\documentclass[reqno]{amsart} \usepackage{hyperref} \AtBeginDocument{{\noindent\small \emph{Electronic Journal of Differential Equations}, Vol. 2013 (2013), No. 249, pp. 1--13.\newline ISSN: 1072-6691. URL: http://ejde.math.txstate.edu or http://ejde.math.unt.edu \newline ftp ejde.math.txstate.edu} \thanks{\copyright 2013 Texas State University - San Marcos.} \vspace{9mm}} \begin{document} \title[\hfilneg EJDE-2013/249\hfil Multiplicity of solutions] {Multiplicity of solutions for quasilinear equations involving critical Orlicz-Sobolev nonlinear terms} \author[J. A. Santos \hfil EJDE-2013/249\hfilneg] {Jefferson A. Santos} % in alphabetical order \address{Jefferson A. Santos \newline Universidade Federal de Campina Grande\\ Unidade Acad\^emica de Matem\'atica e Estat\'istica\\ CEP: 58429-900, Campina Grande - PB, Brazil} \email{jefferson@dme.ufcg.edu.br} \thanks{Submitted October 8, 2013. Published November 20, 2013.} \thanks{Partially supported supported by PROCAD/CAPES-Brazil} \subjclass[2000]{35J62, 35B33, 35A15} \keywords{Quasilinear elliptic equations; critical growth; variational methods} \begin{abstract} In this work, we study the existence and multiplicity of solutions for a class of problems involving the $\phi$-Laplacian operator in a bounded domain, where the nonlinearity has a critical growth. Our main tool is the variational method combined with the genus theory for even functionals. \end{abstract} \maketitle \numberwithin{equation}{section} \newtheorem{theorem}{Theorem}[section] \newtheorem{lemma}[theorem]{Lemma} \allowdisplaybreaks \section{Introduction} In this article, we consider the existence and multiplicity of solutions for the quasilinear problem \begin{equation} \begin{gathered} -\operatorname{div}\big( \phi(|\nabla u|)\nabla u \big) = \lambda \phi_*(|u|)u + f(x, u), \quad\text{in } \Omega\\ u = 0, \quad\text{on } \partial \Omega \\ \end{gathered} \label{prob1} \end{equation} where $ \Omega \subset \mathbb{R}^{N} $ is a bounded domain in $\mathbb{R}^{N}$ with smooth boundary, $ \lambda $ is a positive parameter and $\phi : (0,+\infty) \to \mathbb{R}$ is a continuous function satisfying \begin{equation} (\phi(t)t)'>0 \quad \forall t>0. \label{phi1} \end{equation} There exist $l, m \in (1,N)$ such that \begin{equation} l\leq \frac{\phi(|t|)t^{2}}{\Phi(t)}\leq m \quad \forall t \neq 0, \label{phi2} \end{equation} where \[ \Phi(t)=\int^{|t|}_{0}\phi(s)s\, ds, \quad l\leq m < l^{*},\quad l^{*}=\frac{lN}{N-l},\quad m^{*}=\frac{mN}{N-m}. \] Moreover, $\phi_*(t)t$ is such that Sobolev conjugate function $\Phi_*$ of $\Phi$ is its primitive; that is, $\Phi_*(t)=\int_0^{|t|}\phi_*(s)sds$. Related to function $f:\overline{\Omega} \times \mathbb{R} \to \mathbb{R}$, we assume the following: \begin{itemize} \item[(F1)] $f \in C(\overline{\Omega} \times \mathbb{R},\mathbb{R})$ is odd with respect $ t $ and \begin{gather*} f(x, t) = o\big( \phi(| t|)| t|\big), \quad\text{as $|t| \to 0$ uniformly in $x$}; \\ f(x, t) = o\big( \phi_*(| t|)| t| \big), \quad\text{as $|t|\to +\infty$ uniformly in $x$}; \end{gather*} \item[(F2)] There is $\theta\in(m,l^*)$ such that $ F(x, t) \leq \frac{1}{\theta} f(x, t)t$, for all $ t > 0 $ and a.e. in $ \Omega $, where $ F(x, t) = \int_{0}^{t}f(x, s)ds $. \end{itemize} Problem \eqref{prob1} associated with nonhomogenous nonlinear $\Phi$ arises in various fields of physics \cite{Fukagai3}: \begin{itemize} \item[(i)] in nonlinear elasticity, $\Phi(t)=(1+|t|^2)^{\gamma}-1$ for $\gamma \in (1, \frac{N}{N-2} )$. \item[(ii)] in plasticity, $\Phi(t)=|t|^{p}ln(1+|t|)$ for $10$. \end{itemize} Our main result reads as follows. \begin{theorem} \label{T1} Assume that \eqref{phi1}, \eqref{phi2}, {\rm (F1)} and {\rm(F2)} are satisfied. Then, there exist a sequence $ \{ \lambda_{k} \} \subset (0,+\infty)$ with $ \lambda_{k+1}< \lambda_{k} $, such that, for $ \lambda \in (\lambda_{k+1}, \lambda_{k}) $, problem \eqref{prob1} has at least $ k $ pairs of nontrivial solutions. \end{theorem} The main difficulty to prove Theorem \ref{T1} is related to the fact that the nonlinearity $f$ has a critical growth. In this case, it is not clear that functional energy associated with \eqref{prob1} satisfies the well known (PS) condition, once that the embedding $W^{1,\Phi}(\Omega) \hookrightarrow L_{\Phi_*}(\Omega)$ is not compact. To overcome this difficulty, we use a version of the concentration compactness lemma due to Lions for Orlicz-Sobolev space found in Fukagai, Ito and Narukawa \cite{Fukagai1}. We would like to mention that Theorem \ref{T1} improves the main result found in \cite{xinmin}. We cite the papers of Alves and Barreiro \cite{Barreiro}, Alves, Gon\c{c}alves and Santos \cite{Abrantes}, Bonano, Bisci and Radulescu \cite{BBR}, Cerny \cite{Cerny}, Cl\'ement, Garcia-Huidobro and Man\'asevich \cite{VGMS}, Donaldson \cite{Donaldson}, Fuchs and Li \cite{Fuchs1}, Fuchs and Osmolovski \cite{Fuchs2}, Fukagai, Ito and Narukawa \cite{Fukagai1,Fukagai2}, Gossez \cite{Gossez}, Mihailescu and Raduslescu \cite{MR1, MR2}, Mihailescu and Repovs \cite{MD}, Pohozaev \cite{Pohozaev} and references therein, where quasilinear problems like \eqref{prob1} have been considered in bounded and unbounded domains of $\mathbb{R}^{N}$. In some those papers, the authors have mentioned that this class of problem arises in applications, such as, nonlinear elasticity, plasticity and non-Newtonian fluids. This paper is organized as follows: In Section~\ref{Orlicz-Sobolev}, we collect some preliminaries on Orlicz-Sobolev spaces that will be used throughout the paper, which can be found in \cite{adams}, \cite{adams2}, \cite{Donaldson2} and \cite{Oneill}. In Section 3, we recall an abstract theorem involving genus theory that will use in the proof of Theorem \ref{T1} and prove some technical lemmas, and then we prove Theorem \ref{T1}. \section{Preliminaries on Orlicz-Sobolev spaces}\label{Orlicz-Sobolev} First of all, we recall that a continuous function $A:\mathbb{R} \to [0,+\infty)$ is a $N$-function if: \begin{itemize} \item[(A1)] $A$ is convex. \item[(A2)] $A(t)=0$ if and only if $t=0$. \item[(A3)] $ \frac{A(t)}{t}\to 0 $ as $t \to 0$, and $A(t)/t \to \infty$ as $t \to +\infty$. \item[(A4)] $A$ is an even function. \end{itemize} In what follows, we say that a $N$-function $A$ satisfies the $\Delta_{2}$-condition if, there exists $t_0\geq0$ and $k>0$ such that $$ A(2t)\leq kA(t) \quad \forall t\geq t_0. $$ This condition can be rewritten of the following way: For each $s >0$, there exists $M_s>0$ and $t_0\geq0$ such that \begin{equation} A(st) \leq M_s A(t), \quad\forall t \geq t_0. \label{Delta2} \end{equation} Fixed an open set $\Omega \subset \mathbb{R}^{N}$ and a N-function $A$ satisfying $\Delta_{2}$-condition, the space $L_{A}(\Omega)$ is the vectorial space of the measurable functions $u: \Omega \to \mathbb{R}$ such that $$ \int_{\Omega}A(u) < \infty. $$ The space $L_{A}(\Omega)$ endowed with Luxemburg norm, $$ |u|_{A}= \inf \Big\{\alpha >0: \int_{\Omega}A\big(\frac{u}{\alpha}\big) \leq 1\Big\}, $$ is a Banach space. The complement function of $A$, denoted by $\widetilde{A}(s)$, is given by the Legendre transformation, $$ \widetilde{A}(s)=\max_{t \geq 0}\{st -A(t)\} \quad \text{for } s \geq 0. $$ The functions $A$ and $\widetilde{A}$ are complementary each other. Moreover, we have the Young's inequality \begin{equation} \label{D2} st \leq A(t) + \widetilde{A}(s),\quad \forall t,s \geq 0. \end{equation} Using this inequality, it is possible to prove the H\"{o}lder type inequality \begin{equation} \label{D3} \Bigl|\int_{\Omega}u v\Bigl|\leq 2|u|_{A}|v|_{\widetilde{A}}, \quad \forall u \in L_{A}(\Omega) \text{ and } v \in L_{\widetilde{A}}(\Omega). \end{equation} Another important function related to function $A$ is the Sobolev's conjugate function $A_{*}$ of $A$, defined by $$ A^{-1}_{*}(t)=\int^{t}_{0}\frac{A^{-1}(s)}{s^{(N+1)/N}} ds, \quad \text{for } t>0. $$ When $A(t)=|t|^{p}$ for $1< p 0$ such that \begin{equation}\label{trudinger-emb} | u|_{A_*}\leq S_N|\nabla u|_{A}, \quad u\in W_0^{1,A}(\Omega). \end{equation} Moreover, exist $ C_0>0$ such that \begin{equation}\label{Poincare} \int_\Omega A(u)\leq C_0\int_\Omega A(|\nabla u|), \ u\in W_0^{1,A}(\Omega). \end{equation} This inequality shows the following embedding is continuous $$ W_0^{1,A}(\Omega) \hookrightarrow L_{A_*}(\Omega). $$ If $\Omega$ is a bounded domain and the two limits hold \begin{equation} \label{M1} \limsup_{t \to 0}\frac{B(t)}{A(t)}< +\infty ,\quad \limsup_{|t| \to +\infty}\frac{B(t)}{A_{*}(t)}=0, \end{equation} then the embedding \begin{equation} \label{M2} W_0^{1,A}(\Omega) \hookrightarrow L_{B}(\Omega) \end{equation} is compact. The next four lemmas involving the functions $\Phi, \widetilde{\Phi}$ and $\Phi_{*}$ and theirs proofs can be found in \cite{Fukagai1}. Hereafter, $\Phi$ is the $N$-function given in the introduction and $\widetilde{\Phi},\Phi_{*}$ are the complement and conjugate functions of $\Phi$ respectively. \begin{lemma} Assume \eqref{phi1} and \eqref{phi2}. Then $$ \Phi(t) = \int_0^{|t|} s \phi(s) ds, $$ is a $N$-function with $\Phi, \widetilde{\Phi} \in \Delta_2$. Hence, $L_\Phi(\Omega), W^{1,\Phi}(\Omega)$ and $W_0^{1,\Phi}(\Omega)$ are reflexive and separable spaces. \end{lemma} \begin{lemma} \label{F0} The functions $\Phi$, $\Phi_*$, $\widetilde{\Phi}$ and $\widetilde{\Phi}_*$ satisfy the inequality \begin{equation} \label{D1} \widetilde{\Phi}(\phi(|t|)t) \leq \Phi(2t),\quad \widetilde{\Phi}_*(\phi_*(|t|)t) \leq \Phi_*(2t),\quad \forall t \geq 0. \end{equation} \end{lemma} \begin{lemma} \label{F1} Assume that \eqref{phi1} and \eqref{phi2} hold and let $\xi_{0}(t)=\min\{t^{l},t^{m}\}$, $ \xi_{1}(t)=\max\{t^{l},t^{m}\}$, for all $t\geq 0$. Then \begin{gather*} \xi_{0}(\rho)\Phi(t) \leq \Phi(\rho t) \leq \xi_{1}(\rho)\Phi(t) \quad \text{for } \rho, t \geq 0, \\ \xi_{0}(|u|_{\Phi}) \leq \int_{\Omega}\Phi(u) \leq \xi_{1}(|u|_{\Phi}) \quad \text{for } u \in L_{\Phi}(\Omega). \end{gather*} \end{lemma} \begin{lemma} \label{F3} The function $\Phi_*$ satisfies the inequality $$ l^{*} \leq \frac{\Phi'_*(t)t}{\Phi_{*}(t)} \leq m^{*} \quad \text{for } t > 0. $$ \end{lemma} As an immediate consequence of the Lemma \ref{F3}, we have the following result \begin{lemma} \label{F2} Assume that \eqref{phi1} and \eqref{phi2} hold and let $\xi_{2}(t)=\min\{t^{l^{*}},t^{m^{*}}\}$, $\xi_{3}(t)=\max\{t^{l^{*}},t^{m^{*}}\}$ for all $t\geq 0$. Then \begin{gather*} \xi_{2}(\rho)\Phi_*(t) \leq \Phi_*(\rho t) \leq \xi_{3}(\rho)\Phi_*(t) \quad \text{for } \rho, t \geq 0,\\ \xi_{2}(|u|_{\Phi_*}) \leq \int_{\Omega}\Phi_*(u)dx \leq \xi_{3}(|u|_{\Phi_*}) \quad \text{for } u \in L_{A_*}(\Omega). \end{gather*} \end{lemma} \begin{lemma} \label{lemPhiest} Let $\widetilde{\Phi}$ be the complement of $\Phi$ and put $$ \xi_4(s)=\min\{s^{\frac{l}{l-1}}, s^{\frac{m}{m-1}}\},\quad \xi_5(s)=\max\{s^{\frac{l}{l-1}}, s^{\frac{m}{m-1}}\}, \quad s\geq0. $$ Then the following inequalities hold \begin{gather*} \xi_4(r)\widetilde{\Phi}(s)\leq\widetilde{\Phi}(r s)\leq \xi_5(r)\widetilde{\Phi}(s),\ r,s\geq0;\\ \xi_4(| u|_{\widetilde{\Phi}})\leq \int_\Omega\widetilde{\Phi}(u)dx\leq \xi_5(| u|_{\widetilde{\Phi}}),\ u\in L_{\widetilde{\Phi}}(\Omega). \end{gather*} \end{lemma} \section{An abstract theorem and technical lemmas} In this section we recall an important abstract theorem involving genus theory, which will use in the proof of Theorem \ref{T1}. After, we prove some technical lemmas that will use to show that the energy functional associated with problem~\eqref{prob1} satisfies the hypotheses of the abstract theorem. \subsection{An abstract theorem} Let $ E $ be a real Banach space and $ \Sigma $ the family of sets $ Y \subset E \backslash \{ 0 \} $ such that $ Y $ is closed in $ E $ and symmetric with respect to 0; that is, \[ \Sigma = \{ Y \subset E \backslash \{ 0 \}; Y \text{ is closed in $E$ and } Y = -Y \}. \] Hereafter, let us denote by $ \gamma (Y) $ the genus of $ Y \in \Sigma$ (see \cite[pp. 45]{Rabinowitz1}). Moreover, we set \begin{gather*} K_{c} = \{ u \in E; I(u) = c \text{ and } I'(u) = 0 \},\\ A_{c} = \{ u \in E; I(u) \leq c \}. \end{gather*} Next, we recall a version of the Mountain Pass Theorem for even functionals, whose proof can be found in \cite{Rabinowitz1}. \begin{theorem}\label{genus} Let $ E $ be an infinite dimensional Banach space with \linebreak $ E = V \oplus X $, where $ V $ is finite