\documentclass[reqno]{amsart} \usepackage{hyperref} \AtBeginDocument{{\noindent\small {\em Electronic Journal of Differential Equations}, Vol. 2006(2006), No. 76, pp. 1--12.\newline ISSN: 1072-6691. URL: http://ejde.math.txstate.edu or http://ejde.math.unt.edu \newline ftp ejde.math.txstate.edu (login: ftp)} \thanks{\copyright 2006 Texas State University - San Marcos.} \vspace{9mm}} \begin{document} \title[\hfilneg EJDE-2006/76\hfil Multiplicity of solutions] {Multiplicity of solutions for a class of elliptic systems in $\mathbb{R}^N$} \author[G. M. Figueiredo\hfil EJDE-2006/76\hfilneg] {Giovany M. Figueiredo} \address{Giovany M. Figueiredo \newline Universidade Federal do Par\'a \\ Departamento de Matem\'atica \\ CEP: 66075-110 Bel\'em - Pa, Brazil} \email{giovany@ufpa.br} \date{} \thanks{Submitted May 24, 2005. Published July 12, 2006.} \subjclass[2000]{35J20, 35J50, 35J60} \keywords{Variational methods; Palais-Smale condition; \hfill\break\indent Ljusternik-Schnirelmann theory} \begin{abstract} This article concerns the multiplicity of solutions for the system of equations \begin{gather*} -\Delta u + V(\epsilon x)u = \alpha |u|^{\alpha-2}u|v|^{\beta} , \\ -\Delta v + V(\epsilon x)v = \beta |u|^{\alpha}|v|^{\beta-2}v \end{gather*} in $\mathbb{R}^N$, where $V$ is a positive potential. We relate the number of solutions with the topology of the set where $V$ attains its minimum. The results are proved by using minimax theorems and Ljusternik-Schnirelmann theory. \end{abstract} \maketitle \numberwithin{equation}{section} \newtheorem{theorem}{Theorem}[section] \newtheorem{lemma}[theorem]{Lemma} \newtheorem{proposition}[theorem]{Proposition} \newtheorem{corollary}[theorem]{Corollary} \newtheorem{remark}[theorem]{Remark} \section{Introduction} The purpose of this article is to investigate the multiplicity of solutions for the system \begin{equation} \begin{gathered} -\Delta u+V(\epsilon x)u=\alpha|u|^{\alpha - 2}u|v|^{\beta} \quad \text{in }\mathbb{R}^N, \\ -\Delta v+V(\epsilon x)v=\beta|u|^{\alpha}|v|^{\beta -2}v \quad \text{in }\mathbb{R}^N, \\ u , v \in H^{1}(\mathbb{R}^N),\quad u(x), v(x)> 0 \quad \text{for all } x \in \mathbb{R}^N, \end{gathered}\label{Sep} \end{equation} where $\epsilon>0$, $\alpha, \beta > 1$ such that $\alpha + \beta =p$, $2 < p < 2N/(N-2)$, $N\geq 3$ and the potential $V:\mathbb{R}^N \to \mathbb{R}$ is continuous and satisfies \begin{equation} 0 < V_{0} :=\inf_{x \in \mathbb{R}^N} V(x) < V_{\infty} := \liminf_{|x| \to \infty}V(x). \label{V} \end{equation} In this work, we will consider the cases $V_{\infty}<\infty$ or $V_{\infty}=\infty$. This kind of hypothesis was introduced by Rabinowitz \cite{Rab} in the study of a nonlinear Schr\"odinger equation. We say that $(u,v) \in H^{1}(\mathbb{R}^N)\times H^{1}(\mathbb{R}^N)$ is a weak solution of the system in \eqref{Sep} if $$ \int_{\mathbb{R}^N}\Bigl[\nabla u\nabla \phi +\nabla v\nabla \psi+ V(\epsilon x)(u\phi+v\psi)\Bigl]=\int_{\mathbb{R}^N}\Bigl[\alpha|u|^{\alpha - 2}u|v|^{\beta}\phi +\beta|u|^{\alpha}|v|^{\beta -2}v \psi\Bigl] $$ for all $(\phi , \psi) \in H^{1}(\mathbb{R}^N)\times H^{1}(\mathbb{R}^N)$. In this paper we also relate the number of solutions of \eqref{Sep} with the topology of the set of minima of the potential $V$. In order to present our result we introduce the set of global minima of $V$, given by $$ M = \{ x \in \mathbb{R}^N : V(x) = V_0 \}. $$ Note that, by \eqref{V}, $M$ is compact. For any $\delta>0$, let $M_{\delta} = \{ x \in \mathbb{R}^N : \text{dist}(x,M) \leq \delta \}$ be the closed $\delta$-neighborhood of $M$. Our main result is as follows. \begin{theorem}\label{th1} Suppose that $V$ satisfies \eqref{V}. Then, for any $\delta>0$ given, there exists $\epsilon_{\delta}>0$ such that, for any $\epsilon \in (0,\epsilon_{\delta})$, the system \eqref{Sep} has at least $\mathop{\rm cat}_{M_{\delta}}(M)$ solutions. \end{theorem} We recall that, if $Y$ is a closed set of a topological space $X$, cat$_X(Y)$ is the Ljusternik-Schnirelmann category of $Y$ in $X$, namely the least number of closed and contractible set in $X$ which cover $Y$. Existence and concentration of positive solutions for the problem \begin{equation}\label{ganhapapa} -\epsilon^{2}\Delta u +V(x)u=f(u) \quad\text{in }\mathbb{R}^N \end{equation} have been extensively studied in recent years, see for