\documentclass[reqno]{amsart} \usepackage{hyperref} \AtBeginDocument{{\noindent\small \emph{Electronic Journal of Differential Equations}, Vol. 2016 (2016), No. 158, pp. 1--24.\newline ISSN: 1072-6691. URL: http://ejde.math.txstate.edu or http://ejde.math.unt.edu} \thanks{\copyright 2016 Texas State University.} \vspace{8mm}} \begin{document} \title[\hfilneg EJDE-2016/158\hfil Multiplicity and concentration behavior of solutions] {Multiplicity and concentration behavior of solutions for a quasilinear problem involving $N$-functions via penalization method} \author[C. O. Alves, A. R. da Silva \hfil EJDE-2016/158\hfilneg] {Claudianor O. Alves, Ailton R. da Silva} \address{Claudianor O. Alves \newline Universidade Federal de Campina Grande\\ Unidade Acad\^emica de Matem\'atica - UAMat\\ CEP: 58.429-900 - Campina Grande - PB , Brazil} \email{coalves@dme.ufcg.edu.br} \address{Ailton R. da Silva \newline Universidade Federal de Campina Grande\\ Unidade Acad\^emica de Matem\'atica - UAMat\\ CEP: 58.429-900 - Campina Grande - PB , Brazil} \email{ardsmat@gmail.com} \thanks{Submitted March 15, 2016. Published June 21, 2016.} \subjclass[2010]{35A15, 35B40, 35J62, 46E30} \keywords{Variational method; quasilinear problem, Orlicz-Sobolev space} \begin{abstract} In this work we study the existence, multiplicity and concentration of positive solutions for the quasilinear problem \[ - \Delta_{\Phi}u + V(\epsilon x)\phi(| u|)u = f(u)\quad \text{in } \mathbb{R}^N, \] where $\Phi(t) = \int_0^{| t|}\phi(s)sds$ is an $N$-function, $\Delta_{\Phi}$ is the $\Phi$-Laplacian operator, $\epsilon$ is a positive parameter, and $ N\geq 2$. \end{abstract} \maketitle \numberwithin{equation}{section} \newtheorem{theorem}{Theorem}[section] \newtheorem{lemma}[theorem]{Lemma} \newtheorem{proposition}[theorem]{Proposition} \newtheorem{corollary}[theorem]{Corollary} \allowdisplaybreaks \section{Introduction} Many recent studies have focused on the nonlinear Schr\"{o}dinger equation \begin{equation} i\epsilon \frac{\partial \Psi}{\partial t}=-\epsilon^2\Delta \Psi+(V(z)+E)\Psi-f(\Psi)\quad \text{for }z \in \mathbb{R}^N,\label{eNLS} \end{equation} where $N \geq 1$, $\epsilon > 0$ is a parameter and $V,f$ are continuous function verifying some conditions. This class of equation is one of the main subjects of the quantum physics, because it appears in problems which involve nonlinear optics, plasma physics and condensed matter physics. Knowledge of the solutions for the elliptic equation \begin{equation} \begin{gathered} - \epsilon^2 \Delta{u} + V(z)u=f(u) \quad \text{in } \mathbb{R}^N,\\ u \in H^1(\mathbb{R}^N), \end{gathered} \label{eSe} \end{equation} or equivalently \begin{equation} \begin{gathered} -\Delta{u} + V(\epsilon z)u=f(u) \quad \text{in } \mathbb{R}^N,\\ u \in H^1(\mathbb{R}^N), \end{gathered}\label{eS'e} \end{equation} has a great importance in the study of standing-wave solutions of \eqref{eNLS}. In recent years, the existence and concentration of positive solutions for general semilinear elliptic equations \eqref{eSe} have been extensively studied, see for example, Floer and Weinstein \cite{FW}, Oh \cite{O1,O2}, Rabinowitz \cite{rabinowitz}, Wang \cite{WX}, Ambrosetti and Malchiodi \cite{AM}, Ambrosetti, Badiale and Cingolani \cite{ABC}, Floer and Weinstein \cite{FN}, del Pino and Felmer \cite{DF1} and their references. In the above mentioned papers, existence, multiplicity and concentration of positive solutions have been obtained in connection with the geometry of the function $V$. In some of them, it was proved that the maximum points of the solutions are close to the set $$ \mathcal{V} = \{x \in \mathbb{R}^N : V(x) = \min_{z \in \mathbb{R}^N}V(z) \}, $$ when $\epsilon$ is small enough. Moreover, in a lot of problems, the multiplicity of solutions is related to topology richness of $\mathcal{V}$. By a mountain pass argument, Rabinowitz \cite{rabinowitz} proved the existence of positive solutions of \eqref{eSe}, for $\epsilon > 0$ small, whenever \begin{equation} \liminf_{|z| \to \infty} V(z) > \inf_{z \in \mathbb{R}^N}V(z)=V_0 >0. \label{eR} \end{equation} Later Wang \cite{WX} showed that these solutions concentrate at global minimum points of $V$ as $\epsilon$ tends to 0. Del Pino and Felmer \cite{DF1} found solutions which concentrate around local minimum of $V$ by introducing of a penalization method. More precisely, they assume that \begin{equation} V(x)\geq \inf_{z \in \mathbb{R}^N}V(z)=V_0>0 \quad \text{for all } x \in \mathbb{R}^N\label{eV0} \end{equation} and there is an open and bounded set $\Omega \subset \mathbb{R}^{^N}$ such that \begin{equation} \inf_{z \in \Omega}V(z)< \min_{z \in \partial \Omega}V(z). \label{eV1} \end{equation} Existence, multiplicity and concentration of positive solutions have been also considered for quasilinear problems of the type $$ -\Delta_p{u}+V(\epsilon x)|u|^{p-2}u=f(u) \quad \text{in } \mathbb{R}^N, $$ and $$ -\Delta_p{u}-\Delta_q{u}+V(\epsilon x)(|u|^{p-2}u + |u|^{q-2}u)=f(u) \quad \text{in } \mathbb{R}^N. $$ Related to this subject, we cite the papers by Alves and Figueiredo \cite{AF2005,AG2006}, Benouhiba and Belyacine \cite{BB}, Cammaroto and Vilasi \cite{CV}, Chaves, Ercole and Miyagaki \cite{CEM}, Figueiredo \cite{F1}, Li and Liang \cite{LL} and their references. Recently, Alves and Silva \cite{AS} showed the existence, multiplicity and concentration of positive solutions for the following class of quasilinear problems \begin{align} \begin{gathered} - \Delta_{\Phi}u + V(\epsilon x)\phi(| u|)u = f(u)\quad \text{in } \mathbb{R}^N,\\ u \in W^{1, \Phi}(\mathbb{R}^N), \end{gathered} \label{eP2} \end{align} where $ N\geq 2$, $\epsilon$ is positive parameter, the operator $ \Delta_{\Phi} u = \mathrm{div}(\phi(|\nabla u|)\nabla u) $, where $\Phi(t) = \int_0^{| t|}\phi(s)sds$, called $\Phi$-Laplacian, is a natural extension of the $p$-Laplace operator and $V:\mathbb{R}^N \to \mathbb{R}$ is a continuous function which satisfies \eqref{eR}. This type of operator arises in applications, such as: Nonlinear Elasticity: $\Phi(t) = (1+t^2)^{\alpha}-1$, $\alpha \in (1, \frac{N}{N-2})$; Plasticity: $ \Phi(t) = t^{p}\ln(1+t)$, $1< \frac{-1+\sqrt{1+4N}}{2}1$; Plasma Physics: $\Phi(t) = \frac{1}{p}|t|^{p} + \frac{1}{q}|t|^{q}$ where $10$, $t>0$. \item There exist $l,m \in (1, N)$ such that \begin{gather*} l\leq m 0$ such that \[ | \phi'(t)t| \leq c\phi(t), \quad \forall \ t \in [0, +\infty). \] \end{enumerate} \end{itemize} We remark that the functions $\phi$ associated with each $N$-function mentioned in this introduction fulfill the conditions in (H1). Hereafter, we will say that $\Phi$ belongs to the class $ \mathcal{C}_m$ if $$ \Phi(t) \geq |t|^m , \quad \forall t \in \mathbb{R}.% \leqno{(\mathcal{C}_m)} $$ Also we define $$ \gamma=\begin{cases} m, & \text{if } \Phi \in \mathcal{C}_m, \\ l, & \text{if } \Phi \notin \mathcal{C}_m. \end{cases} $$ We assume that the function $f: \mathbb{R}\to \mathbb{R}$ is $C^1$ and satisfies: \begin{itemize} \item[(H2)] \begin{enumerate} \item There are functions $r,b:[0,+\infty) \to [0,+\infty)$ such that \[ \limsup_{|t|\to 0} \frac{f'(t)}{(r(|t|)|t|)'}=0, \quad \limsup_{|t|\to +\infty} \frac{|f'(t)|}{(b(|t|)|t|)'}<+\infty. \] \item There exists $\theta > m$ such that $ 0< \theta F(t) \leq f(t)t$ for all $t>0$, where $ F(t)= \int_0^{s}f(s)\,ds$. \item The function $f(t)/t^{m -1}$ is increasing for $t>0$. \end{enumerate} \end{itemize} We assume that the function $r $ belongs to $C^1$ and satisfy: \begin{itemize} \item[(H3)] \begin{enumerate} \item $r$ is increasing. \item There exists a constant $\overline{c}>0$ such that $| r'(t)t | \leq \overline{c} r(t)$ for all $t\geq 0$. \item There exist positive constants $r_1$ and $r_2$ such that \[ r_1\leq \frac{r(t)t^2}{R(t)}\leq r_2, \quad \forall t>0, \] where $ R(t) = \int_0^{|t|} r(s)s\,ds$. \item The function $R$ satisfies \[ \limsup_{t\to 0}\frac{R(t)}{\Phi(t)}< +\infty ,\quad \limsup_{|t|\to +\infty}\frac{R(t)}{\Phi_{*}(t)}=0. \] \end{enumerate} \end{itemize} We assume that the function $b$ belongs to $C^1$ and satisfy: \begin{itemize} \item[(H4)] \begin{enumerate} \item $b$ is increasing. \item There exists a constant $\widetilde{c}>0$ such that $| b'(t)t| \leq \widetilde{c}b(t)$ for all $t\geq 0$. \item There exist positive constants $b_1, b_2 \in (1, \gamma^*)$ such that \[ b_1\leq \frac{b(t)t^2}{B(t)}\leq