dimensional and let $ I \in C^{1}(E,\mathbb{R}) $ be a even function with $ I(0) = 0 $, and satisfying: \begin{itemize} \item [(I1)] there are constants $ \beta, \rho > 0 $ such that $ I(u) \geq \beta > 0 $, for each $ u \in \partial B_{\rho} \cap X $; \item [(I2)] there is $ \Upsilon > 0 $ such that $ I $ satisfies the $(PS)_{c}$ condition, for $ 0 < c < \Upsilon$; \item [(I3)] for each finite dimensional subspace $ \widetilde{E} \subset E $, there is $ R = R(\widetilde{E})>0 $ such that $ I(u) \leq 0 $ for all $ u \in \widetilde{E} \backslash B_{R}(0)$. \end{itemize} Suppose $ V $ is $ k$ dimensional and $ V = \operatorname{span}\{e_{1}, \dots , e_{k}\}$. For $ m \geq k $, inductively choose $ e_{m+1} \not\in E_{m} := \operatorname{span}\{e_{1}, \dots , e_{m}\}$. Let $ R_{m} = R(E_{m}) $ and $ D_{m} = B_{R_{m}} \cap E_{m} $. Define \begin{gather} G_{m} := \big\{ h \in C(D_{m}, E); h \text{ is odd and } h(u) = u, \forall u \in \partial B_{R_{m}} \cap E_{m}\},\\ \Gamma_{j} := \big\{ h(\overline{D_{m} \backslash Y}) ; h \in G_{m}, m \geq j, Y \in \Sigma, \text{ and } \gamma(Y) \leq m - j\big\}. \end{gather} For each $ j \in \mathbb{N} $, let \begin{equation} c_{j} = \inf_{K \in \Gamma_{j}} \max_{u \in K} I(u). \end{equation} Then, $ 0 < \beta \leq c_{j} \leq c_{j+1}$ for $ j > k $, and if $ j > k$, $c_{j} < \Upsilon$ and $ c_{j} $ is critical value of $ I $. Moreover, if $ c_{j}=c_{j+1}= \dots = c_{j+l} = c < \Upsilon $ for $ j > k $, then $ \gamma(K_{c}) \geq l + 1 $. \end{theorem} \subsection{Technical lemmas} Associated with problem~\eqref{prob1}, we have the energy functional $ J_{\lambda} : W^{1,\Phi}_{0}(\Omega) \to \mathbb{R} $ defined by $$ J_{\lambda} (u) = \int_{\Omega} \Phi(|\nabla u|) -\lambda \int_{\Omega} \Phi_*(u) - \int_{\Omega} F(x, u). $$ By conditions (F1) and (F2), $ J_{\lambda} \in C^{1} \left( W^{1,\Phi}_{0}(\Omega), \mathbb{R} \right) $ with \[ J'_{\lambda}(u)\cdot v = \int_{\Omega} \phi(|\nabla u|) \nabla u \nabla v - \lambda \int_{\Omega} \phi_*(| u|)u v - \int_{\Omega} f(x, u)v, \] for any $ u, v \in W^{1,\Phi}_{0}(\Omega) $. Thus, critical points of $ J_{\lambda} $ are weak solutions of problem~\eqref{prob1}. \begin{lemma} Under the conditions {\rm (F1)} and {\rm (F2)}, the functional $ J_{\lambda} $ satisfies {\rm (I1)}. \end{lemma} \begin{proof} On the one hand, from (F1) and (F2), for a given $\epsilon >0$, there exists $C_\epsilon >0$ such that \begin{align} |F(x, t)| \leq \epsilon\Phi(t)+C_{\epsilon}\Phi_*(t),\quad \forall (x, t) \in \bar{\Omega} \times \mathbb{R}.\label{cond_G} \end{align} Combining \eqref{Poincare} with \eqref{cond_G}, $$ J_{\lambda}(u) \geq (1-\epsilon C_0)\int_{\Omega}\Phi(|\nabla u|) - (1+C_{\epsilon}) \int_{\Omega} \Phi_*(u). $$ For $\epsilon$ is small enough and $ \|u\| = \rho \simeq 0 $, from \eqref{trudinger-emb} and Lemma \ref{F3}, it follows that \[ J_{\lambda}(u) \geq C_1|\nabla u|_\Phi^m-C_2S_N^{l^*}| \nabla u|_\Phi^{l^*} \] for some positive constants $C_1$ and $C_2$. For $ m < l^* $, if $\rho$ is small enough, there is $\beta >0$ such that \[ J_{\lambda}(u) \geq \beta > 0 \quad \forall u \in \partial B_{\rho}(0), \] which completes the proof. \end{proof} \begin{lemma} Under conditions {\rm (F1)} and {\rm (F2)}, the functional $ J_{\lambda} $ satisfies {\rm (I3)}. \end{lemma} \begin{proof} Suppose (I3) does not hold. Then, there is a finite dimensional subspace $ \widetilde{E} \subset W^{1,\Phi}_{0}(\Omega) $ and a sequence $ (u_n) \subset \widetilde{E} \backslash B_n(0) $ satisfying \begin{equation} \label{NOVAEQUA} J_{\lambda}(u_n) > 0, \quad \forall n \in \mathbb{N} . \end{equation} A direct computation shows that given $\epsilon >0$, there is a constant $ M > 0 $ such that \begin{equation} \label{desM} -M-\epsilon \Phi_*(t)\leq F(x,t)\leq M +\epsilon \Phi_*(t), \quad \forall (x,t)\in \overline{\Omega}\times \mathbb{R}. \end{equation} Consequently, $$ J_{\lambda} (u_n) \leq \int_{\Omega}\Phi(|\nabla u_n|)dx - \lambda \int_{\Omega} \Phi_*(u_n)+ \epsilon\int_{\Omega}\Phi_*(u_n) + M |\Omega|. $$ Fixing $ \epsilon = \lambda/2$, and using Lemma \ref{F2}, we obtain \begin{align} J_{\lambda}(u_n) \leq \int_{\Omega}\Phi(|\nabla u_n|) - \frac{\lambda}{2}\xi_3(|u_n|_{\Phi_*}) + M |\Omega|. \end{align} Using that $ \dim{\tilde{E}} < \infty $, we know that any two norms are equivalent in $ \tilde{E} $. Then, using that $ \|u_n\| \to \infty $, we can assume that $ |u_n|_{\Phi_*}> 1 $. Thereby, from Lemmas \ref{F1} and \ref{F2}, $$ J_{\lambda}(u_n) \leq |\nabla u_n|_{\Phi}^m -\frac{\lambda}{2}|u_n|^{l^*}_{\Phi_*} + M|\Omega|. $$ Using again the equivalence of the norms in $ \tilde{E} $, there is $C>0$ such that $$ J_{\lambda}(u_n) \leq \| u_n\|^m - \frac{\lambda}{2}C\| u_n\|^{l^*} + M|\Omega|. $$ Recalling that $ m < l^*$, the above inequality implies that there is $n_0 \in \mathbb{N}$ such that \begin{align*} J_{\lambda}(u_n) < 0, \quad \forall n \geq n_0, \end{align*} which contradicts \eqref{NOVAEQUA}. \end{proof} \begin{lemma}\label{limitada} Under conditions {\rm (F1)--(F2)}, any (PS) sequence for $J_\lambda$ is bounded in $W_0^{1,\Phi}(\Omega)$. \end{lemma} \begin{proof} Let $ \{ u_n \} $ be a $(PS)_{d} $ sequence of $ J_{\lambda} $. Then, \[ J_{\lambda}(u_n) \to d,\quad J'_{\lambda}(u_n) \to 0 \quad \text{as } n \to +\infty. \] We claim that $ \{ u_n \} $ is bounded. Indeed, note that \begin{align*} J_{\lambda}(u_n) - \frac{1}{\theta}J'_{\lambda}(u_n)u_n &= \int_{\Omega}\Phi(|\nabla u_n|) -\frac{1}{\theta}\int_\Omega\phi(|\nabla u_n|)|\nabla u_n|^2 \\ &\quad -\lambda\int_\Omega\Phi_*(u_n) +\frac{\lambda}{\theta}\int_\Omega \phi_*(|u_n|)u_n^2\\ &\quad -\int_\Omega F(x,u_n)+\frac{1}{\theta}\int_\Omega f(x,u_n)u_n. \end{align*} Consequently, \begin{align*} \lambda \int_{\Omega}\big(\frac{1}{\theta}\phi_*(|u_n|)u_n^2-\Phi_*(u_n)\big) &= J_{\lambda}(u_n) - \frac{1}{\theta}J'_{\lambda}(u_n)u_n - \int_{\Omega}\Phi(|\nabla u_n|) \\ &\quad +\frac{1}{\theta}\int_\Omega\phi(|\nabla u_n|)|\nabla u_n|^2 \\ &\quad +\int_{\Omega}\big( F(x, u_n) - \frac{1}{\theta}f(x, u_n)u_n \big). \end{align*} Then, by \eqref{phi2}, (F2) and Lemma \ref{F3}, for $ n $ sufficiently large, $$ \lambda (\frac{l^*}{\theta}-1)\int_{\Omega}\Phi_*(u_n) \leq