example, Ambrosetti, Badiale and Cingolani \cite{Ambrosetti}, Del Pino \& Felmer \cite{Pino}, Floer \cite{Floer}, Lazzo \cite{Lazzo2}, Oh \cite{Oh1, Oh2, Oh3}, Rabinowitz \cite{Rab} , Wang \cite{Wang} and their references. Cingolani and Lazzo in \cite{CinLaz} studied positive solutions for the Schr\"odinger equation (\ref{ganhapapa}) with $f(u)=|u|^{q-2}u$, $\epsilon >0$, $2 0 \quad \text{for all } x \in \mathbb{R}^N, \mu>0 \,. \end{gathered} \label{ASmu} \end{equation} This fact is used in a lot of papers in the scalar case. The paper is organized as follows: In Section 2 we present the abstract framework of the problem as well as some remarks on the autonomous problem. In Section 3 we obtain some compactness properties of the functional associated to the system \eqref{Sep}. Theorem \ref{th1} is proved in Section 4. \section{The variational framework} Throughout this paper we suppose that the function $V$ satisfies the conditions \eqref{V}. We write only $\int u$ instead of $\int_{\mathbb{R}^N} u(x)\textrm{d}x$. For any $\epsilon >0$, we denote by $X_{\epsilon}$ the Sobolev space \[ X_{\epsilon}=\{(u,v)\in H^{1}(\mathbb{R}^N)\times H^{1}(\mathbb{R}^N): \int V(\epsilon x)(|u|^{2}+|v|^{2})<\infty\} \] endowed with the norm $$ \|(u,v)\|_{\epsilon}^{2}=\int(|\nabla u|^{2}+|\nabla v|^{2})+\int V(\epsilon x)(|u|^{2}+|v|^{2}), $$ We will look for solutions of \eqref{Sep} by finding critical points of the $C^2$-functional $I_{\epsilon}:X_{\epsilon} \to \mathbb{R}$ given by $$ I_{\epsilon}((u,v)) = \frac{1}{2} \int \left[|\nabla u|^2 + |\nabla v|^2+V(\epsilon x)(|u|^2+|v|^2)\right] - \int Q(u,v), $$ where $Q(u,v) = (u^{+})^{\alpha}(v^{+})^{\beta}$ and $w^{\pm} = \max \{\pm w,0\}$ the positive (negative) part of $w$. By definition of $Q$, we see that, if $(u,v)$ is a nontrivial critical point of $I_{\epsilon}$, then $u,v$ are positive in $\mathbb{R}^N$. Indeed, since that \begin{align*} \langle I_{\epsilon}'((u,v)),(\phi,\psi)\rangle &=\int\Bigl[\nabla u\nabla \phi +\nabla v\nabla \psi + V(\epsilon x)(u\phi+v\psi)\Bigl]\\ &\quad -\alpha\int|u|^{\alpha - 2}u|v|^{\beta}\phi -\beta\int|u|^{\alpha}|v|^{\beta -2}v \psi, \end{align*} we have $$ 0 = \langle I_{\epsilon}'((u,v)),(u^-,v^-)\rangle = \Vert (u^-,v^-) \Vert_{\epsilon}^2 $$ and therefore $u , v \geq 0$ in $\mathbb{R}^N$. By the Maximum Principle in $\mathbb{R}^N$, $u , v > 0$ in $\mathbb{R}^N$. We introduce the Nehari manifold of $I_{\epsilon}$ by setting $$ \mathcal{N}_{\epsilon} = \left\{ (u,v) \in X_{\epsilon} \setminus \{(0,0)\} : \langle I_{\epsilon}'((u,v)),(u,v)\rangle =0\right\}. $$ Note that, if $(u,v) \in \mathcal{N}_{\epsilon}$, we have $$ I_{\epsilon}((u,v)) = \frac{1}{2} \Vert (u,v) \Vert_{\epsilon}^2 - \int Q(u,v) = \big( \frac{1}{2} - \frac{1}{p} \big) \Vert (u,v) \Vert_{\epsilon}^2 \geq 0, $$ and therefore the following minimization problem is well defined $$ c_{\epsilon} = \inf_{(u,v) \in \mathcal{N}_{\epsilon}} I_{\epsilon}((u,v)). $$ Moreover, we can easily conclude that there exists $r>0$, independent of $\epsilon$, such that \begin{equation} \Vert (u,v) \Vert_{\epsilon} \geq r > 0\quad \text{for any } \epsilon >0,\;(u,v) \in \mathcal{N}_{\epsilon}. \label{nehari_naozero} \end{equation} We now present some important properties of $c_{\epsilon}$ and $\mathcal{N}_{\epsilon}$. The proofs can be adapted from \cite[Chapter 4]{Wil} (see also \cite[Lemmas 3.1 and 3.2]{Liliane}). First we observe that, for any $(u,v) \in X_{\epsilon} \setminus \{(0,0)\}$ there exists a unique $t_{u,v}>0$ such that $t_{u,v}(u,v) \in \mathcal{N}_{\epsilon}$. The maximum of the function $t \mapsto I_{\epsilon}(t(u,v))$ for $t\geq 0$ is achieved at $t=t_{u,v}$ and the function $(u,v) \mapsto t_{u.v}$ is continuous from $X_{\epsilon} \setminus \{(0,0)\}$ to $(0,\infty)$. Note that by conditions on $\alpha$ and $\beta$, we have \begin{equation}\label{crescimento} Q(u,v)\leq\frac{\alpha}{p}|u|^{p}+\frac{\beta}{p}|v|^{p}. \end{equation} Standard calculations imply that $I_{\epsilon}$ satisfies the geometry of the Mountain Pass theorem. Arguing as in \cite[Theorem 4.2]{Wil} we can prove that $c_{\epsilon}$ is positive, it coincides with the mountain pass level of $I_{\epsilon}$ and satisfies \begin{equation} c_{\epsilon} = \inf_{\gamma \in \Gamma_{\epsilon}} \max_{t \in [0,1]} I_{\epsilon}(\gamma(t)) = \inf_{(u,v) \in X_{\epsilon} \setminus \{ (0,0) \}} \max_{t \geq 0} I_{\epsilon}(t(u,v))> 0, \label{prop_nehari} \end{equation} where $\Gamma_{\epsilon} = \{ \gamma \in C([0,1],X_{\epsilon}) : \gamma(0)=(0,0), ~I_{\epsilon}(\gamma(1)) < 0\}$. We will denote by $\Vert I_{\epsilon}'((u,v))\Vert_*$ the norm of the derivative of $I_{\epsilon}$ restricted to $\mathcal{N}_{\epsilon}$ at the point $(u,v)$. This norm is given by (see \cite[Proposition 5.12]{Wil}) $$ \Vert I_{\epsilon}'((u,v))\Vert_* = \min_{\lambda \in \mathbb{R}} \Vert I_{\epsilon}'((u,v)) - \lambda J_{\epsilon}'((u,v))\Vert_{X_{\epsilon}^*}, $$ where $X_{\epsilon}^*$ denotes the dual space of $X_{\epsilon}$ and $J_{\epsilon} : X_{\epsilon} \to \mathbb{R}$ is defined as \begin{equation} J_{\epsilon}((u,v)) = \Vert( u,v)\Vert_{\epsilon}^2 - p\int Q(u,v). \label{preliminar_0} \end{equation} As we will see, it is important to compare $c_{\epsilon}$ with the minimax level of the autonomous problem \eqref{ASmu}. The solutions of \eqref{ASmu} are precisely the positive critical points of the functional $E_{\mu}:H^{1}(\mathbb{R}^N)\times H^{1}(\mathbb{R}^N)\to \mathbb{R}$ given by $$ E_{\mu}((u,v)) = \frac{1}{2} \int \left( |\nabla u|^2 +|\nabla v|^2\right) + \frac{1}{2}\int \mu\left(|u|^2+|v|^{2}\right)- \int Q(u,v). $$ We also define the autonomous minimization problem $$ m(\mu) = \inf_{(u,v) \in \mathcal{M}_{\mu}} E_{\mu}((u,v)), $$ where $\mathcal{M}_{\mu}$ is the Nehari manifold of $E_{\mu}$, that is $$ \mathcal{M}_{\mu} = \left\{ (u,v) \in H^{1}(\mathbb{R}^N)\times H^{1}(\mathbb{R}^N)\setminus \{(0,0)\} : \langle E_{\mu}'((u,v)),(u,v) \rangle=0\right\}. $$ The number $m(\mu)$ and the manifold $\mathcal{M}_{\mu}$ have properties similar to those of $c_{\epsilon}$ and $\mathcal{N}_{\epsilon}$. Moreover, Alves and Monari in \cite[Theorem 4.11]{AlvesMonari} showed that $m(\mu)$ is attained by a solution $(u,v)\in H^{1}(\mathbb{R}^N)\times H^{1}(\mathbb{R}^N)$ of the problem \eqref{ASmu}. \section{A compactness condition} In this section we obtain some compactness properties of the functional $I_{\epsilon}$. We start by recalling the definition of the Palais-Smale condition. So, let $E$ be a Banach space, $\mathcal{V}$ be a $C^1$-manifold of $E$ and $I:E \to \mathbb{R}$ a $C^1$-functional. We say that $I|_{\mathcal{V}}$ satisfies the Palais-Smale condition at level $c$ ((PS)$_c$) if any sequence $(u_n) \subset \mathcal{V}$ such that $I(u_n) \to c$ and $\Vert I'(u_n) \Vert_* \to 0$ contains a convergent subsequence. The next lemma shows a property involving $(PS)_c$ sequences for $I_{\epsilon}$. Its proof uses well-know arguments and will be omitted. \begin{lemma} \label{lema_pspositiva} Let $((u_n,v_n)) \subset X_{\epsilon}$ be a \emph{(PS)}$_c$ sequence for $I_{\epsilon}$. Then \begin{itemize} \item[\emph{(i)}] $((u_n,v_n))$ is bounded in $X_{\epsilon}$, \item[\emph{(ii)}] there exists $(u,v) \in X_{\epsilon}$ such that, up to a subsequence, $(u_n,v_n) \rightharpoonup (u,v)$ weakly in $X_{\epsilon}$ and $I_{\epsilon}'((u,v))=0$, \item[\emph{(iii)}] $((u_n^+,v_n^+))$ is also a \emph{(PS)}$_c$ sequence for $I_{\epsilon}$. \end{itemize} Moreover, the same holds if we replace $I_{\epsilon}$ and $X_{\epsilon}$ which $E_{\mu}$ and $H^{1}(\mathbb{R}^N)\times H^{1}(\mathbb{R}^N)$, respectively. \end{lemma} \begin{remark} \label{rmk3.2} \rm Let $((u_n,v_n))$ be a Palais-Smale sequence for $I_{\epsilon}$ (or $E_{\mu}$). Since we are always interested in the existence of convergent subsequences, we may use the above lemma to suppose that $u_n \geq 0$ and $v_n \geq 0$ for all $n \in \mathbb{N}$. This will be made from now on. \end{remark} \begin{lemma} \label{lema_lions} Let $((u_n,v_n)) \subset X_{\epsilon}$ be a \emph{(PS)}$_d$ sequence for $I_{\epsilon}$. Then we have either \begin{itemize} \item[\emph{(i)}] $\Vert (u_n,v_n) \Vert_{\epsilon} \to 0$, or \item[\emph{(ii)}] there exist a sequence $(y_n) \subset \mathbb{R}^N$ and constants $R,~\gamma>0$ such that $$ \liminf_{n\to\infty} \int_{B_R(y_n)} (u_n^2 +v_n^2) \geq \gamma > 0. $$ \end{itemize} \end{lemma} The above lemma follows by adapting the arguments of \cite[page 171]{AlvesMonari} (see also \cite[Theorem 2.1]{Liliane}). \begin{remark} \label{lema_lions_remark} \rm For future reference we note that, if $\epsilon_n \to 0$ and $((u_n,v_n)) \subset \mathcal{N}_{\epsilon_n}$ is a bounded