=\inf_{\xi\in\mathbb{R}^3}C(\xi). \] \end{lemma} \begin{proof} Consider $u_n(x)=u_{\varepsilon_n}(x+y_{\varepsilon_n})$. Suppose by contradiction that $(\varepsilon_n y_{\varepsilon_n})$ approaches infinity. It follows from Lemma \ref{awayfromzero} that there exists constants $R,\beta>0$ such that \begin{equation}\label{11} \int_{B_R(0)}u_n^2(x)\, dx \geq \beta>0, \quad \text{for all } n\in\mathbb{N}. \end{equation} Since $u_n(x)$ satisfies \begin{equation}\label{14} -\Delta u_n+ V(\varepsilon_n x+\varepsilon_n y_{\varepsilon_n})u_n +\varepsilon_n^2 K(\varepsilon_n x+\varepsilon_n y_{\varepsilon_n})\phi_{\varepsilon_n,u_n} u_n= |u_n|^{q-2}u_n, \end{equation} it follows that $u_n(x)$ is bounded in $H_{\varepsilon}$. Hence, passing to a subsequence if necessary, $u_n\to \hat{u} \geq 0$ weakly in $H_{\varepsilon}$, strongly in $L_{\rm loc}^p(\mathbb{R}^3)$ for $p\in(2,6)$ and a.e. in $\mathbb{R}^3$. From \eqref{11}, $\hat{u}\neq 0$. Using $\hat{u}$ as a test function in \eqref{14} and taking the limit, we obtain \begin{equation}\label{15} \int_{\mathbb{R}^3}(|\nabla \hat{u}|^2+\mu\hat{u}^2)\,dx \leq \int_{\mathbb{R}^3}(|\nabla \hat{u}|^2+\mu\hat{u}^2)\,dx + \int_{\mathbb{R}^3}\nu \phi_{\hat{u}} \hat{u}^2\,dx \leq \int_{\mathbb{R}^3}|\hat{u}|^q\,dx \end{equation} where, $\mu$ and $\nu$ are positive constantes such that \[ \mu<\liminf_{|x|\to\infty}V(x) \quad \text{and}\quad \nu<\liminf_{|x|\to\infty}K(x). \] Consider the functional $I_{\mu,\nu}:H^1(\mathbb{R}^3)\to \mathbb{R}$ given by \[ I_{\mu,\nu}(u)=\frac{1}{2}\int_{\mathbb{R}^3}(|\nabla u|^2 +\mu u^2)\,dx+\frac{1}{4}\int_{\mathbb{R}^3}\nu\phi_u(x) u^2\,dx-\frac{1}{q}\int_{\mathbb{R}^3}|u|^q\,dx. \] Let $\sigma>0$ be such that $ I_{\mu,\nu}(\sigma \hat{u})=\max_{t>0}I_{\mu,\nu}(t \hat{u})$. We claim that \begin{equation}\label{12} \sigma^2 \int_{\mathbb{R}^3}(|\nabla \hat{u}|^2+\mu\hat{u}^2)\,dx +\sigma^4 \int_{\mathbb{R}^3}\nu \phi_{\hat{u}} \hat{u}^2\,dx = \sigma^q\int_{\mathbb{R}^3}|\hat{u}|^q\,dx. \end{equation} In fact, from \eqref{15} \begin{align*} I_{\mu,\nu}(\sigma \hat{u}) &= \frac{\sigma^2}{2} \int_{\mathbb{R}^3}(|\nabla \hat{u}|^2 +\mu\hat{u}^2)\,dx+\frac{\sigma^4}{4} \int_{\mathbb{R}^3}\nu \phi_{\hat{u}} \hat{u}^2\,dx -\frac{\sigma^q}{q}\int_{\mathbb{R}^3}|\hat{u}|^q\,dx\\ & \leq \frac{\sigma^2}{2} \int_{\mathbb{R}^3}|\hat{u}|^q\,dx +\frac{\sigma^4}{4} \int_{\mathbb{R}^3}\nu \phi_{\hat{u}} \hat{u}^2\,dx - \frac{\sigma^q}{q}\int_{\mathbb{R}^3}|\hat{u}|^q\,dx \end{align*} it follows that $\sigma\leq 1$, and since $\frac{d}{dt}I_{\mu,\nu}(t\hat{u})\Big|_{t=\sigma}=0$, we obtain \[ \frac{d}{dt}I_{\mu,\nu}(t \hat{u})\Big|_{t=\sigma} = \sigma \int_{\mathbb{R}^3}(|\nabla \hat{u}|^2+\mu\hat{u}^2)\,dx +\sigma^3 \int_{\mathbb{R}^3}\nu \phi_{\hat{u}} \hat{u}^2\,dx - \sigma^{q-1}\int_{\mathbb{R}^3}|\hat{u}|^q\,dx=0 \] proving \eqref{12}. From