C+1+\| u_n\|+(\frac{m}{\theta}-1) \int_\Omega\Phi(|\nabla u_n|), $$ which implies that \begin{align*} [\lambda(\frac{l^*}{\theta}-1)] \int_{\Omega}\Phi_*(u_n) \leq C + \|u_n\|, \end{align*} where $ C$ is a positive constant, and so \begin{align}\label{desestr} \int_{\Omega}\Phi_*(u_n) dx \leq C ( 1 + \|u_n\|). \end{align} By \eqref{desM} and \eqref{desestr}, \begin{align*} \int_{\Omega}\Phi(|\nabla u_n|) & \leq J_{\lambda}(u_n) + \lambda \int_{\Omega}\Phi_*(u_n) +\int_{\Omega} F(x, u_n) dx \\ & \leq C + o_n(1) + (\lambda+\epsilon)\int_\Omega\Phi_*(u_n)\\ & \leq C(1+\| u_n\|)+o_n(1).\\ \end{align*} Therefore, for $ n $ sufficiently large, $$ \int_{\Omega}\Phi(|\nabla u_n|) \leq C\left(1 + \|u_n\| \right). $$ If $ \|u_n\| > 1 $, from Lemma \ref{F2}, it follows that \begin{align*} \| u_n\|^{l} \leq C(1 + \|u_n\| ). \end{align*} Using that $ l > 1 $, the above inequality gives that $ \{ u_n\} $ is bounded in $ W_{0}^{1, \Phi}(\Omega) $. \end{proof} As a consequence of the above result, if $\{u_n\}$ is a (PS) sequence for $J_\lambda$, we can extract a subsequence of $\{u_n\}$, still denoted by $ \{u_n\} $ and $u \in W_0^{1,\Phi}(\Omega)$, such that \begin{itemize} \item $ u_n \rightharpoonup u $ in $ W_{0}^{1, \Phi}(\Omega) $; \item $ u_n \rightharpoonup u $ in $ L_{\Phi_*}(\Omega) $; \item $ u_n \to u $ in $ L_{\Phi}(\Omega) $; \item $u_n(x) \to u(x)$ a.e. in $\Omega$; \end{itemize} From the concentration compactness lemma of Lions in Orlicz-Sobolev space found in \cite{Fukagai1}, there exist two nonnegative measures $ \mu, \nu \in \mathcal{M}(\mathbb{R}^N) $, a countable set $\mathcal{J}$, points $ \{ x_{j} \}_{j \in \mathcal{J}} $ in $ \overline{\Omega} $ and sequences $ \{ \mu_{j}\}_{j \in \mathcal{J}}, \{ \nu _{j}\}_{j \in \mathcal{J}} \subset [0, +\infty)$, such that \begin{gather} \Phi(|\nabla u_n|) \to \mu \geq \Phi(|\nabla u|) + \sum_{j \in \mathcal{J}}\mu_{j}\delta_{x_{j}} \quad \text{in } \mathcal{M}( \mathbb{R}^N) \\ \Phi_*(u_n) \to \nu = \Phi_*(u) + \sum_{j \in \mathcal{J}}\nu_{j}\delta_{x_{j}} \quad \text{in } \mathcal{M}( \mathbb{R}^N) \\ \nu_j\leq\max\{S_N^{l^*}\mu_j^{\frac{l^*}{l}}, S_N^{m^*}\mu_j^{\frac{m^*}{l}},S_N^{l^*}\mu_j^{\frac{l^*}{m}}, S_N^{m^*}\mu_j^{\frac{m^*}{m}}\}, \end{gather} where $S_N$ satisfies \eqref{trudinger-emb}. Next, we will show an important estimate for $\{\nu_i\}$, from below. Firs, we prove a technical lemma. \begin{lemma} \label{NU} Under the conditions of Lemma~\ref{limitada}. If $\{u_n\}$ is a $(PS)$ sequence for $J_\lambda$ and $\{\nu_j\}$ as above, then for each $j \in \mathcal{J}$, $$ \nu_j \geq\big(\frac{l}{\lambda m^*}\big)^{\frac{\beta}{\beta-1}} S_N^{-\frac{\alpha}{\beta-1}} \quad \text{or} \quad \nu_j = 0, $$ for some $\alpha \in\{l^*,m^*\}$ and $\beta\in \{\frac{l^*}{l},\frac{m^*}{l},\frac{l^*}{m},\frac{m^*}{m}\}$. \end{lemma} \begin{proof} Let $ \psi \in C^{\infty}_{0}(\mathbb{R}^{N}) $ such that $$ \psi (x)= 1 \text{ in } B_{1/2}(0) ,\quad \operatorname{supp}\psi \subset B_{1}(0), \quad 0 \leq \psi(x) \leq 1 \; \forall x \in \mathbb{R}^{N}. $$ For each $j\in \Gamma$ and $ \epsilon > 0 $, let