sequence in $H^{1}(\mathbb{R}^N)\times H^{1}(\mathbb{R}^N)$ such that $I_{\epsilon_n}((u_n,v_n)) \to d$, then we can argue along the same lines of the above proof to conclude that either $\Vert (u_n,v_n) \Vert_{\epsilon_n} \to 0$ or (ii) holds. We also have a similar result if $((u_n,v_n)) \subset H^{1}(\mathbb{R}^N)\times H^{1}(\mathbb{R}^N)$ is a \emph{(PS)}$_d$ sequence for the autonomous functional $E_{\mu}$. \end{remark} \begin{lemma} \label{lema_PalaisSmale} Consider $V_{\infty}<\infty$ and let $((u_n,v_n)) \subset X_{\epsilon}$ be a \emph{(PS)}$_d$ sequence for $I_{\epsilon}$ such that $(u_n,v_n) \rightharpoonup (0,0)$ weakly in $X_{\epsilon}$. If $(u_n,v_n) \not\to (0,0)$ in $X_{\epsilon}$, then $d \geq m(V_{\infty})$. \end{lemma} \begin{proof} Let $(t_n) \subset (0,+\infty)$ be such that $(t_n(u_n,v_n)) \subset \mathcal{M}_{V_{\infty}}$. We start by proving that $\limsup_{n\to \infty} t_n \leq 1$. Arguing by contradiction, we suppose that there exist $\lambda>0$ and a subsequence, which we also denote by $(t_n)$, such that \begin{equation} t_n \geq 1+\lambda\quad \textrm{for all } n \in \mathbb{N}. \label{lema_PS_eq1} \end{equation} Since $((u_n,v_n))$ in bounded in $X_{\epsilon}$, $\langle I_{\epsilon}'((u_n,v_n)),(u_n,v_n) \rangle \to 0$, that is, \[ \int\Bigl[| \nabla u_{n}|^{2}+| \nabla v_{n}|^{2} + V(\epsilon x)(| u_{n}|^{2}+| v_{n}|^{2})\Bigl] =p \int Q(u_{n},v_{n}) + o_{n}(1). \] Moreover, recalling that $(t_n(u_n,v_n)) \subset \mathcal{M}_{V_{\infty}}$, we get \[ \int\Bigl[| \nabla u_{n}|^{2} +| \nabla v_{n}|^{2} + V_{\infty}(| u_{n}|^{2}+| v_{n}|^{2})\Bigl] =p(t_{n}^{p-2}) \int Q(u_{n},v_{n}). \] These two equalities imply \begin{equation}\label{lema_PS_eq2} p(t_{n}^{p-2}-1)\int Q(u_{n},v_{n})= \int[V_{\infty} - V(\epsilon x)](| u_{n}| ^{2}+ | v_{n}| ^{2})+ o_{n}(1). \end{equation} Using the condition \eqref{V}, we have that given $\delta>0$, there exists $R>0$ such that \begin{equation} V(\epsilon x) \geq V_{\infty} - \delta\quad \textrm{for any } |x| \geq R. \label{lema_PS_eq3} \end{equation} Let $C>0$ be such that $\|(u_{n},v_{n})\|_{\epsilon}\leq C$. Since $\|(u_{n},v_{n})\|_{\epsilon}\to 0$ in $H^{1}(B_{R}(0))\times H^{1}(B_{R}(0))$ we can use (\ref{lema_PS_eq2}) and (\ref{lema_PS_eq3}) to obtain \begin{equation}\label{temquedacerto} p(t_{n}^{p-2}-1)\int Q(u_{n},v_{n})\leq \delta C + o_{n}. \end{equation} for any $\delta>0$. % Since $(u_n,v_n) \not\to (0,0)$, we may invoke Lemma \ref{lema_lions} to obtain $(y_n) \subset \mathbb{R}^N$ and $R,\gamma>0$ such that \begin{equation} \int_{B_R(y_n)} (u_n^2 + v_n^2) \geq \gamma >0. \label{lema_PS_eq5} \end{equation} If we define $(\tilde{u}_n(x),\tilde{v}_n(x)) = (u_n(x+y_n),v_n(x+y_n))$ we may suppose that, up to a subsequence, \begin{gather*} (\tilde{u}_n,\tilde{v}_n) \rightharpoonup (u,v)\quad \textrm{weakly in }X_{\epsilon}, \\ (\tilde{u}_n,\tilde{v}_n) \to (u,v)\quad \textrm{in }L^p(B_R(0))\times L^p(B_R(0)), \\ (\tilde{u}_n(x),\tilde{v}_n(x)) \to (u(x),v(x)) \quad \textrm{for a.e. }x \in \mathbb{R}^N, \end{gather*} for some nonnegative functions $u,v$. Moreover, in view of (\ref{lema_PS_eq5}), there exists a subset $\Omega \subset \mathbb{R}^N$ with positive measure such that $u,v$ are strictly positive in $\Omega$. We can use (\ref{lema_PS_eq1}) to rewrite (\ref{temquedacerto}) as \[ 0 0. \] for any $\delta>0$. Letting $n\to \infty$, using Fatou's lemma, we obtain \begin{align*} 0 0$. We obtain a contradiction by taking $\delta \to 0$. Thus, $\limsup_{n\to \infty} t_n \leq 1$, as claimed. Setting $t_0 = \limsup_{n \to \infty} t_n$, we consider two complementary cases: \noindent\textbf{Case 1: $t_0<1$.} In this case we may suppose, without loss of generality, that $t_n<1$ for all $n \in \mathbb{N}$. Thus, \begin{align*} m(V_{\infty}) &\leq E_{V_{\infty}}(t_{n}(u_{n},v_{n}))-\frac{1}{2}E'_{V_{\infty}}\langle (t_{n}(u_{n},v_{n}))(t_{n}(u_{n},v_{n})) \rangle \\ &= (\frac{p}{2}-1)t_{n}^{p}\int Q(u_{n},v_{n})\leq (\frac{p}{2}-1)\int Q(u_{n},v_{n})\\ &= I_{\epsilon}((u_{n},v_{n}))-\frac{1}{2}\langle I'_{\epsilon} ((u_{n},v_{n})),(u_{n},v_{n})\rangle \\ &= d+o_{n}(1). \end{align*} Taking the limit we conclude that $d \geq m(V_{\infty})$. \noindent\textbf{Case 2: $t_0 = 1$.