Lemma \ref{beta0}, equation \eqref{12} and the fact that $\sigma\leq 1$, we have \begin{align*} c_{\mu,\nu} &= \inf_{u\neq 0}\max_{t>0}I_{\mu,\nu}(tu) =\inf_{u\neq 0}I_{\mu,\nu}(\sigma u)\leq I_{\mu,\nu}(\sigma \hat{u})\\ &=\frac{\sigma^2}{2} \int_{\mathbb{R}^3}(|\nabla \hat{u}|^2 +\mu\hat{u}^2)\,dx+\frac{\sigma^4}{4} \int_{\mathbb{R}^3}\nu \phi_{\hat{u}} \hat{u}^2\,dx -\frac{\sigma^q}{q}\int_{\mathbb{R}^3}|\hat{u}|^q\,dx\\ &= \frac{\sigma^2}{4} \int_{\mathbb{R}^3}(|\nabla \hat{u}|^2+\mu\hat{u}^2)\,dx + \frac{\sigma^q(q-4)}{4q}\int_{\mathbb{R}^3}|\hat{u}|^q\,dx\\ &\leq \frac{1}{4} \int_{\mathbb{R}^3}(|\nabla \hat{u}|^2+\mu\hat{u}^2)\,dx + \frac{q-4}{4q}\int_{\mathbb{R}^3}|\hat{u}|^q\,dx\\ &\leq \liminf_{n\to\infty}\Big(\mathcal{I}_{\varepsilon_n}(u_n) -\frac{1}{4}\mathcal I'_{\varepsilon_n}(u_n)u_n\Big)\\ &=\liminf_{n\to\infty}c_{\varepsilon_n} \leq \limsup_{n\to\infty}c_{\varepsilon_n} \leq \inf_{\xi\in\mathbb{R}^3}C(\xi) \end{align*} hence, $c_{\mu,\nu}\leq \inf_{\xi\in\mathbb{R}^3}C(\xi) $. If we consider \[ \mu\to \liminf_{|x|\to\infty}V(x)=V_{\infty} \quad \text{and}\quad \nu\to \liminf_{|x|\to\infty}K(x)=K_{\infty}, \] then by the continuity of the function $(\mu,\nu)\mapsto c_{\mu\nu}$ we obtain $ C_\infty\leq \inf_{\xi\in\mathbb{R}^3}C(\xi)$, which contradicts condition $(C^\infty)$. Therefore, $(\varepsilon y_\varepsilon)$ is bounded and there exists a subsequence of $(\varepsilon y_\varepsilon)$ such that $\varepsilon_n y_{\varepsilon_n}\to y^*$. Now we proceed to prove that $ C(y^*)=\inf_{\xi\in\mathbb{R}^3}C(\xi)$. Recalling that $u_n(x)=u_{\varepsilon_n}(x+y_{\varepsilon_n})$ and from the arguments above, $\hat{u}$ satisfies the equation \begin{equation} \label{13} -\Delta u+V(y^*)u+K(y^*)\phi_u u=|u|^{q-2}u \end{equation} The Euler-Lagrange functional associated to this equation is $I_{y^*}: H_{y^*}(\mathbb{R}^3)$, defined as in \eqref{Ixi} with $\xi=y^*$. Using $\hat{u}$ as a test function in \eqref{13} and taking the limit, we obtain \[ \int_{\mathbb{R}^3}(|\nabla \hat{u}|^2+V(y^*)\hat{u}^2)\,dx \leq \int_{\mathbb{R}^3}|\hat{u}|^q\,dx. \] Then \[ I_{y^*}(\sigma \hat{u})=\max_{t>0}I_{y^*}(t\hat{u}). \] Finally, from Lemma \ref{beta0} and since $0<\sigma\leq 1$ we have \begin{align*} &\inf_{\xi\in\mathbb{R}^3}C(\xi) \\ &\leq C(y^*) \leq I_{y^*}(\sigma \hat{u}) \\ & = \frac{\sigma^2}{4} \int_{\mathbb{R}^3}(|\nabla \hat{u}|^2+V(y^*)\hat{u}^2)\,dx + \frac{\sigma^q(q-4)}{4q}\int_{\mathbb{R}^3}|\hat{u}|^q\,dx\\ &\leq \frac{1}{4} \int_{\mathbb{R}^3}(|\nabla \hat{u}|^2+V(y^*)\hat{u}^2)\,dx + \frac{q-4}{4q}\int_{\mathbb{R}^3}|\hat{u}|^q\,dx\\ &\leq \liminf_{n\to\infty} \Big[ \frac{1}{4} \int_{\mathbb{R}^3}\Big(|\nabla u_n|^2+V(\varepsilon_n x +\varepsilon_n y_{\varepsilon_n})u_n^2\Big)\,dx +\frac{q-4}{4q}\int_{\mathbb{R}^3}|u_n|^q\,dx\Big]\\ &\leq \liminf_{n\to\infty}\Big(\mathcal