b_2, \quad \forall t> 0, \] where $\gamma^*=N\gamma/(N-\gamma)$ and $ B(t) = \int_0^{|t|} b(s)s\,ds$. \item The function $B$ satisfies \[ \limsup_{t\to 0}\frac{B(t)}{\Phi(t)}< +\infty, \quad \limsup_{|t|\to +\infty}\frac{B(t)}{\Phi_{*}(t)}=0, \] where $\Phi_{*}$ is the Sobolev conjugate function, which is defined by the inverse function of \[ G_{\Phi}(t) = \int_0^{t}\frac{\Phi^{-1}(s)}{s^{1+\frac{1}{N}}}ds. \] \end{enumerate} \end{itemize} Using the above hypotheses, we are able to state our main result. \begin{theorem}\label{thm1} Suppose that {\rm(H1)--(H4)}, \eqref{eV0} and \eqref{eV1} hold. Then, for any $\delta > 0$ small enough, there exists $\epsilon_{\delta}> 0$ such that \eqref{eP2} has at least $\operatorname{cat}_{M_{\delta}}(M)$ positive solutions, for any $0 < \epsilon < \epsilon_{\delta}$, where \[ M = \{x \in \Omega :V(x)=V_0\},\quad M_{\delta} = \big\{x \in \mathbb{R}^N :\operatorname{dist}(x,M) \leq \delta \big\}. \] Moreover, if $u_{\epsilon}$ denotes one of these solutions and $x_{\epsilon} \in \mathbb{R}^N$ is a global maximum point of $u_{\epsilon}$, we have that \[ \lim_{\epsilon \to 0}V(\epsilon x_{\epsilon}) = V_0. \] \end{theorem} We would like point out that, if $Y$ is a closed subset of a topological space $X$, the Lusternik-Schnirelman category $\operatorname{cat}_{X}(Y)$ is the least number of closed and contractible sets in $X$ which cover $Y$. The plan of this article is as follows: In Section 2, we will prove the existence and multiplicity of solutions for an auxiliary problem, more precisely, by using of the Lusternik-Schnirelman category theory, we show that the auxiliary problem has at last $\operatorname{cat}_{M_{\delta}}(M)$ positive solutions for $\epsilon$ small enough. In Section 3, we make some estimates to prove that the solutions found are solutions of the original problem for $\epsilon$ small enough. Finally, we write an Appendix A, where we show the existence of a special function used in Section 2. \section{An auxiliary problem} In this is section, motivated by some arguments explored in Alves and Figueiredo \cite{AF2005}, and mainly in del Pino and Felmer \cite{DF1}, we will show the existence and multiplicity of positive solutions for an auxiliary problem. To this end, we need to fix some notations, however if the reader does not know the main properties involving the Orlicz-Sobolev spaces, we suggest \cite[Section 2]{AS} for a brief review and \cite{Adams1,Adams2,DT,MU,Rao} for a more complete study. Since we intend to find positive solutions, we will assume that \begin{equation} \label{posf} f(t) = 0 \quad \text{for all } t< 0. \end{equation} Let $a, k > 0$ satisfy $$ k>\frac{(\theta - l)}{(\theta - m)}\frac{m}{l} \quad \text{and} \quad \frac{f(a)}{\phi(a)a} = \frac{V_0}{k}, $$ where $\theta$ is the number given in (H2)(3). Using the above numbers, we define the function \[ \widehat{f}(s) = \begin{cases} f(s) &\text{if } s\leq a\\ \frac{V_0}{k}\phi(s)s &\text{if } s>a. \end{cases} \] Fixing $t_0 < a < t_1$ with $t_0,t_1 \approx a$, it is possible to find a function $\eta \in C^1([t_0, t_1])$ which satisfies: \begin{itemize} \item[(H5)] \begin{enumerate} \item $\eta(s) \leq \widehat{f}(s)$ for all $s \in [t_0, t_1]$, \item $\eta(t_0) = \widehat{f}(t_0)$ and $\eta(t_1)= \widehat{f}(t_1)$, \item $\eta'(t_0) = (\widehat{f})'(t_0)$ and $\eta'(t_1)= (\widehat{f})'(t_1)$, \item The function $s \to \frac{\eta(s)}{\phi(s)s}$ is nondecreasing for all $s \in [t_0, t_1]$. \end{enumerate} \end{itemize} In Appendix A, we shown the existence of $\eta$. Using the functions $\eta$ and $\widehat{f}$, we consider two new functions \begin{gather*} \widetilde{f}(s) = \begin{cases} \widehat{f}(s) & \text{if } s \not\in [t_0, t_1],\\ \eta(s) &\text{if } s \in [t_0, t_1], \end{cases} \\ g(x, s) = \chi_{\Omega}(x)f(s) + (1-\chi_{\Omega}(x))\widetilde{f}(s), \end{gather*} where $\chi_{\Omega}$ is the characteristic function related to the set $\Omega$. From definition of $g$, we see that $g$ is a Carath\'eodory function with \begin{gather}\label{posg} g(x, s) = 0, \quad \forall (x,s) \in \mathbb{R}^N \times (-\infty, 0],\\ \label{gf} g(x, s) \leq f(s), \quad \forall (x,s) \in \mathbb{R}^N \times \mathbb{R}. \end{gather} Moreover, for each $x \in \mathbb{R}^N$, the function $s\to g(x, s)$ is of class $C^1$ and it satisfies the following conditions: \begin{itemize} \item[(H6)] \begin{enumerate} \item $\limsup_{|s|\to 0}\frac{g(x, s)}{\phi(|s|)|s|}=0$, uniformly in $x \in \mathbb{R}^N$. \item $\limsup_{|s|\to +\infty}\frac{ g(x, s)}{b(|s|)|s|}< +\infty$, uniformly in $x \in \mathbb{R}^N$. \item $0\leq\theta G(x, s) = \theta\int_0^{s}g(x, t)dt \leq g(x, s)s$ for all $(x,s) \in \Omega \times (0,+\infty)$. \item $0 0$. \end{enumerate} \end{itemize} Using the function $g$, we can consider the auxiliary problem \begin{equation} \begin{gathered} - \Delta_{\Phi}u + V(\epsilon x)\phi(|u|)u = g(\epsilon x, u) \quad \text{in } \mathbb{R}^N,\\ u \in W^{1, \Phi}(\mathbb{R}^N). \end{gathered} \label{ePA2} % \tag{$ \widetilde{P}_{\epsilon} $} \end{equation} Here, we would like to point out that if $\Omega_\epsilon$ denotes the set $\Omega / \epsilon$, that is, $$ \Omega_\epsilon=\{x \in \mathbb{R}^N:\epsilon x \in \Omega\} $$ and $u$ is a positive solution of \eqref{ePA2} with $u(x)\leq t_0$ for all $x \in \mathbb{R}^N\backslash \Omega_\epsilon$, then $u$ is also a positive solution of \eqref{eP2}. \subsection{Preliminary results} In what follows, we denote by $J_{\epsilon} : X_{\epsilon} \to \mathbb{R}$ the energy functional related to \eqref{ePA2} given by \[ J_{\epsilon}(u)= \int_{\mathbb{R}^N}\Phi(|\nabla u|)dx + \int_{\mathbb{R}^N}V(\epsilon x)\Phi(| u|)dx - \int_{\mathbb{R}^N}G(\epsilon x, u)dx, \] where $X_{\epsilon}$ denotes the subspace of $W^{1, \Phi}(\mathbb{R}^N)$ given by \[ X_{\epsilon}= \Big\{ u \in W^{1, \Phi}(\mathbb{R}^N): \int_{\mathbb{R}^N}V(\epsilon x)\Phi(| u |)dx < +\infty\Big\}, \] endowed with the norm \[ \| u\|_{\epsilon} = \| \nabla u\|_{\Phi} + \| u\|_{\Phi, V_{\epsilon}} \] where \begin{gather*} \| \nabla u \|_{\Phi}:= \inf\big\{ \lambda > 0; \int_{\mathbb{R}^N}\Phi\Big(\frac{| \nabla u|}{\lambda}\Big)dx \leq 1\big\},\\ \| u\|_{\Phi, V_{\epsilon}}:= \inf\big\{ \lambda > 0; \int_{\mathbb{R}^N}V(\epsilon x)\Phi\Big(\frac{| u|}{\lambda}\Big)dx \leq 1\big\}. \end{gather*} From \eqref{eV0}, the embeddings $X_{\epsilon}\hookrightarrow L^{\Phi}( \mathbb{R}^N)$ and $X_{\epsilon}\hookrightarrow L^{B}( \mathbb{R}^N)$ are continuous. Using the above embeddings, a direct computation yields $J_{\epsilon} \in C^1(X_{\epsilon}, \mathbb{R})$ with \[ J'_{\epsilon}(u)v = \int_{\mathbb{R}^N}\phi(|\nabla u|)\nabla u \nabla v \, dx + \int_{\mathbb{R}^N}V(\epsilon x)\phi(| u|)uv \, dx - \int_{\mathbb{R}^N}g(\epsilon x, u)v \, dx, \] for all $u,v \in X_{\epsilon}$. Thereby, $u \in X_\epsilon$ is a weak solution of \eqref{ePA2} if, and only if, $u$ is a critical point of $J_{\epsilon}$. Furthermore, by \eqref{posg}, the critical points of $J_{\epsilon}$ are nonnegative. \begin{lemma}\label{Lmtd21} Let $(u_n)$ be a sequence $(PS)_{c}$. Then, $(u_n)$ is a bounded sequence in $X_{\epsilon}$. \end{lemma} \begin{proof} Since $(u_n)$ is a $(PS)_{c}$ sequence for $J_{\epsilon}$, there is $C_1>0$ such that \begin{align*} J_{\epsilon}(u_n) - \frac{1}{\theta}J_{\epsilon}'(u_n)u_n \leq C_1(1 + \| u_n\|_{\epsilon}), \quad \forall n \in \mathbb{N}. \end{align*} On the other hand, by (H1)(2), (H6)(3) and (H6)(4), $$ J_{\epsilon}(u_n) - \frac{1}{\theta}J_{\epsilon}'(u_n)u_n \geq C\Big(\int_{\mathbb{R}^N}\Phi(|\nabla u_n|)dx + \int_{\mathbb{R}^N}V(\epsilon x)\Phi(u_n)dx\Big), $$ where $C = \big[(1-\frac{m}{\theta}) -\big(1-\frac{l}{\theta}\big)\frac{m}{kl}\big]>0$. Hence, by \cite[Lemma 2.1]{FN}, \[ J_{\epsilon}(u_n) - \frac{1}{\theta}J_{\epsilon}'(u_n)u_n \geq C_1\big( \xi_0(\| \nabla u_n\|_{\Phi}) + \xi_0(\| u_n\|_{\Phi, V_{\epsilon}})\big), \quad \forall n \in \mathbb{N}. \] Now, the proof follows as in \cite[Lemma 4.2]{AGJ}. \end{proof} \begin{lemma}\label{Lema22} Let $(u_n)$ be a $(PS)_d$ sequence for $J_\epsilon$. Then for each $\tau>0$, there exists $\rho_0 = \rho_0(\tau) > 0$ such that \[ \limsup_{n \to +\infty}\int_{\mathbb{R}^N\backslash B_{\rho_0}(0)} \big[ \Phi(| \nabla u_n|) + V(\epsilon x)\Phi(| u_n|)\big]dx < \tau. \] \end{lemma} \begin{proof} For each $\rho>0$, let $\xi_\rho \in C^{\infty}(\mathbb{R}^N)$ satisfy \[ \xi_{\rho}(x) = \begin{cases} 0, & x \in B_{\frac{\rho}{2}}(0)\\ 1, & x \notin B_{\rho}(0)\\ \end{cases} \] with $0\leq \xi_{\rho}(x)\leq 1$ and $| \nabla \xi_{\rho}| \leq \frac{C}{\rho}$, where $C$ is a constant independent of $\rho$. Note that \begin{align*} J'_{\epsilon}(u_n)(\xi_\rho u_n) &= \int_{\mathbb{R}^N}\phi(| \nabla u_n|)\nabla u_n\nabla(\xi_\rho u_n)dx + \int_{\mathbb{R}^N}V(\epsilon x)\phi(| u_n|)u_n^2\xi_\rho dx\\ &\quad - \int_{\mathbb{R}^N}g(\epsilon x, u_n)u_n\xi_\rho dx. \end{align*} Choosing $\rho>0$ such that $\Omega_{\epsilon} \subset B_{\frac{\rho}{2}}(0)$, the condition (H1)(2) ensures that \begin{align*} &l\int_{\mathbb{R}^N}\xi_\rho\Big[\Phi(| \nabla u_n|) + V(\epsilon x)\Phi(| u_n|)\Big]dx \\ &\leq J'_{\epsilon}(u_n)(\xi_\rho u_n) - \int_{\mathbb{R}^N}u_n\phi(| \nabla u_n|)\nabla u_n\nabla \xi_\rho dx + \int_{\mathbb{R}^N\backslash \Omega_{\epsilon}} g(\epsilon x, u_n)u_n\xi_\rho dx. \end{align*} Gathering (H6)(4) and (H1)(2), \begin{align*} &l\int_{\mathbb{R}^N}\xi_\rho\Big[\Phi(| \nabla u_n|) + V(\epsilon x)\Phi(| u_n|)\Big]dx \\ &\leq J'_{\epsilon}(u_n)(\xi_\rho u_n) - \int_{\mathbb{R}^N}u_n\phi(| \nabla u_n|)\nabla u_n\nabla \xi_\rho\, dx + \frac{m}{k}\int_{\mathbb{R}^N}V(\epsilon x)\Phi(| u_n|)\xi_\rho \,dx. \end{align*} Since $(\xi_\rho u_n)$ is bounded in $X_{\epsilon}$ and $k>\frac{m}{l}$, by H\"{o}lder inequality there is a constant $C_1>0$ such that \[ \int_{\mathbb{R}^N}\xi_\rho \big[\Phi(| \nabla u_n|) + V(\epsilon x)\Phi(| u_n|)\big]dx \leq o_n(1)+ \frac{C_1}{\rho}. \] Now, fixed $\tau>0$, there exists $\rho_0>0$ such that $\frac{C_1}{\rho_0}<\tau$. Then, \[ \int_{\mathbb{R}^N\backslash B_{\rho_0}(0)} \big[\Phi(| \nabla u_n|) + V(\epsilon x)\Phi(| u_n|)\big]dx \leq o_n(1)+ \tau. \] Passing to the limit in the last inequality, it follows that \[ \limsup_{n \to +\infty}\int_{\mathbb{R}^N\backslash B_{\rho_0}(0)} \big[\Phi(| \nabla u_n|) + V(\epsilon x)\Phi(| u_n|)\big]dx \leq \tau. \] \end{proof} The lemma below establishes an important property of the $(PS)$ sequences of $J_{\epsilon}$. Since the proof follows as in \cite[Lemma 4.3]{AGJ}, we will omit it. \begin{lemma}\label{LemaP} Let $(u_n)$ be a $(PS)_d$ sequence for $J_\epsilon$ with $u_n\rightharpoonup u$ in $X_{\epsilon}$. Then, \begin{equation}\label{cgrad} \nabla u_n(x)\to \nabla u(x)\quad \text{a. e. in } \mathbb{R}^N. \end{equation} As a consequence of the above limit, we deduce that $u$ is a critical point for $J_\epsilon$. \end{lemma} \begin{proposition}\label{prop2.1} The functional $J_\epsilon$ satisfies the $(PS)$ condition. \end{proposition} \begin{proof} Let $(u_n)$ be a $(PS)_c$ sequence for $J_{\epsilon}$. From Lemma \ref{Lmtd21}, there exists $u \in X_{\epsilon}$ such that \begin{equation}\label{In} u_n \rightharpoonup u \quad \text{in } X_{\epsilon}. \end{equation} Moreover, by Lemma \ref{Lema22}, given $\tau>0$ there is $\rho_0>0$ satisfying \[ \limsup_{n \to +\infty}\int_{\mathbb{R}^N\backslash B_{\rho_0}(0)} \big[\Phi(| \nabla u_n|) + V(\epsilon x)\Phi(| u_n|)\big]dx < \tau. \] Increasing $\rho_0>0$ if necessary, the above limit together with $\Delta_2$-condition gives \begin{equation} \label{gradiente} \limsup_{n \to +\infty}\int_{\mathbb{R}^N}\Phi(| \nabla u_n - \nabla u|)dx \leq \limsup_{n \to +\infty}\int_{B_{\rho_0}(0)}\Phi(| \nabla u_n - \nabla u|)dx + 2\tau \end{equation} and \[ \limsup_{n \to +\infty}\int_{\mathbb{R}^N}V(\epsilon x)\Phi(| u_n - u|)dx \leq \limsup_{n \to +\infty}\int_{B_{\rho_0}(0)}V(\epsilon x)\Phi(| u_n - u|)dx + 2\tau. \] By \eqref{In}, up to a subsequence, $u_n \to u$ in $L^{\Phi}(B_{\rho_0}(0))$. This information combined with the last limit guarantees that $$ \limsup_{n \to +\infty}\int_{\mathbb{R}^N}V(\epsilon x)\Phi(| u_n - u|)dx \leq 2\tau. $$ As $\tau$ is arbitrary, we conclude that \begin{equation}\label{cg4} \limsup_{n \to +\infty}\int_{\mathbb{R}^N}V(\epsilon x)\Phi(| u_n - u|)dx=0. \end{equation} Now, we will show that \begin{align*} \limsup_{n \to +\infty}\int_{B_{\rho_0}(0)}\Phi(| \nabla u_n - \nabla u|)dx =0. \end{align*} By Lemma \ref{LemaP}, \[ \Phi(| \nabla u_n(x) -\nabla u(x)|) \to 0 \quad \text{a. e. in } B_{\rho_0}(0). \] Moreover, from $\Delta_2$-condition and (H1)(2), there exist constants $c_1, c_2>0$ such that \[ \Phi(| \nabla u_n -\nabla u|) \leq c_1\phi(|\nabla u_n |)|\nabla u_n |^2 + c_2\phi(|\nabla u |)|\nabla u |^2. \] Using again Lemma \ref{LemaP} \[c_1\phi(|\nabla u_n |)|\nabla u_n |^2 + c_2\phi(|\nabla u |)|\nabla u |^2 \to (c_1+c_2)\phi(|\nabla u |)|\nabla u |^2 \quad \text{a. e. in }\, B_{\rho_0}(0). \] On the other hand, as in \cite[Lemma 4.3]{AGJ}, \[ \int_{B_{\rho_0}(0)}(\phi(| \nabla u_n|)\nabla u_n - \phi(|\nabla u|)\nabla u)(\nabla u_n - \nabla u)dx = o_n(1), \] and so, \[ \int_{B_{\rho_0}(0)}\phi(| \nabla u_n|)|\nabla u_n|^2 dx\to\int_{B_{\rho_0}(0)}\phi(| \nabla u |)|\nabla u|^2 dx. \] Therefore, \[ \int_{B_{\rho_0}(0)}\Big[c_1\phi(| \nabla u_n|)| \nabla u_n|^2 + c_2\phi(| \nabla u|)|\nabla u|^2\Big]dx \to (c_1 + c_2)\int_{B_{\rho_0}(0)}\phi(| \nabla u|)|\nabla u|^2dx. \] Applying the Lebesgue's Theorem, we see that \[ \lim_{n\to +\infty}\int_{B_{\rho_0}(0)}\Phi(| \nabla u_n -\nabla u|)dx = 0. \] Combining the last limit with \eqref{gradiente}, we obtain \begin{equation}\label{cg5} \lim_{n\to +\infty}\int_{\mathbb{R}^N}\Phi(| \nabla u_n -\nabla u|)dx = 0. \end{equation} From \eqref{cg4} and \eqref{cg5}, we have the limit $u_n\to u$ in $ X_{\epsilon}$, which shows the $(PS)$ condition. \end{proof} Next, we show some results involving $J_{\epsilon}$ and its Nehari manifold. We recall that the Nehari manifold associated to $J_{\epsilon}$ is given by \[ \mathcal{N}_{\epsilon} = \big\{ u \in X_{\epsilon}\backslash \{0 \} : J'_{\epsilon}(u)u = 0 \big\}. \] Our first lemma ensures that the Nehari manifold has a positive distance from the origin in $X_\epsilon$. Once it follows by standard arguments, we omit its proof. \begin{lemma} \label{Nehari2} For all $u \in \mathcal{N}_{\epsilon}$, there exists $\sigma>0$, which is independent of $\epsilon$, such that $\| u\|_{\epsilon} > \sigma$. \end{lemma} The next lemma establishes an important characterization of the mountain pass level, which is useful in a lot of problems. In what follows, we denote by $c_{\epsilon, 1}$ and $c_{\epsilon, 2}$ the following numbers \[ c_{\epsilon, 1} = \inf_{u \in \mathcal{N}_{\epsilon}}J_{\epsilon}(u) \quad \text{and} \quad c_{\epsilon, 2} = \inf_{u \in X_{\epsilon}\backslash\{0\}}\max_{t\geq 0}J_{\epsilon}(tu). \] Fixing the subset \[ \mathcal{A}_{\epsilon} = \big\{ u \in X_{\epsilon} \, : \, u^{+}\neq0 \quad \text{and}\quad |supp(u)\cap\Omega_{\epsilon}|>0\big\} \] and the number \[ \overline{c}_{\epsilon, 2} = \inf_{u \in \mathcal{A}_{\epsilon}}\max_{t\geq 0}J_{\epsilon}(tu), \] it is easy to see that $c_{\epsilon, 2} = \overline{c}_{\epsilon, 2}$. \begin{lemma}\label{Lcar} Assume that {\rm (H1)--(H4)}, \eqref{eV0} and \eqref{eV1} hold. Then, for each $u \in \mathcal{A}_{\epsilon}$, there exists a unique $t_{u}>0$ such that $t_{u}u \in \mathcal{N}_{\epsilon}$ and $J_{\epsilon}(t_{u}u) = \max_{t \geq 0}J_{\epsilon}(tu)$. Moreover, \[ c_{\epsilon} = c_{\epsilon, 1} = c_{\epsilon, 2}, \] where $c_{\epsilon}$ denotes the mountain pass level associated with $J_{\epsilon}$. \end{lemma} \begin{proof} For each $u \in \mathcal{A}_{\epsilon}$, we define $h_{\epsilon}(t)= J_{\epsilon}(tu)$; that is, \[ h_{\epsilon}(t) = \int_{\mathbb{R}^N}\Phi(|\nabla (tu)|)dx + \int_{\mathbb{R}^N}V(\epsilon x)\Phi(| tu|)dx - \int_{\mathbb{R}^N}G(\epsilon x, tu)\, dx. \] \noindent\textbf{Existence.} By a direct computation, $h_{\epsilon}(t)> 0$ for $t$ enough small and $h_{\epsilon}(t)<0$ for $t$ sufficiently large. Thus, there is $t_{u}>0$ such that \[ h_{\epsilon}(t_{u}) = \max_{t\geq 0}h_{\epsilon}(t) = \max_{t\geq 0}J_{\epsilon}(tu). \] From this, $h_{\epsilon}'(t_{u}) = 0$, and so, $t_{u}u \in \mathcal{N}_{\epsilon}$. \noindent\textbf{Uniqueness.