} Up to a subsequence, we may suppose that $t_n \to 1$. We first note that \begin{align*} d + o_{n}(1) \geq m(V_{\infty})+I_{\epsilon}((u_{n},v_{n}))- E_{V_{\infty}}(t_{n}(u_{n},v_{n})). \end{align*} Note that \begin{align*} &I_{\epsilon}((u_{n},v_{n}))- E_{V_{\infty}}(t_{n}(u_{n},v_{n}))\\ &=\int\frac{(1-t_{n}^{2})}{2}(|\nabla u_{n}|^{2}+|\nabla v_{n}|^{2}) + \frac{1}{2}\int V(\epsilon x)(|u_{n}|^{2}+|v_{n}|^{2})\\ &\quad -\frac{t_{n}^{2}}{2}\int V_{\infty}(|u_{n}|^{p}+|v_{n}|^{p}) -(1-t_{n}^{p})\int Q(u_{n},v_{n}). \end{align*} Since $(\|(u_{n},v_{n})\|_{\epsilon})$ is bounded, we have \begin{gather*} \int \frac{(1-t_{n}^{2})}{2}(|\nabla u_{n}|^{2}+|\nabla v_{n}|^{2})=o_{n}(1),\\ (1-t_{n}^{p})\int Q(u_{n},v_{n})=o_{n}(1). \end{gather*} Using the condition \eqref{V}, we obtain $$ d + o_n(1) \geq m(V_{\infty}) - \delta C + o_n(1), $$ for any $\delta>0$. By taking $n\to \infty$ and $\delta \to 0$, we conclude that $d \geq m(V_{\infty})$. \end{proof} We present below two compactness results which we will need for the proof of the main theorem. \begin{proposition} The functional $I_{\epsilon}$ satisfies the $(PS)_{c}$ condition at any level $c < m({V_{\infty}})$ if $V_{\infty}<\infty$ and at any level $c\in \mathbb{R}$ if $V_{\infty}=\infty$. \label{prop_PS} \end{proposition} \begin{proof} Let $((u_n,v_n)) \subset X_{\epsilon}$ be such that $I_{\epsilon}((u_n,v_n)) \to c$ and $I_{\epsilon}'((u_n,v_n)) \to 0$ in $X_{\epsilon}^*$. By Lemma \ref{lema_pspositiva} the weak limit $(u,v)$ of $((u_n,v_n))$ is such that $I_{\epsilon}'((u,v))=0$. Thus, \[ I_{\epsilon}(u,v)= I_{\epsilon}(u,v)- \frac{1}{2}I'_{\epsilon}((u,v))(u,v) = (\frac{p}{2}-1)\int Q(u,v)\geq 0. \] Let $\tilde{u}_n = u_n - u$ and $\tilde{v}_n = v_n - v$. Arguing as in \cite[Lemma 3.3]{AlvCarMed} we can show that $I_{\epsilon}'(\tilde{u}_n,\tilde{v}_n) \to 0$ and $$ I_{\epsilon}((\tilde{u}_n,\tilde{v}_n)) \to c - I_{\epsilon}((u,v)) = d < m(V_{\infty}), $$ where we used that $c0$, we denote by $\Vert \cdot \Vert_{H_{\mu}}$ the following norm in $H^{1}(\mathbb{R}^N)\times H^{1}(\mathbb{R}^N)$ $$ \Vert (u,v) \Vert_{H_{\mu}} = \big\{ \int \Bigl[ |\nabla u|^2 + |\nabla v|^2+\mu (|u|^2+|v|^2) \Bigl]\big\}^{1/2} $$ which is well defined and equivalent to the standard norm of $H^{1}(\mathbb{R}^N)\times H^{1}(\mathbb{R}^N)$. Let $(w_{1},w_{2})$ be a ground state solution of the problem $(AP_{V_0})$ and consider $\eta:[0,\infty) \to \mathbb{R}$ a cut-off function such that $0 \leq \eta \leq 1$, $\eta(s) =1$ if $0 \leq s \leq \delta/2$ and $\eta(s)=0$ if $s \geq \delta$. We recall that $M$ denotes the set of global minima points of $V$ and define, for each $y \in M$, $\Psi_{i,\epsilon,y}:\mathbb{R}^N \to \mathbb{R}$ by setting $$ \Psi_{i,\epsilon,y}(x) = \eta(|\epsilon x-y|)w_{i}\big(\frac{\epsilon x -y}{\epsilon}\big), \quad i=1,2. $$ Let $t_{\epsilon}$ be the unique positive number satisfying \begin{align*} \max_{t \geq 0}I_{\epsilon}(t( \Psi_{1, \epsilon , y },\Psi_{2, \epsilon , y }))= I_{\epsilon}(t_{\epsilon}(\Psi_{1, \epsilon , y },\Psi_{2, \epsilon , y })), \end{align*} and define the map $\Phi_{\epsilon} : M \to \mathcal{N}_{\epsilon}$ in the following way \begin{equation} \Phi_{\epsilon}(y) = \Phi_{\epsilon,y} = (t_{\epsilon}(\Psi_{1, \epsilon , y },\Psi_{2, \epsilon , y })). \end{equation} In view of the definition of $t_{\epsilon}$ we have that the above map is well defined. Moreover, the following holds. \begin{lemma} $\lim_{\epsilon \to 0}I_{\epsilon}(\Phi_{\epsilon , y})= m(V_{0})$, uniformly in $y \in M$. \label{lema_funcional_phi} \end{lemma} \begin{proof} Suppose, by contradiction, that the lemma is false. Then there exist $\lambda>0$, $(y_n) \subset M$ and $\epsilon_n \to 0$ such that \begin{equation} |I_{\epsilon_n}(\Phi_{\epsilon_n,y_n}) - m(V_0)| \geq \lambda > 0. \label{lema_funcional_phi_eq1} \end{equation} Since $\langle I_{\epsilon_n}'(\Phi_{\epsilon{_n},y_{n}}), \Phi_{\epsilon{_n},y_{n}} \rangle =0$, we have that \[ \|(\Psi_{1,\epsilon_{n},y_{n}},\Psi_{2,\epsilon_{n},y_{n}}) \|^{2}_{\epsilon_{n}}=t_{\epsilon_{n}}^{p-2}p \int Q (\Psi_{1,\epsilon_{n},y_{n}},\Psi_{2,\epsilon_{n},y_{n}}). \] Moreover, making the change of variables $z=(\epsilon_n x-y_n)/\epsilon_n$ and using the Lebesgue theorem, we can check that \begin{gather*} \lim_{n\to \infty}\|(\Psi_{1,\epsilon_{n},y_{n}},\Psi_{2,\epsilon_{n},y_{n}}) \|^{2}_{\epsilon_{n}} = \| (w_{1},w_{2})\|^{2}_{H_{V_{0}}}, \\ \lim_{n\to \infty}\int