I_{\varepsilon_n}(u_n) -\frac{1}{4}\mathcal I'_{\varepsilon_n}(u_n)u_n\Big)\\ &= \liminf_{n\to\infty}c_{\varepsilon_n}\leq \inf_{\xi\in\mathbb{R}^3}C(\xi) \end{align*} which implies that $ C(y^*)=\inf_{\xi\in\mathbb{R}^3}C(\xi)$. \end{proof} As a consequence of the previous lemma, there exists a subsequence of $(\varepsilon_n y_{\varepsilon_n})$ such that $\varepsilon_n y_{\varepsilon_n}\to y^*$. Let $u_{\varepsilon_n}(x+y_{\varepsilon_n})=u_n(x)$ and consider $\tilde{u}\in H^1$ such that $u_n\rightharpoonup \tilde{u}$. \begin{lemma} \label{lem5.4} $u_n\to \tilde{u}$ in $H^{1}(\mathbb{R}^3)$, as $n\to\infty$. Moreover, there exists $\varepsilon^*>0$ such that $\lim_{|x|\to\infty}u_\varepsilon(x)=0$ uniformly on $\varepsilon\in (0,\varepsilon^*)$. \end{lemma} \begin{proof} By Lemmas \ref{beta0} and \ref{lema1}, we have \begin{align*} &\inf_{\xi\in\mathbb{R}^3}C(\xi) \\ &= C(y^*)\leq I_{y^*}(\tilde{u})-\frac{1}{4}I'_{y^*}(\tilde{u})\tilde{u}\\ &=\frac{1}{4}\int_{\mathbb{R}^3}(|\nabla \tilde{u}|^2+V(y^*)\tilde{u}^2)\,dx+ \Big(\frac{q-4}{4q}\Big)\int_{\mathbb{R}^3}|\tilde{u}|^q\,dx\\ &\leq \liminf_{n\to\infty}\Big\{\frac{1}{4} \int_{\mathbb{R}^3}(|\nabla u_n|^2+V(\varepsilon_n x +\varepsilon_n y_{\varepsilon_n})u_n^2)\,dx +\Big(\frac{q-4}{4q}\Big)\int_{\mathbb{R}^3}|u_n|^q\,dx\Big\}\\ &\leq \limsup_{n\to\infty} \Big\{\frac{1}{4}\int_{\mathbb{R}^3}(|\nabla u_n|^2+V(\varepsilon_n x+\varepsilon_n y_{\varepsilon_n})u_n^2)\,dx +\Big(\frac{q-4}{4q}\Big)\int_{\mathbb{R}^3}|u_n|^q\,dx\Big\}\\ &= \limsup_{n\to\infty}\Big\{\mathcal I_{\varepsilon_n}(u_{\varepsilon_n}) -\frac{1}{4}\mathcal I'_{\varepsilon_n}(u_{\varepsilon_n})u_{\varepsilon_n}\Big\} \\ &= \limsup_{n\to\infty} c_{\varepsilon_n} \leq \inf_{\xi\in\mathbb{R}^3}C(\xi)\,. \end{align*} Then \[ \lim_{n\to\infty}\int_{\mathbb{R}^3}(|\nabla u_n|^2+V(\varepsilon_n x+\varepsilon_n y_{\varepsilon_n})u_n^2)\,dx= \int_{\mathbb{R}^3}(|\nabla \tilde{u}|^2+V(y^*)\tilde{u}^2)\,dx. \] Now observe that \begin{align*} c_{\varepsilon_n} &= \mathcal I_{\varepsilon_n}(u_{\varepsilon_n})-\frac{1}{4}\mathcal I'_{\varepsilon_n}(u_{\varepsilon_n})u_{\varepsilon_n}\\ &= \frac{1}{4}\int_{\mathbb{R}^3}(|\nabla u_{\varepsilon_n}|^2+V(\varepsilon_n x)u_{\varepsilon_n}^2)\,dx+ \Big(\frac{q-4}{4q}\Big)\int_{\mathbb{R}^3}|u_{\varepsilon_n}|^q\,dx\\ &= \frac{1}{4}\int_{\mathbb{R}^3}(|\nabla u_{n}|^2+V(\varepsilon_n x+\varepsilon_n y_{\varepsilon_n})u_{n}^2)\,dx+ \Big(\frac{q-4}{4q}\Big)\int_{\mathbb{R}^3}|u_{n}|^q\,dx \\ &:= \alpha_n; \end{align*} hence, \[ \limsup_{n\to\infty}\alpha_n= \limsup_{n\to\infty}c_{\varepsilon_n}\leq C(y^*). \] On the other hand, using Fatou's Lemma, \begin{align*} \liminf_{n\to\infty}\alpha_n &\geq \frac{1}{4}\int_{\mathbb{R}^3}(|\nabla \tilde{u}|^2+V(y^*)\tilde{u}^2)\,dx + \Big(\frac{q-4}{4q}\Big)\int_{\mathbb{R}^3}|\tilde{u}|^q\,dx\\ &= I_{y^*}(\tilde{u})-\frac{1}{4}I'_{y^*}(\tilde{u})\tilde{u}\\ &\geq C(y^*); \end{align*} then $ \lim_{n\to\infty}\alpha_n=C(y^*)$. Therefore, since $\tilde{u}$ is the weak limit of $(u_n)$ in $H^1(\mathbb{R}^3)$, we conclude that $u_n\to \tilde{u}$ strongly in $H^1(\mathbb{R}^3)$. In particular, we have \begin{equation}\label{*} \lim_{R\to\infty}\int_{|x|\geq R}u_n^{2^*}\,dx=0 \quad \text{uniformly on } n. \end{equation} Applying Proposition \ref{R1} with $b(x)=V(\varepsilon_n x+\varepsilon_n y_{\varepsilon_n})+\varepsilon_n^2 K(\varepsilon_n x+\varepsilon_n y_{\varepsilon_n})\phi_{u_n}$, we obtain $u_n\in L^t(\mathbb{R}^3)$, $t\geq 2$ and \[ \|u_n\|_t\leq C\|u_n\|, \] where $C$ does not depend on $n$. Now consider \begin{align*} -\Delta u_n &\leq -\Delta u_n+V(\varepsilon_n x+\varepsilon_n y_{\varepsilon_n})u_n +\varepsilon_n^2 K(\varepsilon_n x+\varepsilon_n y_{\varepsilon_n}) \phi_{u_n}u_n\\ &= |u_n|^{q-2}u_n := g_n(x). \end{align*} For some $t>3$, $\|g_n\|_{\frac{t}{2}}\leq C$, for all $n$. Using Proposition \ref{R2}, we have \[ \sup_{B_{R}(y)}u_n \leq C \Big(\|u_n\|_{L^2(B_{2R}(y))}+\|g_n\|_{L^{1/2}(B_{2R}(y)) }\Big) \] for every $y\in\mathbb{R}^3$, which implies that $\|u_n\|_{L^\infty(\mathbb{R}^3)}$ is uniformly bounded. Then, from \eqref{*}, \[ \lim_{|x|\to\infty}u_n(x)=0 \quad \text{uniformly on } n\in\mathbb{N}. \] Consequently, there exists $\varepsilon^*>0$ such that \[ \lim_{|x|\to\infty}u_\varepsilon(x)=0 \quad \text{uniformly on } \varepsilon\in(0, \varepsilon^*). \] \end{proof} To complete the proof of Theorem \ref{principal}, it remains to show that the solutions of \eqref{SPe} have at most one local (hence global) maximum point $y^*$ such that $C(y^*)=\min_{\xi\in\mathbb{R}^3} C(\xi)$. From the previous Lemma, we can focus our attention only in a fixed ball $B_R(0)\subset \mathbb{R}^3$. If $w\in L^{\infty}(\mathbb{R}^3)$ is the limit in $C^2_{\rm loc}(\mathbb{R}^3)$ of \[ w_n(x)=u_n(x+y_n) \] then, from Gidas, Ni and Nirenberg \cite{Gidas-Ni-Nirenberg}, $w$ is radially symmetric and has a unique local maximum at zero which is a non-degenerate global maximum. Therefore, there exists $n_0\in\mathbb{N}$ such that $w_n$ does not have two critical points in $B_R(0)$ for all $n\geq n_0$. Consider $p_\varepsilon\in\mathbb{R}^3$ this local (hence global) maximum of $w_\varepsilon$. Recall that if $u_\varepsilon$ is a solution of $(S_\varepsilon)$, then \[ v_\varepsilon(x)=u_\varepsilon(\frac{x}{\varepsilon}) \] is a solution of \eqref{SPe}. Since $p_\varepsilon$ is the unique maximum of $w_\varepsilon$, then $\hat{y_\varepsilon}=p_\varepsilon+y_\varepsilon$ is the unique maximum of $u_\varepsilon$. Hence, $\tilde{y}_\varepsilon=\varepsilon p_\varepsilon+\varepsilon y_\varepsilon$ is the unique maximum of $v_\varepsilon$. 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