} Suppose that there exist $t_1, t_2 >0$ such that $t_1u, t_2u \in \mathcal{N}_{\epsilon}$ and $t_10]}g(\epsilon x, t_1u)t_1u\,dx, \\ \int_{\mathbb{R}^N}\phi(|\nabla (t_2u)|)|\nabla(t_2u)|^2dx + \int_{\mathbb{R}^N}V(\epsilon x)\phi(|t_2u|)|t_2u|^2dx = \int_{[u>0]}g(\epsilon x, t_2u)t_2udx. \end{gather*} Setting $\upsilon(t) = \phi(t)/t^{m-2}$ for all $t>0$, we have \begin{align*} &\int_{\mathbb{R}^N}\big(\upsilon(| t_1|\nabla u||) - \upsilon(| t_2|\nabla u||)\big)|\nabla u|^m dx + \int_{\mathbb{R}^N}V(\epsilon x)\big(\upsilon(| t_1| u||) - \upsilon(| t_2|u||)\big)|u|^m dx\\ &= \int_{[u>0]}\Big[\frac{g(\epsilon x, t_1u)}{(t_1u)^{m-1}} -\frac{g(\epsilon x, t_2u)}{(t_2u)^{m-1}}\Big]u^m dx. \end{align*} Thus, from \eqref{eV0} and (H1)(3), \begin{align*} &\int_{\mathbb{R}^N}\big(\upsilon(| t_1|\nabla u||) - \upsilon(| t_2|\nabla u||)\big)|\nabla u|^m dx\\ &+ \int_{(\mathbb{R}^N\backslash\Omega_{\epsilon})\cap[u>0]} \Big[\Big(\frac{V_0}{k}\upsilon(| t_1| u||) -\frac{\widetilde{f}(t_1u)}{(t_1u)^{m-1}}\Big) - \Big(\frac{V_0}{k}\upsilon(| t_2| u||) -\frac{\widetilde{f}(t_2u)}{(t_2u)^{m-1}}\Big)\Big]u^m dx\\ &\leq \int_{\Omega_{\epsilon}\cap[u>0]}\Big[\frac{f(t_1u)}{(t_1u)^{m-1}} -\frac{f(t_2u)}{(t_2u)^{m-1}}\Big]u^m \,dx. \end{align*} Now, consider the function \[ h(t) = \frac{V_0}{k}\upsilon(t) - \frac{\widetilde{f}(t)}{t^{m-1}} \] and note that $ h(t) = \upsilon(t)h_1(t)$, where $$ h _1(t) = \frac{V_0}{k}-\frac{\widetilde{f}(t)}{\phi(t)t}. $$ As $\upsilon$, $h_1$ are non increasing and nonnegative, $h$ is nonincreasing. Hence, $h(t_1u)\geq h(t_2u)$, and so, \begin{align*} 0&\leq \int_{\mathbb{R}^N}\big(\upsilon(| t_1|\nabla u||) - \upsilon(| t_2|\nabla u||)\big)|\nabla u|^m dx \\ &\quad +\int_{(\mathbb{R}^N\backslash\Omega_{\epsilon})\cap[u>0]}\big(h(t_1u) - h(t_2u)\big)u^m dx\\ &\leq \int_{\Omega_{\epsilon}\cap[u>0]}\Big[\frac{f(t_1u)}{(t_1u)^{m-1}} -\frac{f(t_2u)}{(t_2u)^{m-1}}\Big]u^m dx<0, \end{align*} which is an absurd. Therefore, $t_1 = t_2$. Now the proof follows as in \cite[Theorem 4.2]{Willen}. \end{proof} \subsection{ $(PS)$ condition for $J_\epsilon$ on $\mathcal{N}_{\epsilon}$} In this subsection, our main goal is to study the $(PS)$ condition for $J_\epsilon$ on $\mathcal{N}_{\epsilon}$, which will be used later on. In what follows, without loss of generality, we will assume that $$ V(0)=\min_{z \in \mathbb{R}^N}V(z)=V_0. $$ Using the above number, we fix the quasilinear problem \begin{equation} \begin{gathered} - \Delta_{\Phi}u + V_0\phi(| u|)u = f(u)\quad \text{in } \mathbb{R}^N,\\ u \in W^{1, \Phi}(\mathbb{R}^N). \end{gathered} \label{PAu2} % (P_0) \end{equation} We recall that the weak solutions of \eqref{PAu2} are critical points of the functional \[ E_0(u) = \int_{\mathbb{R}^N}\Phi(| \nabla u |)\,dx + V_0\int_{\mathbb{R}^N}\Phi(| u |)\,dx -\int_{\mathbb{R}^N}F(u)\,dx, \] which is well defined in $Y= W^{1, \Phi}(\mathbb{R}^N)$ endowed with the norm \[ \| u\|_{Y} = \| \nabla u\|_{\Phi} + V_0 \| u\|_{\Phi}. \] For the rest of this article, we denote by $d_0$ the mountain pass level of $E_0$, and by $\mathcal{M}_0$ the Nehari manifold \[ \mathcal{M}_0 = \big\{u \in Y \backslash \{0\}: E'_0(u)u=0 \big\}. \] The next lemma will be used to prove that $J_\epsilon$ satisfies the $(PS)$ condition on $\mathcal{N}_{\epsilon}$. \begin{lemma}\label{lema2.4} Consider $U=\{u \in \mathcal{N}_{\epsilon} : J_{\epsilon}(u)< d_0 +1\}$. Then, there are $\sigma_1, \sigma_2 >0$ independents of $\epsilon$, for $\epsilon$ small enough, such that \begin{itemize} \item[(a)] $\int_{\mathbb{R}^N}\Phi(u)dx \leq \sigma_1$ for all $u \in U$, \item[(b)] $\int_{\Omega_{\epsilon}}(f'(u)u^2 - (m-1)f(u)u)dx\geq \sigma_2$ for all $u \in U$. \end{itemize} \end{lemma} \begin{proof} (a) For any $u \in U$, \[ J_\epsilon(u) - \frac{1}{\theta}J_\epsilon'(u)u = J_\epsilon(u)< d_0 + 1. \] On the other hand, as in the proof of Lemma \ref{Lmtd21}, there is $C>0$ such that $$ J_{\epsilon}(u) - \frac{1}{\theta}J_{\epsilon}'(u)u \geq CV_0\int_{\mathbb{R}^N}\Phi(u)dx. $$ From this, $$ CV_0\int_{\mathbb{R}^N}\Phi(u)dx \leq d_0 + 1 \quad \forall u \in U, $$ and $(a)$ is proved. (b) The claim follows by proving that if $\epsilon_n \to 0$ and $u_n \in \mathcal{N}_{\epsilon_n}$, then \begin{equation} \label{LIM1} \liminf_{n \to +\infty}\int_{\Omega_{\epsilon_n}}(f'(u_n)u_n^2 - (m-1)f(u_n)u_n)dx > 0. \end{equation} By a Lions' type arguments, it is possible to show that there are $z_n \in \mathbb{R}^N$ and $r,\beta>0$ satisfying $$ \int_{\Omega_{\epsilon_n} \cap B_r(z_n)}\Phi(u_n)dx \geq \beta \quad \forall n \in \mathbb{N}. $$ Setting $\tilde{u}_n(x)=u_n(x+z_n)$, the above inequality together with $(f_3)$ permits to conclude that $$ \liminf_{n \to +\infty}\int_{\Omega_{\epsilon_n}-z_n}(f'(\tilde{u}_n)\tilde{u}_n^2 - (m-1)f(\tilde{u}_n)\tilde{u}_n)dx>0. $$ Once $$ \int_{\Omega_{\epsilon_n}}(f'(u_n)u_n^2 - (m-1)f(u_n)u_n)dx =\int_{\Omega_{\epsilon_n}-z_n}(f'(\tilde{u}_n)\tilde{u}_n^2 - (m-1)f(\tilde{u}_n)\tilde{u}_n)dx, $$ we obtain \eqref{LIM1}. This completes the proof of (b). \end{proof} \begin{proposition} The functional $J_{\epsilon}$ restricted to $\mathcal{N}_{\epsilon}$ satisfies the $(PS)_c$ condition for $c \in (0, d_0 + 1)$. \end{proposition} \begin{proof} Let $(u_n)$ be a $(PS)_{c}$ sequence on $\mathcal{N}_{\epsilon}$; that is, $J_\epsilon(u_n)\to c$ and $\| J'_{\epsilon}(u_n)\|_{*} = o_n(1)$. Then, there exists $(\lambda_n)\subset \mathbb{R}$ such that \begin{align*} J'_{\epsilon}(u_n) = \lambda_nL_\epsilon'(u_n) + o_n(1), \end{align*} where $L_\epsilon(v) = J'_{\epsilon}(v)v$ for all $v \in X_{\epsilon}$. Thus, \begin{equation}\label{2.3} \lambda_nL'_{\epsilon}(u_n)u_n = o_n(1). \end{equation} We claim that $\lambda_n = o_n(1)$. In fact, note that \begin{align*} L_\epsilon'(u_n)u_n &= \int_{\mathbb{R}^N}\big[\phi'(| \nabla u_n|)| \nabla u_n| + 2\phi(| \nabla u_n|)\big]| \nabla u_n|^2dx\\ &\quad + \int_{\mathbb{R}^N}V(\epsilon x)\big[\phi'(| u_n|)| u_n| + 2\phi(| u_n|)\big]| u_n|^2dx\\ &\quad - \int_{\mathbb{R}^N}\big[g'(\epsilon x, u_n)u_n^2 + g(\epsilon x, u_n)u_n\big]dx. \end{align*} By (H1)(3), \begin{align*} L_\epsilon'(u_n)u_n &\leq m\Big[\int_{\mathbb{R}^N}\phi(| \nabla u_n|)| \nabla u_n|^2dx + \int_{\mathbb{R}^N}V(\epsilon x)\phi(| u_n|)| u_n|^2dx\Big]\\ &\quad -\int_{\mathbb{R}^N}\big[g'(\epsilon x, u_n)u_n^2 + g(\epsilon x, u_n)u_n\big]dx, \end{align*} which implies \begin{align*} L_\epsilon'(u_n)u_n &\leq \int_{\mathbb{R}^N} \big[(m-1)g(\epsilon x, u_n)u_n - g'(\epsilon x, u_n)u_n^2\big]dx\\ &= \int_{\Omega_{\epsilon}\cup [u_n< t_0]} \big[(m-1)f(u_n)u_n - f'(u_n)u_n^2\big]dx\\ &\quad + \int_{(\mathbb{R}^N\backslash\Omega_{\epsilon})\cap [t_0\leq u_n\leq t_1]} \big[(m-1)\eta(u_n)u_n - \eta'(u_n)u_n^2\big]dx\\ &\quad +\int_{(\mathbb{R}^N\backslash\Omega_{\epsilon})\cap [u_n>t_1]} \big[(m-1)\frac{V_0}{k}\phi(u_n)u_n^2 - \frac{V_0}{k}(\phi(u_n)(u_n))' u_n^2\big]dx. \end{align*} Since $\eta'(t), (\phi(t)t)'\geq 0$ for $t>0$, it follows that, \begin{equation}\label{2.4} \begin{aligned} L_\epsilon'(u_n)u_n &\leq \int_{\Omega_{\epsilon}\cup [u_n< t_0]} \big[(m-1)f(u_n)u_n - f'(u_n)u_n^2\big]dx \\ &\quad + \int_{(\mathbb{R}^N\backslash\Omega_{\epsilon}) \cap [t_0\leq u_n\leq t_1]}(m-1)\eta(u_n)u_ndx \\ &\quad +\int_{(\mathbb{R}^N\backslash\Omega_{\epsilon})\cap [u_n>t_1]}(m-1)\frac{V_0}{k}\phi(u_n)u_n^2\,dx. \end{aligned} \end{equation} Now, the inequality $$ \eta(t) \leq \frac{V_0}{k}\phi(t)t \quad \forall t \in [t_0, t_1], $$ leads to \begin{align*} L_\epsilon'(u_n)u_n &\leq \int_{\Omega_{\epsilon}\cup [u_n< t_0]} \Big[(m-1)f(u_n)u_n - f'(u_n)u_n^2\Big]dx \\ &\quad + \int_{(\mathbb{R}^N\backslash\Omega_{\epsilon}) \cap [t_0\leq u_n\leq t_1]}(m-1)\frac{V_0}{k}\phi(u_n)u_n^2dx\\ &\quad +\int_{(\mathbb{R}^N\backslash\Omega_{\epsilon}) \cap [u_n>t_1]}(m-1)\frac{V_0}{k}\phi(u_n)u_n^2dx, \end{align*} and so, \[ L_\epsilon'(u_n)u_n \leq \int_{\Omega_{\epsilon}}\Big[(m-1)f(u_n)u_n - f'(u_n)u_n^2\Big]dx +\frac{V_0}{k}\int_{\mathbb{R}^N}\phi(u_n)u_n^2dx. \] Then, by Lemma \ref{lema2.4}, \[ -L_\epsilon'(u_n)u_n \geq \sigma_2 - \frac{m V_0\sigma_1}{k}. \] Therefore, by increasing of $k$ if necessary, we deduce that \[ -L_\epsilon'(u_n)u_n\geq C \quad \forall n \in \mathbb{N}. \] for some $C>0$. The last inequality combine with \eqref{2.3} to give $\lambda_n = o_n(1)$. Hence $J'_{\epsilon}(u_n)= o_n(1)$, which permits us to conclude that $(u_n)$ is a $(PS)_{c}$ sequence for $J_{\epsilon}$ in $X_{\epsilon}$. Now, the result follows from Proposition \ref{prop2.1}. \end{proof} As a byproduct of the arguments used in the proof of the last proposition, we have the following result. \begin{corollary}\label{cor2.1} The critical points of functional $J_{\epsilon}$ on $\mathcal{N}_{\epsilon}$ are critical points