Q(\Psi_{1,\epsilon_{n},y_{n}},\Psi_{2,\epsilon_{n},y_{n}}) =\int Q(w_{1},w_{2}). \end{gather*} Thus, up to a subsequence, we have $t_n \to t_0>0$ and \[ \|(w_{1},w_{2})\|^{2}_{H_{V_{0}}}=t_{0}^{p-2} \int Q(w_{1},w_{2}). \] Since $(w_{1},w_{2})\in \mathcal{M}_{V_{0}}$, we obtain $t_{0}=1$. Letting $n\to \infty$, we get \[ \lim_{n\to \infty}I_{\epsilon_{n}} (\Phi_{\epsilon_{n},y_{n}})= E_{V_{0}}(w_{1},w_{2})= m(V_{0}), \] which contradicts (\ref{lema_funcional_phi_eq1}) and proves the lemma. \end{proof} For any $\delta>0$, let $\rho=\rho_{\delta}>0$ be such that $M_{\delta} \subset B_{\rho}(0)$. Let $\chi:\mathbb{R}^N \to \mathbb{R}^N$ be defined as $\chi(x)=x$ for $|x| < \rho$ and $\chi(x)=\rho x/|x|$ for $|x| \geq \rho$. Finally, let us consider the barycenter map $\beta_{\epsilon}:\mathcal{N}_{\epsilon} \to \mathbb{R}^N$ given by $$ \beta_{\epsilon}(u,v)=\frac{\int\chi(\epsilon x)|u(x)|^{2}}{\int | u(x)|^{2}} +\frac{\int\chi(\epsilon x)| v(x)|^{2}}{\int|v(x)|^{2}}. $$ \begin{lemma} $\lim_{\epsilon \to 0} \beta_{\epsilon}(\Phi_{\epsilon,y}) = y$ uniformly for $y \in M$. \label{lema_beta_funcional} \end{lemma} \begin{proof} Arguing by contradiction, we suppose that there exist $\lambda>0$, $(y_n) \subset M$ and $\epsilon_n \to 0$ such that \begin{equation} \left| \beta_{\epsilon_n}(\Phi_{\epsilon_n,y_n}) - y_n \right| \geq \lambda > 0. \label{lema_beta_funcional_eq1} \end{equation} By using the change of variables $z=(\epsilon_nx-y_n)/\epsilon_n$, we get \begin{align*} \beta_{\epsilon}(\Phi_{\epsilon_{n},y_{n}}) &= y_{n} + \frac{\int [\chi(\epsilon_{n}z + y_{n})-y_{n}]|\eta(|\epsilon_{n}z|)w_{1}(z)|^{2}} {\int|\eta(|\epsilon_{n}z|)w_{1}(z)|^{2}}\\ &\quad +\frac{\int[\chi(\epsilon_{n}z + y_{n})-y_{n}]|\eta(|\epsilon_{n}z|)w_{2}(z)|^{2}} {\int|\eta(|\epsilon_{n}z|)w_{2}(z)|^{2}}. \end{align*} Since $(y_n) \subset M \subset B_{\rho}(0)$ we have that $ \chi(\epsilon_nz+y_n) - y_n =o_n(1)$. Hence, by the Lebesgue theorem, we conclude that \[ \beta_{\epsilon_n}(\Phi_{\epsilon_n,y_n}) - y_n =o_n(1), \] which contradicts (\ref{lema_beta_funcional_eq1}) and proves the lemma. \end{proof} \begin{lemma}[A Compactness Lemma]\label{lema_distancia_translado_aux} Let $((u_{n},v_{n})) \subset {\mathcal{M}}_{\mu}$ be a sequence satisfying $E_{\mu}(u_{n},v_{n})\to m({\mu})$. Then, \begin{itemize} \item[(a)] $((u_{n},v_{n}))$ has a subsequence strongly convergent in $H^{1}(\mathbb{R}^N)\times H^{1}(\mathbb{R}^N)$, or \item[(b)] there exists a sequence $(\tilde{y}_{n}) \subset \mathbb{R}^N$ such that, up to a subsequence, $$ (\tilde{u}_{n}(x),\tilde{v}_{n}(x))=(u_{n}(x + \tilde{y}_{n}),v_{n}(x + \tilde{y}_{n})) $$ converges strongly in $H^{1}(\mathbb{R}^N)\times H^{1}(\mathbb{R}^N)$. \end{itemize} In particular, there exists a minimizer for $m({\mu})$. \end{lemma} \begin{proof} Applying Ekeland's variational principle \cite[Theorem 8.5 ]{Wil}, we may suppose that $((u_{n},v_{n}))$ is a (PS)$_{m(\mu)}$ sequence for $E_{\mu}$. Thus, going to a subsequence if necessary, we have that $(u_n,v_{n}) \rightharpoonup (u,v)$ weakly in $H^{1}(\mathbb{R}^N)\times H^{1}(\mathbb{R}^N)$ with $(u,v)$ being a critical point of $E_{\mu}$. If $(u,v)\neq (0,0)$, it is easy to check that $(u,v)$ is a ground state solution of the autonomous problem \eqref{ASmu}, that is, $E_{\mu}(u,v)=m(\mu)$. We now consider the complementary case $(u,v) =(0,0)$. In this case, by Remark \ref{lema_lions_remark}, there exist a sequence $(\tilde{y}_n) \subset \mathbb{R}^N$ and constants $R,\gamma>0$ such that $$ \liminf_{n\to\infty} \int_{B_R(\tilde{y}_n)}(|u_n|^{2}+ |v_n|^2) \geq \gamma > 0. $$ Defining $\tilde{u}_n(x) = u_n(x+\tilde{y}_n)$ and $\tilde{v}_n(x) = v_n(x+\tilde{y}_n)$ we have that $((\tilde{u}_n,\tilde{v}_n))$ is also a (PS)$_{m(\mu)}$ sequence of $E_{\mu}$ such that $(\tilde{u}_n,\tilde{v}_n) \rightharpoonup (\tilde{u},\tilde{v}) \neq (0,0)$. It follows from the first part of the proof that, up to a subsequence, $((\tilde{u}_n,\tilde{v}_n))$ converges in $H^{1}(\mathbb{R}^N)\times H^{1}(\mathbb{R}^N)$. The lemma is proved. \end{proof} \begin{lemma} Let $\epsilon_n \to 0$ and $((u_{n},v_{n})) \subset \mathcal{N}_{\epsilon_n}$ be such that $I_{\epsilon_n}((u_{n},v_{n}))\to m(V_0)$. Then there exists a sequence $(\tilde{y}_n) \subset \mathbb{R}^N$ such that $(\tilde{u}_{n},\tilde{v}_{n})(x)=(u_{n}(x+\tilde{y}_{n}), v_{n}(x+\tilde y_{n}))$ has a convergent subsequence in $H^{1}(\mathbb{R}^N)\times H^{1}(\mathbb{R}^N)$. Moreover, up to a subsequence, $(y_n) = (\epsilon_n\tilde{y}_n)$ is such that $y_n \to y \in M$. \label{lema_distancia_translado} \end{lemma} \begin{proof} Arguing as in Remark \ref{lema_lions_remark}, we obtain a sequence $(\tilde{y}_n) \subset \mathbb{R}^N$ such that \[ \tilde{u}_{n}\rightharpoonup \tilde{u} \quad\text{in }H^{1}(\mathbb{R}^N) \ and \ \tilde{v}_{n}\rightharpoonup \tilde{v} \quad\text{in } H^{1}(\mathbb{R}^N), \] where $\tilde{u}_{n}={u}_{n}(x + \tilde{y}_{n})$ and $\tilde{v}_{n}={v}_{n}(x + \tilde{y}_{n})$ with $\tilde{u}\not\equiv 0$ and $\tilde{v}\not\equiv 0$. Let $(t_n) \subset (0,+\infty)$ be such that $(\hat{u}_{n},\hat{v}_{n})=t_{n}(\tilde{u}_{n},\tilde{v}_{n}) \in {\mathcal{M}}_{V_{0}}$. Defining $y_n = \epsilon_n\tilde{y}_n$, changing variables and recalling that $(u_n,v_{n}) \in \mathcal{N}_{\epsilon_n}$, we get \begin{align*} &E_{V_0}((\hat{u}_{n},\hat{v}_{n})) \\ &\leq \frac{1}{2} \int \left[|\nabla \hat{u}_{n}|^{2}+|\nabla \hat{v}_{n}|^{2}+ V(\epsilon_{n}x+y_{n})\left( |\hat{u}_n|^2 +|\hat{v}_n|^2\right)\right] -\int Q(\hat{u}_{n},\hat{v}_{n})\\ &=I_{\epsilon_n}((\hat{u}_{n},\hat{v}_{n})) \\ &=I_{\epsilon_n}(t_{n}(\tilde{u}_{n},\tilde{v}_{n}))\\ &\leq I_{\epsilon_n}((u_{n},v_{n}))= m(V_0) +o_n(1). \end{align*} Hence $$ E_{V_{0}}((\hat{u}_{n},\hat{v}_{n}))\to m({V_{0}}). $$ Since $(t_{n})$ is bounded, the sequence $((\hat{u}_{n},\hat{v}_{n}))$ is also bounded in $H^{1}(\mathbb{R}^N)\times H^{1}(\mathbb{R}^N)$, thus for some subsequence, $(\hat{u}_{n},\hat{v}_{n})\rightharpoonup (\hat{u},\hat{v})$ in $H^{1}(\mathbb{R}^N)\times H^{1}(\mathbb{R}^N)$. Moreover, reasoning as in \cite{giovany}, up to some subsequence, still denote by $(t_{n})$,we can assume that $t_{n}\to t_{0}>0$, and this limit implies that $(\hat{u},\hat{v})\not\equiv (0,0)$. From Lemma \ref{lema_distancia_translado_aux} $(\hat{u}_{n},\hat{v}_{n})\to (\hat{u},\hat{v})$ in $H^{1}(\mathbb{R}^N)\times H^{1}(\mathbb{R}^N)$ and so, $(\tilde{u}_{n},\tilde{v}_{n})\to (\tilde{u},\tilde{v})$ in $H^{1}(\mathbb{R}^N)\times H^{1}(\mathbb{R}^N)$. To complete the proof of the lemma, it suffices to check that $(y_n) =(\epsilon_n \tilde{y}_n)$ has a subsequence such that $y_n \to y \in M$. Indeed, suppose by contradiction that $(y_{n})$ is not bounded, then there exists a subsequence, still denoted by $(y_{n})$, such that $|y_{n}|\to \infty$. Considering firstly the case $V_{\infty}=\infty$, the inequality \begin{align*} &\int V(\epsilon_{n} x + y_{n})(| u_{n}|^{2}+v_{n}|^{2})\\ &\leq \int(| \nabla u_{n}|^{2}+\nabla v_{n}|^{2}) +\int V(\epsilon_{n} x + y_{n})(| u_{n}|^{2}+|v_{n}|^{2})\\ &=p \int Q(u_{n},v_{n}), \end{align*} together with Fatou's Lemma imply \begin{align*} \infty = p\liminf_{n\to \infty}\int Q(u_{n},v_{n}), \end{align*} which is an absurd, because the sequence $Q(u_{n},v_{n})$ is bounded in $L^{1}(\mathbb{R}^N)$. Now, let us consider the case $V_{\infty}< \infty$. Since $(\hat{u}_{n},\hat{v}_{n})\to(\hat{u},\hat{v})$ in $H^{1}(\mathbb{R}^N)\times H^{1}(\mathbb{R}^N)$ and $V_{0} < V_{\infty}$, we have \begin{align*} m({V_{0}})&= \frac{1}{2}\int(| \nabla \hat{u}|^{2}+\nabla \hat{v}|^{2})+ \frac{1}{2}\int V_{0}(|\hat{u}|^{2} +| \hat{v}|^{2})- \int Q(\hat{u},\hat{v})\\ &< \frac{1}{2}\int(| \nabla \hat{u}|^{2}+\nabla \hat{v}|^{2})+ \frac{1}{2}\int V_{\infty}(|\hat{u}|^{2} +| \hat{v}|^{2})- \int Q(\hat{u},\hat{v})\\ &\leq \liminf_{n\to \infty}\Bigl[\frac{1}{2}\Bigl(\int((| \nabla \hat{u}_{n}|^{2}+\nabla \hat{v}_{n}|^{2})+ V(\epsilon_{n} x +y_{n})(|\hat{u}_{n}|^{2} +| \hat{v}_{n}|^{2}))\Bigl)\\ &\quad -\int Q(\hat{u}_{n},\hat{v}_{n})\Bigl], \end{align*} or, equivalently, \begin{align*} m({V_{0}})&<\liminf_{n\to \infty}\Bigl[ \frac{t_{n}^{2}}{2}\Bigl(\int((| \nabla \tilde{u}_{n}|^{2}+\nabla \tilde{v}_{n}|^{2})+ V(\epsilon_{n} x + y_{n})(|\tilde{u}_{n}|^{2} +|\tilde{v}_{n}|^{2}))\Bigl)\\ &\quad -\int Q(t_{n}\tilde{u}_{n},t_{n}\tilde{v}_{n}) \Bigl]. \end{align*} The last inequality implies, \[ m({V_{0}})< \liminf_{n\to \infty}I_{\epsilon_{n}}((t_{n}u_{n},t_{n}v_{n}))\leq \liminf_{n\to \infty}I_{\epsilon_{n}}((u_{n},v_{n}))= m({V_{0}}), \] which is impossible. Hence, $(y_{n})$ is bounded and, up to a subsequence, $y_{n}\to y\in\mathbb{R}^N$. If $y\not\in M$, then $V(y)>V_{0}$ and we obtain