of $J_{\epsilon}$ in $X_{\epsilon}$. \end{corollary} \subsection{Multiplicity of solutions to \eqref{ePA2}} After the previous subsection, we are able to show the existence of multiple positive solutions for \eqref{ePA2}, by using of Lusternik-Schnirelman theory. Furthermore, we also study the behavior of the maximum points these solutions in relation to $M$. For each $\delta>0$ small enough, we consider $\vartheta \in C_0^{\infty}([0, +\infty),[0,1])$ with \[ \vartheta(s) = \begin{cases} 1, &\text{if } 0 \leq s \leq \frac{\delta}{2}\\ 0, &\text{if } s \geq \delta. \end{cases} \] Using the function above, for each $y \in M$ we set \[ \Psi_{\epsilon, y}(x)=\vartheta(|\epsilon x -y|)w(\frac{\epsilon x -y}{\epsilon}), \] where $w\in W^{1, \Phi}(\mathbb{R}^N)$ denotes a positive ground state solution of problem $(P_0)$, which exists according to \cite[Theorem 3.4]{AS}. By Lemma \ref{Lcar}, there exists $t_{\epsilon}>0$ such that $t_{\epsilon}\Psi_{\epsilon, y} \in {\mathcal N}_{\epsilon}$ and \[ J_{\epsilon}(t_{\epsilon}\Psi_{\epsilon, y}) = \max_{t\geq 0}J_{\epsilon}(t\Psi_{\epsilon, y}). \] From this, we can define $\widetilde{\Psi}_{\epsilon}: M \to {\mathcal N}_{\epsilon}$ by $\widetilde{\Psi}_{\epsilon}(y) = t_{\epsilon}\Psi_{\epsilon, y}$. \begin{lemma}\label{lemapsi2.1} The function $\widetilde{\Psi}_{\epsilon}$ satisfies \[ \lim_{\epsilon\to 0}J_{\epsilon}(\widetilde{\Psi}_{\epsilon}(y)) = d_0, \quad \text{uniformly in } y \in M. \] \end{lemma} \begin{proof} It is sufficient to show that for each $(y_n) \subset M$ and $(\epsilon_n)\subset \mathbb{R^{+}}$ with $\epsilon_n \to 0$, there is a subsequence such that \[ J_{\epsilon}(\widetilde{\Psi}_{\epsilon_n}(y_n)) \to d_0. \] Recall firstly that $J'_{\epsilon_n}(\widetilde{\Psi}_{\epsilon_n}(y_n)) \widetilde{\Psi}_{\epsilon_n}(y_n)= 0 $; that is, \begin{align*} &\int_{\mathbb{R}^N}\widehat{\phi}(|\nabla(\widetilde{\Psi}_{\epsilon_n}(y_n)|)dx + \int_{\mathbb{R}^N}V(\epsilon_nx)\widehat{\phi} (| \widetilde{\Psi}_{\epsilon_n}(y_n)|)dx \\ &= \int_{\mathbb{R}^N}g(\epsilon_n x, \widetilde{\Psi}_{\epsilon_n}(y_n)) \widetilde{\Psi}_{\epsilon_n}(y_n)dx, \end{align*} where $\widehat{\phi}(s)=\phi(s)s^2$ for all $s \geq 0$. Using (H1)(2) and \cite[Lemma 2.1]{FN}, \begin{equation}\label{psit1} \begin{aligned} &\int_{\mathbb{R}^N}\widehat{\phi}(|\nabla(\widetilde{\Psi}_{\epsilon_n}(y_n)|)dx +\int_{\mathbb{R}^N}V(\epsilon_nx)\widehat{\phi} (| \widetilde{\Psi}_{\epsilon_n}(y_n)|)dx \\ &\leq m \xi_1(t_{\epsilon_n})\Big[ \int_{\mathbb{R}^N} \Phi(|\nabla(\Psi_{\epsilon_n, y_n})|)dx + \int_{\mathbb{R}^N}V(\epsilon_nx)\Phi(|\Psi_{\epsilon_n, y_n}|)dx\Big], \end{aligned} \end{equation} where $\xi_1(t)=\max\{t^{l}, t^m \}$. On the other hand, the change of variable $z = \frac{\epsilon_nx - y_n}{\epsilon_n}$ leads to \begin{align*} &\int_{\mathbb{R}^N}g(\epsilon_n x, \widetilde{\Psi}_{\epsilon_n}(y_n)) \widetilde{\Psi}_{\epsilon_n}(y_n)dx \\ &=\int_{\mathbb{R}^N}g(\epsilon_nz + y_n, t_{\epsilon_n} \vartheta(|\epsilon_nz|)w(z))t_{\epsilon_n}\vartheta(|\epsilon_nz|)w(z)dx. \end{align*} Note that, if $z \in B_{\frac{\delta}{\epsilon_n}}(0)$, then $\epsilon_nz + y_n \in B_{\delta}(y_n) \subset M_{\delta}\subset \Omega$. Since $f=g$ in $\Omega$, $\vartheta\equiv1$ on $B_{\delta/2}(0)$ and $B_{\delta/2}(0) \subset B_{\frac{\delta}{2\epsilon_n}}(0)$, it follows that \begin{equation}\label{psit2} \begin{aligned} &\int_{\mathbb{R}^N}g(\epsilon_n x, \widetilde{\Psi}_{\epsilon_n}(y_n))\widetilde{\Psi}_{\epsilon_n}(y_n)dx \\ & \geq \int_{\mathbb{R}^N}f( t_{\epsilon_n}\vartheta(|\epsilon_nz|)w(z)) t_{\epsilon_n}\vartheta(|\epsilon_nz|)w(z)dx \\ &\geq \int_{B_{\delta/2}(0)}f( t_{\epsilon_n}w(z))t_{\epsilon_n}w(z)dx. \end{aligned} \end{equation} Combining \eqref{psit1} with \eqref{psit2}, we find \begin{align*} &\int_{B_{\delta/2}(0)}\frac{f(t_{\epsilon_n}w(z))} {(t_{\epsilon_n}w(z))^{m-1}}| t_{\epsilon_n}w(z)|^m dx \\ &\leq m\xi_1(t_{\epsilon_n})\Big[\int_{\mathbb{R}^N} \Phi(|\nabla(\Psi_{\epsilon_n, y_n})|)dx + \int_{\mathbb{R}^N}V(\epsilon_nx)\Phi(|\Psi_{\epsilon_n, y_n}|)dx\Big]. \end{align*} By Proposition \ref{L1}, we know that $w$ is a continuous function. Then, there is $z_0 \in \mathbb{R}^N$ such that \[ w(z_0)=\min_{z\in B_{\delta/2}(0)}w(z), \] and so, from (H2)(3), \begin{align*} &\frac{f(t_{\epsilon_n}w(z_0))}{(t_{\epsilon_n}w(z_0))^{m-1}} \int_{B_{\delta/2}(0)}| t_{\epsilon_n}w(z)|^m dx \\ &\leq m\xi_1(t_{\epsilon_n}) \Big[\int_{\mathbb{R}^N}\Phi(|\nabla(\Psi_{\epsilon_n, y_n})|)dx +\int_{\mathbb{R}^N}V(\epsilon_nx)\Phi(|\Psi_{\epsilon_n, y_n}|)dx\Big]. \end{align*} By (H2)(2), there are $c_1, c_2>0$ such that \begin{align*} &\big[c_1(t_{\epsilon_n}w(z_0))^{\theta-m} - c_2(t_{\epsilon_n}w(z_0))^{-m}\big]t_{\epsilon_n}^m \int_{B_{\delta/2}(0)}| w(z)|^m dx\\ &\leq m\xi_1(t_{\epsilon_n})\Big[\int_{\mathbb{R}^N} \Phi(|\nabla(\Psi_{\epsilon_n, y_n})|)dx + \int_{\mathbb{R}^N}V(\epsilon_nx)\Phi(|\Psi_{\epsilon_n, y_n}|)dx\Big]. \end{align*} Now, we show that $(t_{\epsilon_n}$ is bounded. To do this, we will suppose that for some subsequence $t_{\epsilon_n}\to +\infty$ and $ t_{\epsilon_n} \geq 1$ for all $n \in \mathbb{N}$. Thereby, $\xi_1(t_{\epsilon_n})=t_{\epsilon_n}^m $ and \begin{align*} &\big[c_1(t_{\epsilon_n}w(z_0))^{\theta-m} - c_2(t_{\epsilon_n}w(z_0))^{-m}\big] \int_{B_{\delta/2}(0)}| w(z)|^m \,dx\\ &\leq m\Big[\int_{\mathbb{R}^N}\Phi(|\nabla(\Psi_{\epsilon_n, y_n})|)dx + \int_{\mathbb{R}^N}V(\epsilon_nx)\Phi(|\Psi_{\epsilon_n, y_n}|)dx\Big]. \end{align*} The change of variable $z = \frac{\epsilon_nx - y_n}{\epsilon_n}$ together with the Lebesgue's Theorem ensures that \begin{gather*} \int_{\mathbb{R}^N}\Phi(|\nabla(\Psi_{\epsilon_n, y_n})|)dx \to \int_{\mathbb{R}^N}\Phi(|\nabla w|)\,dx ,\\ \int_{\mathbb{R}^N}V(\epsilon_nz + y_n)\Phi(|\Psi_{\epsilon_n, y_n}|)dx \to \int_{\mathbb{R}^N}V_0\Phi(|w|)dx. \end{gather*} Since $\theta> m$, we have $$ [c_1(t_{\epsilon_n}w(z_0))^{\theta-m} - c_2(t_{\epsilon_n}w(z_0))^{-m}\big] \to +\infty, $$ which contradicts the bove inequality. Therefore $(t_{\epsilon_n})$ is bounded, and for some subsequence, there exists $t_0\geq 0$ such that $t_{\epsilon_n} \to t_0$. Now, as $\widetilde{\Psi}_{\epsilon_n}(y_n) \in \mathcal{N}_{\epsilon_n}$, we know that $\|\widetilde{\Psi}_{\epsilon_n}(y_n)\|_{\epsilon_n} \geq \sigma$ for all $n \in \mathbb{N}$. Using again the Lebesgue's Theorem, it is possible to prove that $$ E'_0(t_0w)(t_0w)=0 \quad \text{and} \quad \| t_0w \|_{Y}\geq \sigma. $$ Thus, $t_0>0$ and $t_0w \in \mathcal{M}_0$. Furthermore, as $w$ is a ground state solution, we must have $t_0=1$. Then, the limit $t_n \to 1$ together with the Lebesgue's Theorem implies that \[ \lim_{n\to +\infty}J_{\epsilon}(\widetilde{\Psi}_{\epsilon_n}(y_n)) = E_0(w)= d_0, \] and the proof is complete. \end{proof} In the sequel, for any $\delta > 0$, we let $\rho = \rho(\delta)>0$ be such that $M_{\delta} \subset B_{\rho}(0)$. Let the function $\chi: \mathbb{R}^N \to \mathbb{R}^N$ be given by \[ \chi(x) =\begin{cases} x, &\text{if } x \in B_{\rho}(0),\\ \frac{\rho x}{| x|}, & \text{if } x \in B^{c}_{\rho}(0)\\ \end{cases} \] and $\beta: \mathcal{N}_{\epsilon} \to \mathbb{R}^N$ the barycenter map given by $$ \beta(u)= \frac{\int_{\mathbb{R}^N}\chi(\epsilon x)\Phi(| u|)dx} {\int_{\mathbb{R}^N}\Phi(| u|)dx}. $$ \begin{lemma} \label{lem2.11} The function $\widetilde{\Psi}_{\epsilon}$ satisfies \[ \lim_{\epsilon \to 0}\beta(\widetilde{\Psi}_{\epsilon}(y)) = y, \quad \text{uniformly in } M. \] \end{lemma} \begin{proof} The lemma follows by using of the definition of $\widetilde{\Psi}_{\epsilon}(y)$ together with the Lebesgue's Theorem. \end{proof} Hereafter, we consider the function $h: \mathbb{R}^{+} \to \mathbb{R}^{+}$ given by \[ h(\epsilon)=\sup_{y \in M}|J_\epsilon(\widetilde{\Psi}_{\epsilon}(y))-d_0|, \] which satisfies $ \lim_{\epsilon \to 0}h(\epsilon)=0$. Moreover, we set \[ \widetilde{\mathcal{N}}_{\epsilon}:= \{u \in \mathcal{N}_{\epsilon} : J_{\epsilon}(u) \leq d_0 + h(\epsilon) \}. \] From Lemma \ref{lemapsi2.1}, it follows that $\widetilde{\mathcal{N}}_{\epsilon}\neq\emptyset$ because $\widetilde{\Psi}_{\epsilon}(y) \in \widetilde{\mathcal{N}}_{\epsilon}$. Using the above notation, we have the following result. \begin{lemma}\label{lemapsi2.2} Let $\delta>0$ and $M_{\delta}= \big\{ x \in \mathbb{R}^N : \operatorname{dist}(x, M)\leq \delta \big\}$. Then, the limit below holds \[ \lim_{\epsilon\to 0}\sup_{u \in \widetilde{\mathcal{N}}_{\epsilon}} \inf_{y \in M_{\delta}} \big|\beta(u) - y \big| = 0. \] \end{lemma} The proof of the above lemma follows the proof in \cite[Lemma 3.7]{AF2005}. The next theorem is a result of multiplicity for the auxiliary problem. \begin{theorem}\label{thm2.1} For any $\delta > 0$ there exists $\epsilon_{\delta}>0$ such that \eqref{ePA2} has at least $\operatorname{cat}_{M_{\delta}}(M)$ positive solutions, for any $0 < \epsilon<\epsilon_{\delta}$. \end{theorem} \begin{proof} We fix a small $\epsilon > 0$. Then, by Lemmas \ref{lemapsi2.1} and \ref{lemapsi2.2}, $\beta\circ\widetilde{\Psi}_{\epsilon}$ is homotopic to inclusion map $\operatorname{id} : M \to M_{\delta}$, and so, \[ \operatorname{cat}_{\widetilde{\mathcal{N}}_{\epsilon}} (\widetilde{\mathcal{N}}_{\epsilon})\geq \operatorname{cat}_{M_{\delta}}(M). \] Since that functional $J_{\epsilon}$ satisfies the $(PS)_{c}$ condition for $c \in (d_0, d_0 + h(\epsilon))$, by the Lusternik-Schnirelman theory of critical points (\cite{Willen}), we can conclude that $J_{\epsilon}$ has at least $\operatorname{cat}_{M_{\delta}}(M)$ critical points on $\mathcal{N}_{\epsilon}$. Consequently by Corollary \ref{cor2.1}, $J_{\epsilon}$ has at least $\operatorname{cat}_{M_{\delta}}(M)$ critical points in $X_{\epsilon}$. \end{proof} Using the same approach explored in \cite[Section 3]{AS}, it is possible to show the following result. \begin{proposition}\label{L1} If $u_\epsilon \in W^{1, \Phi}(\mathbb{R}^N)$ is a nontrivial solution of \eqref{ePA2}, then $u_\epsilon$ is positive, $u_\epsilon \in L^{\infty}(\mathbb{R}^N) \cap C^{1, \alpha}_{\rm loc}(\mathbb{R}^N)$ and $\lim_{|x| \to +\infty}u_\epsilon(x) =0$. \end{proposition} \section{Multiplicity of solutions for the original problem} After the study made in Section 2, the main goal this section is to prove that the solutions obtained are solutions for the original problem when $\epsilon$ is small enough. To do this, we will show three technical results. \begin{proposition}\label{Propimp2.1} Let $\epsilon_n\to 0$ and $(u_n)\subset {\mathcal N}_{\epsilon_n}$ be such that $J_{\epsilon_n}(u_n)\to d_0$. Then, there exists a sequence $(\widetilde{y}_n)\subset\mathbb{R}^N$, such that $v_n(x) = u_n(x+\widetilde{y}_n)$ has a convergent subsequence in $W^{1,\Phi}(\mathbb{R}^N)$. Moreover, up to a subsequence, $y_n\to y \in M$, where $y_n=\epsilon_n\widetilde{y}_n$. \end{proposition} The proof of the above propositions follows from the proof of \cite[Proposition 5.3]{AS}. \begin{lemma}\label{Lemaunif} Let $(x_j)\subset \overline{\Omega}_{\epsilon_j}$ and $(\epsilon_j)$ be sequences with $\epsilon_j\to 0$ as $j\to +\infty$. If $v_j(x) = u_{\epsilon_j}(x + x_j)$ where $u_{\epsilon_j}$ is a solution of $(\widetilde{P}_{\epsilon_j})$ given by Theorem \ref{thm2.1}, then $(v_j)$ converges uniformly on compact subsets of $\mathbb{R}^N$. \end{lemma} \begin{proof} First of all, note that $v_j$ is a solution to the problem \begin{equation} \begin{gathered} - \Delta_{\Phi}v_j + V_j(x)\phi(| v_j|)v_j = g(\epsilon_j x + \overline{x}_j, v_j)\quad \text{in } \mathbb{R}^N,\\ v_j \in W^{1, \Phi}(\mathbb{R}^N), \\ v_j > 0 \quad \text{in } \mathbb{R}^N, \end{gathered} \label{Pj}% P_j \end{equation} where $V_j(x) = V(\epsilon_jx + \overline{x}_j)$ and $\overline{x}_j = \epsilon_jx_j$. Next, let $x_0 \in \mathbb{R}^N$, $R_0>1$, $0< t< s <1 \zeta \big\}$. Using $\eta_j$ as a test function and combining \eqref{eV0} with (H1)(2), we obtain \begin{align*} lQ_j&\leq m\int_{A_{j, \zeta, s}}\phi(| \nabla v_j |)|\nabla v_j |\, |\nabla\xi|\xi^{m - 1}(v_j-\zeta)_{+} dx\\ &\quad - V_0\int_{A_{j, \zeta, s}}\phi(| v_j |)v_j\xi^m (v_j-\zeta)_{+} dx + \int_{A_{j, \zeta, s}}g(\epsilon_{j }x + \overline{x}_j, v_j)\xi^m (v_j-\zeta)_{+} dx. \end{align*} Now, by repeating of the same arguments found in \cite[Lemma 3.5]{AS}, we obtain \[ Q_j \leq c_1\Big( \int_{A_{j, \zeta, s}} \big| \frac{v_j-\zeta}{s-t}\big|^{\gamma^{*}} dx + (\zeta^{\gamma^{*}} + 1)| A_{j, \zeta, s}|\Big). \] Using the condition $(\mathcal{C}_{m})$ and the definition of $\xi$, we obtain \[ \int_{A_{j, \zeta, t}}| \nabla v_j |^m dx \leq c_2 \Big( \int_{A_{j, \zeta, s}}\big| \frac{v_j-\zeta}{s-t}\big|^{\gamma^{*}} dx + (\zeta^{\gamma^{*}} + 1)| A_{j, \zeta, s}|\Big), \] where the constant $c_2$ does not depend of $\zeta$ and $\zeta\geq \zeta_0\geq 1$, for some constant $\zeta_0$. Now, fix $R_1>0$ and define \begin{gather*} \sigma_n = \frac{R_1}{2} + \frac{R_1}{2^{n +1}},\quad \overline{\sigma}_n = \frac{\sigma_n + \sigma_{n+1}}{2}, \\ \zeta_n = \frac{\zeta_0}{2}\Big(1 - \frac{1}{2^{n+1}}\Big),\quad Q_{n, j} = \int_{A_{j, \zeta_n, \sigma_n}}\big((v_j-\zeta_n)_{+} \big)^{\gamma^{*}}dx. \end{gather*} Arguing as in proof of \cite[Lemma 3.6]{AS}, we see that for each $j \in\mathbb{N}$, \[ Q_{n, j} \leq CA^{\eta}Q_{n, j}^{1 + \eta} \quad \forall n \in \mathbb{N}, \] where $C, \eta>0$ are independent of $n$ and $A>1$. Now, we claim that \[ Q_{0, j} \leq C^{\frac{1}{\eta}}A^{-1/\eta^2},\quad \text{for } j\approx +\infty. \] Indeed, by Proposition \ref{Propimp2.1}, we have $v_j \to v$ in $W^{1, \Phi}(\mathbb{R}^N)$. Therefore, \[ \limsup_{\zeta_0\to +\infty}\Big(\limsup_{j\to +\infty}Q_{0, j}\Big) =\limsup_{\zeta_0\to +\infty}\Big(\limsup_{j\to +\infty} \int_{A_{j, \zeta_0, \sigma_0}}(v_j - \frac{\zeta_0}{4})_{+}^{\gamma^*}dx \Big)=0. \] Then, there exist $j_0 \in \mathbb{N}$ and $\zeta^{*}_0 >0$ such that \[ Q_{0, j} \leq C^{\frac{1}{\eta}}A^{-1/\eta^2} \quad \text{for } j\geq j_0 \text{ and } \zeta_0 \geq \zeta^{*}_0. \] By \cite[Lemma 4.7]{LU}, $\lim_{n \to +\infty}Q_{n, j}=0$ for $j \geq j_0$. On the other hand, \[ \lim_{n \to +\infty}Q_{n, j} = \lim_{n \to +\infty}\int_{A_{j,K_n, \sigma_n}}(( v_j-\zeta_n)_{+})^{\gamma^*}dx =\int_{A_{j,\frac{\zeta}{2}, \frac{R_1}{2}}} (( v_j-\frac{\zeta_0}{2})_{+})^{\gamma^*}dx. \] Then \[ \int_{A_{j,\frac{\zeta_0}{2}, \frac{R_1}{2}}} (( v_j-\frac{\zeta_0}{2})_{+})^{\gamma^*}dx = 0,\quad \forall j \geq j_0 , \] and so, \[ v_j(x)\leq \frac{\zeta_0}{2}\quad \text{a.e in } B_{\frac{R_1}{2}}(x_0),\quad \forall j \geq j_0. \] Since $x_0 \in \mathbb{R}^N$ is arbitrary, we deduce that $v_j(x)\leq \zeta_0/2$ a.e in $\mathbb{R}^N$ for all $j \geq j_0$; that is, $$ \|v_j\|_\infty \leq \frac{\zeta_0}{2}, \quad \forall j \geq j_0. $$ Setting $C=\max\{\frac{\zeta_0}{2},\|v_1\|_\infty, \dots ,\|v_{j_0-1}\|_\infty \}$, we derive that $\|v_j\|_\infty \leq C$ for all $j \in \mathbb{N}$. Combining the above estimate with regularity theory, we deduce that $(v_j) \subset C^{1, \alpha}_{\rm loc}(\mathbb{R}^N)$. Moreover, there is $v \in C^{1, \alpha}_{\rm loc}(\mathbb{R}^N)$ such that \[ v_j \to v \quad \text{in} \quad C^{1, \alpha}(B_{\rho_0}(0)) \quad \forall \rho_0>0. \] \end{proof} \begin{lemma}\label{lema2.5} Let $(\epsilon_n)$ be a sequence with $\epsilon_n\to 0$ and let $(x_n)\subset \overline{\Omega}_{\epsilon_n}$ be a sequence such that $u_{\epsilon_n}(x_n)\geq \tau_0 >0$, for all $n \in \mathbb{N}$ and some $\tau_0 >0$, where $u_{\epsilon_n}$ is the solution of \eqref{ePA2} given by Theorem \ref{thm2.1}. Then $\lim_{n \to +\infty}V(\overline{x}_n) = V_0$, where $\overline{x}_n=\epsilon_nx_n$. \end{lemma} \begin{proof} As $\Omega$ is bounded and $\overline{x}_n \in \overline{\Omega}$, there exists $x_0 \in \overline{\Omega}$ such that, up to a subsequence, $\overline{x}_n\to x_0$. Then, the continuity of $V$ leads to \begin{equation}\label{2.6} \lim_{n \to +\infty}V(\overline{x}_n) = V(x_0)\geq V_0. \end{equation} In the sequel, we will argue by contradiction, by supposing that \begin{equation}\label{2.7} V(x_0)> V_0. \end{equation} From Theorem \ref{thm2.1}, $(u_{\epsilon_n}) \subset \mathcal{\widetilde{N}}_{\epsilon_n}$. Thus, \[ c_{\epsilon_n}\leq J_{\epsilon_n}(u_{\epsilon_n})0$ combine to give $v\not\equiv0$. For all $n \in \mathbb{N}$, let $t_n>0$ such that $t_nv_n \in \mathcal{M}_0$. Repeating the same arguments used in the proof of Lemma \ref{lemapsi2.1}, we obtain $t_n\to t_0$. Setting $\widetilde{v}_n = t_nv_n$, we see that \begin{align*} E_0(\widetilde{v}_n) &= \int_{\mathbb{R}^N}\Phi(| \nabla (t_nv_n)|)dx + V_0\int_{\mathbb{R}^N}\Phi(| t_nv_n|)dx - \int_{\mathbb{R}^N}F(t_nv_n)dx\\ &\leq \int_{\mathbb{R}^N}\Phi(| \nabla (t_nv_n)|)dx + \int_{\mathbb{R}^N}V(\epsilon_nz + \overline{x}_n)\Phi(| t_nv_n|)dx\\ &\quad - \int_{\mathbb{R}^N}G(\epsilon_nz+\overline{x}_n, t_nv_n)dx\\ &= \int_{\mathbb{R}^N}\Phi(| \nabla (t_nu_{\epsilon_n}|)dx + \int_{\mathbb{R}^N}V(\epsilon_nz)\Phi(| t_nu_{\epsilon_n}|)dx\\ &\quad - \int_{\mathbb{R}^N}G(\epsilon_nz, t_nu_{\epsilon_n})dx\\ &= J_{\epsilon_n}(t_nu_{\epsilon_n})\\ &\leq\max_{t\geq 0}J_{\epsilon_n}(tu_{\epsilon_n})=J_{\epsilon_n}(u_{\epsilon_n}). \end{align*} Thereby, \[ d_0\leq E_0(\widetilde{v}_n)\leq d_0 + o_n(1), \] or equivalently, $E_0(\widetilde{v}_n)\to d_{V_0}$. Applying \cite[Proposition 5.3]{AS}, we obtain \begin{equation}\label{2.8} \widetilde{v}_n\to \widetilde{v}\quad \text{in}\quad W^{1, \Phi}(\mathbb{R}^N), \end{equation} with $\widetilde{v} = t_0v\not\equiv 0$. Moreover, $E_0(\widetilde{v})= d_0$ and \[ d_0 < \int_{\mathbb{R}^N}\Phi(| \nabla \widetilde{v}|)dx + \int_{\mathbb{R}^N}V(x_0)\Phi(| \widetilde{v}|)dx - \int_{\mathbb{R}^N}F(\widetilde{v})dx. \] By \eqref{2.8} and Fatou's Lemma, \begin{align*} d_0&< \liminf_{n \to +\infty}\Big[\int_{\mathbb{R}^N} \big(\Phi(| \nabla \widetilde{v}_n|) + V(\epsilon_nz + \overline{x}_n)\Phi(| \widetilde{v}_n|) - F(\widetilde{v}_n)\big)dx\Big]\\ &\leq \liminf_{n \to +\infty}\Big[\int_{\mathbb{R}^N} \big(\Phi(| \nabla (t_nv_n)|) + V(\epsilon_nz + \overline{x}_n)\Phi(| t_nv_n|)\\ &\quad - G(\epsilon_nz + \overline{x}, t_nv_n)\big)dx\Big]\\ &= \liminf_{n \to +\infty}J_{\epsilon_n}(t_nu_{\epsilon_n})\\ &\leq \liminf_{n \to +\infty}J_{\epsilon_n}(u_{\epsilon_n}) = d_0 \end{align*} which is an absurd. Hence, from \eqref{2.6}, $\lim_{n \to +\infty}V(\overline{x}_n) = V_0$. \end{proof} Our next lemma will permit us to conclude that the solutions of the auxiliary problem are solutions for the original problem for $\epsilon$ small enough. \begin{lemma}\label{lemakapa} If $\kappa_{\epsilon} = \sup\big\{\max_{\partial \Omega_{\epsilon}}u_{\epsilon}: u_{\epsilon} \in \mathcal{\widetilde{N}}_{\epsilon} \, \text{is a solution of}\,\,\, (\widetilde{P}_{\epsilon})\big\}$, then \begin{equation}\label{pc2} \lim_{\epsilon \to0}\kappa_{\epsilon} = 0. \end{equation} \end{lemma} \begin{proof} Arguing by contradiction, we will assume that $$ \liminf_{\epsilon \to 0}\kappa_{\epsilon} > \tau_0>0, $$ for some $\tau_0>0$. From this, there is $(\epsilon_n)\subset (0, +\infty)$ and $x_n \in \partial \Omega_{\epsilon_n}$ such that \[ u_{\epsilon_n}(x_n) = \max_{x \in \partial \Omega_{\epsilon_n}}u_{\epsilon_n}(x)> \tau_0 \quad \forall n \in \mathbb{N}. \] Applying the Lemma \ref{lema2.5}, $\lim_{n \to +\infty}V(\overline{x}_n) = V_0$, where $\overline{x}_n=\epsilon_nx_n$. Since $(\overline{x}_n) \subset \partial \Omega$, there exist $x_0 \in \partial \Omega$ such that, up to a subsequence, $\overline{x}_n \to x_0$. Then, by continuity of $V$, we have $V(x_0) = V_0$, which contradicts \eqref{eV1}. Thereby, $\lim_{\epsilon \to 0}\kappa_{\epsilon}=0$. \end{proof} \begin{proof}[Proof of Theorem \ref{thm1}] (i) Multiplicity of positive solutions. From Theorem \ref{thm2.1}, for any $\delta > 0$ there exists $\epsilon_{\delta}>0$ such that \eqref{ePA2} has at least $\operatorname{cat}_{M_{\delta}}(M)$ positive solutions, for any $0 < \epsilon<\epsilon_{\delta}$. Let $u_{\epsilon}$ be a these solutions. By Lemma \ref{lemakapa} there exists $\overline{\epsilon}>0$ such that \[ \kappa_{\epsilon}< t_0, \quad \forall \epsilon \in (0, \overline{\epsilon}). \] Thus, $(u_{\epsilon} - t_0)_{+} \in W_0^{1, \Phi}(\mathbb{R}^N\backslash \Omega_{\epsilon})$ and \[ \omega_{\epsilon}(x)=\begin{cases} 0, & \text{if } x \in \Omega_{\epsilon} \\ (u_{\epsilon} - t_0)_{+}, &\text{if } x \in \mathbb{R}^N\backslash \Omega_{\epsilon} \end{cases} \] belongs to $ W^{1, \Phi}(\mathbb{R}^N)$. Using $\omega_{\epsilon}$ as text function, we have \begin{align*} &\int_{\mathbb{R}^N\backslash \Omega_{\epsilon}} \phi(| \nabla u_{\epsilon}|)\nabla u_{\epsilon}\nabla (u_{\epsilon} - t_0)_{+}dx + \int_{\mathbb{R}^N\backslash \Omega_{\epsilon}}V(\epsilon x) \phi(| u_{\epsilon}|) u_{\epsilon} (u_{\epsilon} - t_0)_{+}dx\\ &= \int_{\mathbb{R}^N\backslash \Omega_{\epsilon}} g(\epsilon x, u_{\epsilon})(u_{\epsilon} - t_0)_{+}dx, \end{align*} which implies \begin{align*} &\big(1-\frac{1}{k}\big)\Big[\int_{\mathbb{R}^N\backslash \Omega_{\epsilon}}\phi(|\nabla (u_{\epsilon} - t_0)_{+}|) |\nabla (u_{\epsilon} - t_0)_{+}|^2 dx \\ & +V_0\int_{\mathbb{R}^N\backslash \Omega_{\epsilon}}\phi(| u_{\epsilon}|) u_{\epsilon} (u_{\epsilon} - t_0)_{+}dx\Big]\leq 0. \end{align*} By Proposition \ref{L1}, we know that $u_{\epsilon}>0$, then the above inequality gives \[ \int_{\mathbb{R}^N\backslash \Omega_{\epsilon}} \phi(| u_{\epsilon}|) u_{\epsilon} (u_{\epsilon} - t_0)_{+}dx = 0, \] from where it follows that $u_{\epsilon} \leq t_0$ in $\mathbb{R}^N\backslash \Omega_{\epsilon}$. Hence, $u_{\epsilon}$ is a solution of \eqref{eP2} and we can conclude that \eqref{eP2} has at least $\operatorname{cat}_{M_{\delta}}(M)$ positive solutions for $\epsilon \in (0, \epsilon_{\delta})$. \smallskip \noindent(ii) Behavior of maximum points. Finally, if $u_{\epsilon_n}$ is a solution of problem \eqref{eP2} with $\epsilon_n$ instead of $\epsilon$. Then, $v_n(x) = u_{\epsilon_n}(x + \widetilde{y}_n)$ is a solution of the problem \begin{equation} \begin{gathered} - \Delta_{\Phi}v_n + V_n(x)\phi(| v_n|)v_n = f(v_n) \quad \text{in } \mathbb{R}^N,\\ v_n \in W^{1, \Phi}(\mathbb{R}^N), \\ v_n > 0 \quad \text{in } \mathbb{R}^N, \end{gathered} \label{P3}% \right. \tag{$ P_n $} \end{equation} where $V_n(x) = V(\epsilon_nx + \epsilon_n\widetilde{y}_n)$ and $(\widetilde{y}_n)$ is the sequence obtained in Proposition \ref{Propimp2.1}. Moreover, up to a subsequence, $v_n \to v$ in $W^{1, \Phi}(\mathbb{R}^N)$ and $y_n \to y$ in $M$, where $y_n=\epsilon_n\widetilde{y}_n$. Applying \cite[Proposition 6.1]{AS} and \cite[Lemma 6.4]{AS}, there are $R_0>0$ and $q_n \in B_{R_0}(0)$ such that $v_n(q_n) = \max_{z \in \mathbb{R}^N}v_n(z)$. Hence, $x_n = q_n + \widetilde{y}_n$ is a global maximum point of $u_{\epsilon_n}$ and $\epsilon_nx_n \to y$. Since $V$ is a continuous function, it follows that \[ \lim_{n \to +\infty}V(\epsilon_nx_n)=V(y) = V_0, \] and the concentration behavior is proved. \end{proof} \section{Appendix A: Existence of the function $\eta$.} In this appendix, we show the existence of $\eta$ which was used in Section 2. In what follows, we fix $\varrho$ small enough, such that the number $\lambda = a-\varrho$ satisfies \begin{equation}\label{varrho} f'(\lambda)>\frac{V_0}{k}(m-1)\phi(\lambda). \end{equation} Considering $h(s) = f(s)/\phi(s)$, we have that $h'(\lambda)> V_0/k$ and $h(\lambda)a \end{cases} \] satisfies $\widehat{h}(s) = \widehat{f}(s)/\phi(s)$ and \begin{itemize} \item $\frac{h(a)}{a} = \frac{f(a)}{\phi(a)a}= \frac{V_0}{k}$, \item $\frac{h(s)}{s} = \frac{f(s)}{\phi(s)s} = \frac{s^{m-2}}{\phi(s)}\frac{f(s)}{s^{m-1}}\uparrow$ for $s>0$, \item $h'(\lambda)> \frac{V_0}{k}$, \item $B:=\frac{V_0}{k}\lambda - h(\lambda)>0$. \end{itemize} The next lemma is a key step to get the function $\eta$. \begin{lemma} \label{lem4.1} There exist $t_0, t_1 \in (0,+\infty)$ such that $t_0 < a < t_1$ and $\widetilde{\eta} \in C^1([t_0, t_1])$, satisfying \begin{enumerate} \item $\widetilde{\eta}(t) \leq \widehat{h}(t)$, for all $t \in [t_0, t_1]$, \item $\widetilde{\eta}(t_0) = \widehat{h}(t_0)$ and $\widetilde{\eta}(t_1)= \widehat{h}(t_1)$, \item $(\widetilde{\eta})'(t_0) = (\widehat{h})'(t_0)$ and $(\widetilde{\eta})'(t_1)= (\widehat{h})'(t_1)$, \item The function $t \to \frac{\widetilde{\eta}(t)}{t}$ is nondecreasing for all $s \in [t_0, t_1]$. \end{enumerate} \end{lemma} \begin{proof} In what follows, for each $\delta>0$ small enough, we fix the numbers $$ \lambda=a-\delta,\quad B=h'(\lambda)>\frac{V_0}{k},\quad D=\frac{V_0 }{k}\lambda-h(\lambda) $$ where $\frac{V_0}{k}=\frac{h(a)}{a}$. Setting the function \[ y(t)=At^2+Bt, \] we have $ y(0)=0$ and $y'(0)=B$. Next, our goal is proving that there are $A<0$ and $T>\delta$ such that $$ y(T)=\frac{V_0}{k}T+D \quad \text{and} \quad y'(T)=\frac{V_0}{k}. $$ The above equalities are equivalent to the system \begin{gather*} AT^2+BT=\frac{V_0}{k}T+D\\ 2AT+B=\frac{V_0}{k} \end{gather*} whose solution is \begin{gather*} T=\frac{2D}{B-\frac{V_0}{k}} =\frac{2(\frac{V_0}{k}\lambda-h(\lambda))}{B-\frac{V_0}{k}}>\delta, \quad \text{if } \delta \approx 0^+, \\ A=-\frac{1}{4}\frac{(B-\frac{V_0}{k})^2}{D}. \end{gather*} Now, we set $\widetilde{\eta} :\mathbb{R} \to \mathbb{R}$ by $$ \widetilde{\eta}(t)=y(t-\lambda)+h(\lambda). $$ Note that $$ \widetilde{\eta}(\lambda)=h(\lambda), \quad \widetilde{\eta}'(\lambda)=h'(\lambda), \quad \widetilde{\eta}(T+\lambda)=\frac{V_0}{k}(T+\lambda), \quad \widetilde{\eta}'(T+\lambda)=\frac{V_0}{k}. $$ A simple computation gives \begin{gather*} \widetilde{\eta}(t) \leq \widehat{h}(t), \quad \forall t \in \mathbb{R},\\ \widetilde{\eta}'(t)t-\widetilde{\eta}(t)=At^2-A\lambda^2 +B\lambda -h(\lambda). \end{gather*} Thus, $$ \widetilde{\eta}'(t)t-\widetilde{\eta}(t)>0 \;\Leftrightarrow\; At^2-A\lambda^2 +B\lambda -h(\lambda)>0. $$ Moreover, $$ At^2-A\lambda^2 +B\lambda -h(\lambda)>0 \;\Leftrightarrow\; -t_* < t < t_* $$ where $$ t_*=\sqrt{\lambda^2-\frac{(B\lambda-f(\lambda))}{A}}=T+\lambda. $$ Therefore, $$ \widetilde{\eta}'(t)t-\widetilde{\eta}(t)>0 \quad \forall t \in [\lambda, T+\lambda), $$ from where it follows that $\widetilde{\eta}(t)/t$ is increasing in $[a-\delta,a+\tau]$, where $\tau=T-\delta>0$. \end{proof} Using the above lemma, the function $\eta(t) = \phi(t)\widetilde{\eta}(t)$ satisfies the following conditions: \begin{itemize} \item $\eta(t) \leq \phi(t) \widehat{h}(t) = \widehat{f}(t)$, for all $ t \in [t_0, t_1]$; \item $\eta(t_0) = \phi(t_0)\widehat{h}(t_0) = \widehat{f}(t_0)$ and $\eta(t_1) = \phi(t_1)\widehat{h}(t_1) = \widehat{f}(t_1)$; \item \begin{align*} \eta'(t_0) &= \phi'(t_0)\widetilde{\eta}(t_0) + \phi(t_0)(\widetilde{\eta})'(t_0)\\ &= \phi'(t_0)\widehat{h}(t_0) + \phi(t_0)(\widehat{h})'(t_0)\\ &= \phi'(t_0)h(t_0) + \phi(t_0)h'(t_0)\\ &= \big(\phi(t)h(t)\big)'(t_0) = f'(t_0); \end{align*} \item $\eta'(t_1)= (\widehat{f})'(t_1)$; \item $\frac{\eta(s)}{\phi(s)s} = \frac{\phi(s)\widetilde{\eta}(s)}{\phi(s)s} = \frac{\widetilde{\eta}(s)}{s}$; \end{itemize} From this, the function $\eta$ satisfies conditions (H5) mentioned in Section 2. \subsection*{Acknowledgments} C. O. Alves was partially supported by CNPq/Brazil 304036/ 2013-7 and INCT-MAT. \begin{thebibliography}{99} \bibitem{Adams1} A. Adams, J.F. Fournier; \emph{Sobolev Spaces}, 2nd ed., Academic Press (2003). \bibitem{Adams2} A. Adams, L. I. Hedberg; \emph{Function Spaces and Potential Theory, Grundlehren der Mathematischen Wissenschaften}, vol. 314, Springer-Verlag, Berlin, 1996. \bibitem {AF2005} C. O. Alves, G. M. Figueiredo; \emph{Multiplicity of Positive Solutions for a Quasilinear Problem in $\mathbb{R}^N$ Via Penalization Method}, Adv. Nonlinear Stud. 5 (2005), 551-572. \bibitem {AG2006} C. O. Alves, G.M. Figueiredo; \emph{Existence and multiplicity of positive solutions to a p-Laplacian equation in $\mathbb{R}^N$}, Differential Integral Equations 19 (2006), 143-162. \bibitem{AGJ} C. O. Alves, G. M. Figueiredo, J. A. Santos; \emph{Strauss and Lions type results for a class of Orlicz-Sobolev spaces and applications}, Topol. Methods Nonlinear Anal. 44 (2014), 435-456. \bibitem {AF} C. O. Alves, M. C. Ferreira; \emph{Multi-bump solutions for a class of quasilinear problems involving variable exponents}, Ann. Mat. Pura Appl. (2014). DOI 10.1007/s10231-014-0434-2. \bibitem{AS} C. O. Alves, A. R. Silva; \emph{Multiplicity and concentration of positive solutions for a class of quasilinear problems through Orlicz-Sobolev space}, arXiv:1506.01669v1. \bibitem{ABC} A. Ambrosetti, M. Badiale, S. Cingolani; \emph{Semiclassical states of nonlinear Schr\"{o}dinger equations}, Arch. Rational Mech. Anal. 140 (1997), 285-300. \bibitem{AM} A. Ambrosetti, A. Malchiodi; \emph{Concentration phenomena for NLS: recent results and new perspectives}. Perspectives in nonlinear partial differential equations, Contemp. Math., 446, Amer. Math. Soc., Providence, RI (2007), 19-30. \bibitem{BB} N. Benouhiba, Z. Belyacine; \emph{On the solutions of the $(p,q)$-Laplacian problem at resonance}, Nonlinear Anal. 77 (2013), 74-81 . \bibitem{BBR2} G. Bonanno, G. M. Bisci, V. Radulescu; \emph{Infinitely many solutions for a class of nonlinear eigenvalue problem in Orlicz-Sobolev spaces}, C. R. Math. Acad. Sci. Paris, Ser. I 349 (2011), 263-268. \bibitem{CV} F. Cammaroto, L. Vilasi; \emph{Multiple solutions for a nonhomogeneous Dirichlet problem in Orlicz-Sobolev spaces}, Appl. Math. Comput. 218 (2012), 11518-11527. \bibitem{CEM} M. F. Chaves, G. Ercole, O. H. Miyagaki; \emph{Existence of a nontrivial solution for the $(p, q)$-Laplacian in $\mathbb{R}^N$ without the Ambrosetti-Rabinowitz condition}, Nonlinear Anal. 114 (2015), 133-141. \bibitem{DF1}M. del Pino, P.L. Felmer; \emph{Local mountain pass for semilinear elliptic problems in unbounded domains}, Calc. Var. Partial Differential Equations 4 (1996), 121-137. \bibitem{Db} E. DiBenedetto; \emph{$C^{1, \gamma}$ local regularity of weak solutions of degenerate elliptic equations}, Nonlinear Anal. 7 (1985), 827-850. \bibitem{DT} T. K. Donaldson, N. S. Trudinger; \emph{Orlicz-Sobolev spaces and imbedding theorems}, J. Funct. Anal. 8 (1971), 52-75. \bibitem{F1} G. M. Figueiredo; \emph{Existence and multiplicity of solutions for a class of $p\&q$ elliptic problems with critical exponent}, Math. Nachr. 286 (2013), 1129-1141. \bibitem{FW} A. Floer, A. Weinstein; \emph{Nonspreading wave packets for the cubic Schr\"{o}dinger equation with a bounded potential}, J. Funct. Anal. 69 (1986), 397-408. \bibitem{FN} N. Fukagai, M. Ito, K. Narukawa; \emph{Positive solutions of quasilinear elliptic equations with critical Orlicz-Sobolev nonlinearity on $\mathbb{R}^N$}, Funkcial. Ekvac. 49 (2006), 235-267. \bibitem{FN1}N. Fukagai, M. Ito and K. Narukawa, Quasilinear elliptic equations with slowly growing principal part and critical Orlicz-Sobolev nonlinear term, Proc. Roy. Soc. Edinburgh Sect. A 139 (2009), 73-106. \bibitem{FN2}N. Fukagai, K. Narukawa; \emph{On the existence of multiple positive solutions of quasilinear elliptic eigenvalue problems}, Ann. Mat. Pura Appl. 186 (2007), 539-564. \bibitem{FUCHS} M. Fuchs, L. Gongbao; \emph{$L^{\infty}$-bounds for elliptic equations on Orlicz-Sobolev spaces}, Arch. Math. (Basel) 72 (1999), 293-297. \bibitem{Fusco} N. Fusco, C. Sbordone; \emph{Some remarks on the regularity of minima of anisotropic integrals}, Comm. Partial Differential Equations 18 (1993), 153-167. \bibitem{LU} O. A. Ladyzhenskaya, N. N. Ural'tseva; \emph{Linear and quasilinear elliptic equations}, Acad. Press (1968). \bibitem{LL} G. Li, X. Liang; \emph{The existence of nontrivial solutions to nonlinear elliptic equation of $p$-$q$-Laplacian type on $\mathbb{R}^N$}, Nonlinear Anal. 71 (2009), 2316-2334. \bibitem{MR1} M. Mihailescu, V. Radulescu; \emph{Nonhomogeneous Neumann problems in Orlicz-Sobolev spaces}, C. R. Acad. Sci. Paris, Ser. I 346 (2008), 401-406. \bibitem{MR2} M. Mihailescu, V. Radulescu; \emph{Existence and multiplicity of solutions for a quasilinear non- homogeneous problems: An Orlicz-Sobolev space setting.}, J. Math. Anal. Appl. 330 (2007), 416-432. \bibitem{MU} J. Musielak; \emph{Orlicz Spaces and Modular Spaces, Lecture Notes in Mathematics}, Vol. 1034, Springer, Berlin, 1983. \bibitem{O1} Y. G. Oh; \emph{Existence of semi-classical bound states of nonlinear Schr\"odinger equations with potentials of the class $(V)_\alpha$}, Comm. Partial Differential Equations 13 (1988), 1499-1519. \bibitem{O2} Y. G. Oh; \emph{On positive multi-lump bound states of nonlinear Schr\"odinger equations under multiple well potential}, Comm. Math. Phys. 131 (1990), 223-253. \bibitem {Rao} M. N. Rao, Z. D. Ren; \emph{Theory of Orlicz Spaces}, Marcel Dekker, New York (1985). \bibitem{rabinowitz} P. H. Rabinowitz; \emph{On a class of nonlinear Schr\"odinger equations}, Z. Angew. Math. Phys. 43 (1992), 270-291. \bibitem {J} J. A. Santos; \emph{Multiplicity of solutions for quasilinear equations involving critical Orlicz-Sobolev nonlinear terms}, Electron. J. Differential Equations 2013 (2013), 1-13. \bibitem{ZF} Z. Tan, F. Fang; \emph{Orlicz-Sobolev versus Holder local minimizer and multiplicity results for quasilinear elliptic equations}, J. Math. Anal. Appl. 402 (2013), 348-370. \bibitem{WX} X. Wang; \emph{On concentration of positive bound states of nonlinear Schr\"odinger equations}, Comm. Math. Phys., {53} (1993), 229-244. \bibitem{Willen} M. Willem; \emph{Minimax Theorems}, Birkhauser, (1996). \end{thebibliography} \end{document}