a contradiction arguing as above. Thus, $y\in M$ and the lemma is proved. \end{proof} Following \cite{CinLaz}, we introduce a subset of $\mathcal{N}_{\epsilon}$ which will be useful in the future. We take a function $h:[0,\infty) \to [0,\infty)$ such that $h(\epsilon) \to 0$ as $\epsilon \to 0$ and set $$ \Sigma_{\epsilon} = \{ (u,v) \in \mathcal{N}_{\epsilon} : I_{\epsilon}((u,v)) \leq m(V_0) + h(\epsilon) \}. $$ Given $y \in M$, we can use Lemma \ref{lema_funcional_phi} to conclude that $h(\epsilon) = |I_{\epsilon}(\Phi_{\epsilon,y}) - m(V_0)|$ is such that $h(\epsilon) \to 0$ as $\epsilon \to 0$. Thus, $\Phi_{\epsilon,y} \in \Sigma_{\epsilon}$ and we have that $\Sigma_{\epsilon} \neq \emptyset$ for any $\epsilon >0$. \begin{lemma} For any $\delta>0$ we have that $$ \lim_{\epsilon \to 0} \sup_{(u,v) \in \Sigma_{\epsilon}} \mathop{\rm dist}(\beta_{\epsilon}(u,v),M_{\delta}) = 0. $$ \label{distancia} \end{lemma} \begin{proof} Let $(\epsilon_n) \subset \mathbb{R}$ be such that $\epsilon_n \to 0$. By definition, there exists $((u_n,v_n)) \subset \Sigma_{\epsilon_n}$ such that $$ \mathop{\rm dist}(\beta_{\epsilon_n}(u_n,v_n),M_{\delta}) = \sup_{(u,v) \in \Sigma_{\epsilon_n}} \mathop{\rm dist}(\beta_{\epsilon_n}(u,v), M_{\delta}) + o_n(1). $$ Thus, it suffices to find a sequence $(y_n) \subset M_{\delta}$ such that \begin{equation} |\beta_{\epsilon_n}(u_n,v_n) - y_n| = o_n(1). \label{distanciaeq1} \end{equation} To obtain such sequence, we note that $((u_n,v_n)) \subset \Sigma_{\epsilon_n} \subset \mathcal{N}_{\epsilon_n}$. Thus, recalling that $m(V_0) \leq c_{\epsilon_n}$, we get $$ m(V_0)\leq c_{\epsilon_n} \leq I_{\epsilon_n}((u_n,v_n)) \leq m(V_0) + h(\epsilon_n), $$ from which follows that $I_{\epsilon_n}((u_n,v_n)) \to m(V_0)$. We may now invoke the Lemma \ref{lema_distancia_translado} to obtain a sequence $(\tilde{y}_n) \subset \mathbb{R}^N$ such that $(y_n) = (\epsilon_n \tilde{y}_n) \subset M_{\delta}$ for $n$ sufficiently large. Hence, $$ \beta_{\epsilon}(u_{n},v_{n})= y_{n}+\frac{\int[\chi (\epsilon_{n}z + y_{n})- y_{n}]|\tilde{u}_{n}(z)|^{2}}{\int| \tilde{u}_{n}(z)|^{2}}+\frac{\int[\chi (\epsilon_{n}z + y_{n})- y_{n}]| \tilde{v}_{n}(z)|^{2}}{\int| \tilde{v}_{n}(z)|^{2}}, $$ Since $\epsilon_nz+y_n \to y \in M$, we have that $ \beta_{\epsilon_n}(u_n,v_n) = y_n + o_n(1) $ and therefore the sequence $(y_n)$ verifies (\ref{distanciaeq1}). The lemma is proved. \end{proof} We are now ready to present the proof of the multiplicity result and the technique used here is due to Benci and Cerami \cite{BenCer}. \begin{proof}[Proof of Theorem \ref{th1}] Given $\delta>0$ we can use Lemmas \ref{lema_funcional_phi}, \ref{lema_beta_funcional}, \ref{distancia} and argue as in \cite[Section 6]{CinLaz} to obtain $\epsilon_{\delta}>0$ such that, for any $\epsilon \in (0,\epsilon_{\delta})$, the diagram $$ M \stackrel{\Phi_{\epsilon}}{\longrightarrow} \Sigma_{\epsilon} \stackrel{\beta}{\longrightarrow} M_{\delta} $$ is well defined and $\beta_{\epsilon} \circ \Phi_{\epsilon}$ is homotopically equivalent to the embedding $\iota:M\to M_{\delta}$. Moreover, using the definition of $\Sigma_{\epsilon}$ and taking $\epsilon_{\delta}$ small if necessary, we may suppose that $I_{\epsilon}$ satisfies the Palais-Smale condition in $\Sigma_{\epsilon}$. Standard Ljusternik-Schnirelmann theory and Corollary \ref{lema_pcNehari} provide at least cat$_{\Sigma_{\epsilon}}(\Sigma_{\epsilon})$ solutions of the problem \eqref{Sep}. The inequality \begin{equation*} \mbox{cat}_{\Sigma_{\epsilon}}(\Sigma_{\epsilon})\geq \mbox{cat}_{M_{\delta}}(M) \end{equation*} follows from arguments used in \cite[Lemma 4.3]{BenCer}. \end{proof} \subsection*{Acknowledgments} The author would like to thank Professor F. J. S. A. Corr\^{e}a for his help and encouragement. \begin{thebibliography}{00} \bibitem{AlvCarMed} {C. O. Alves, P. C. Carri\~ao} and {E. S. Medeiros}, \emph{Multiplicity of solutions for a class of quasilinear problems in exterior domains with Newmann conditions}, Abs. Appl. Analysis \textbf{3} (2004), 251--268. \bibitem{AlvFig} {C. O. Alves} and {G. M. Figueiredo}, \emph{Existence and multiplicity of positive solutions to a p-Laplacian equation in $\mathbb{R}^N$}, Differential Integral Equations, 19-2(2006)143-162. \bibitem{AlvesMonari}{C. O. 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