us define $$ \psi_{\epsilon}(x) = \psi\big(\frac{x-x_j}{\epsilon} \big), \quad \forall x \in \mathbb{R}^{N}. $$ Then $\{ \psi_{\epsilon} u_n\} $ is bounded in $ W_{0}^{1, \Phi}(\Omega) $. Since $J'_\lambda(u_n)\to 0$, we have $$ J'_\lambda(u_n)(\psi_{\epsilon}u_n)=o_n(1), $$ or equivalently, \begin{equation} \label{der_nabla_un_phi} \begin{aligned} \int_{\Omega}\phi( |\nabla u_n|)\nabla u_n\nabla \left(u_n \psi_{\epsilon}\right) &=o_n(1)+ \lambda \int_{\Omega} \phi_*(|u_n|)u_n^2 \psi_{\epsilon} + \int_{\Omega} f(x,u_n)u_n \psi_{\epsilon} \\ &\leq o_n(1)+\lambda m^*\int_\Omega \Phi_*(u_n)\psi_\epsilon +\int_\Omega f(x,u_n)u_n \psi_\epsilon. \end{aligned} \end{equation} Using that \[ \lim_{t\to +\infty}\frac{ f(x, t) t\psi_{\epsilon}(x)}{\Phi_*(t)}=0, \quad \text{uniformly in } x\in \overline{\Omega} \] and that $\lim_{n\to +\infty} f(x, u_n)u_n\psi_{\epsilon}=0$ a.e. on $\overline{\Omega}$, we have by compactness Lemma of Strauss \cite{chabro} (note that this result is still true when we replace $\mathbb{R}^N$ by $\overline{\Omega}$) \begin{equation}\label{conv_grad3} \lim_{n \to \infty} \int_{\Omega} f(x, u_n)u_n\psi_{\epsilon} = \int_{\Omega} f(x, u)u\psi_{\epsilon}. \end{equation} On the other hand, by \eqref{phi2} \begin{equation} \label{eq est03} \begin{aligned} \int_{\Omega}\phi( |\nabla u_n|)\nabla u_n\nabla (u_n \psi_{\epsilon}) & =\int_{\Omega}\phi( |\nabla u_n|)|\nabla u_n|^2 \psi_{\epsilon} + \int_{\Omega}\phi( |\nabla u_n|) (\nabla u_n\nabla \psi_{\epsilon})u_n \\ &\geq l\int_\Omega \Phi(|\nabla u|) \psi_{\epsilon} + \int_{\Omega}\phi( |\nabla u_n|) (\nabla u_n\nabla \psi_{\epsilon})u_n. \end{aligned} \end{equation} By Lemmas \ref{F0} and \ref{lemPhiest}, the sequence $\{|\phi(|\nabla u_n|)\nabla u_n|_{\widetilde{\Phi}}\}$ is bounded. Thus, there is a subsequence $\{u_n\}$ such that $$ \phi(|\nabla u_n|)\nabla u_n\rightharpoonup \widetilde{w}_1 \quad \text{weakly in }L_{\widetilde{\Phi}}(\Omega,\mathbb{R}^N), $$ for some $\widetilde{w}_1\in L_{\widetilde{\Phi}}(\Omega,\mathbb{R}^N)$. Since $u_n \to u$ in $L_\Phi( \Omega)$, $$ \int_\Omega \phi(|\nabla u_n|)(\nabla u_n\nabla \psi_\epsilon)u_n \to \int_\Omega (\widetilde{w}_1\nabla\psi_\epsilon)u. $$ Thus, combining \eqref{der_nabla_un_phi}, \eqref{conv_grad3}, \eqref{eq est03}, and letting $n\to \infty$, we have \begin{equation}\label{eq est02} l\int_\Omega \psi_\epsilon d\mu+\int_\Omega (\widetilde{w}_1\nabla \psi_\epsilon)u \leq \lambda m^*\int_\Omega \psi_\epsilon d\nu +\int_\Omega f(x,u)u\psi_\epsilon. \end{equation} Now we show that the second term of the left-hand side converges 0 as $\epsilon \to 0$. \noindent\textbf{Claim 1:} $\{f(x,u_n)\}$ is bounded in $L_{\widetilde{\Phi}_*}(\Omega)$. In fact, by (F1) and Lemma \ref{F0} we have \begin{align*} \int_\Omega\widetilde{\Phi}_*( f(x,u_n)) &\leq c_1\int_\Omega \widetilde{\Phi}_*(\phi_*(|u_n|)u_n) +c_2\int_{[|u_n|>1]}\widetilde{\Phi}_*(\phi(|u_n|)u_n)\\ &\quad +c_3\int_{[|u_n|\leq 1]}\widetilde{\Phi}_*(\phi(|u_n|)u_n)\\ &\leq c_1\int_\Omega\Phi_*(u_n)+c_2\int_{[|u_n|>1]} \widetilde{\Phi}_*(\phi(|u_n|)u_n)+c_3|\Omega|. \end{align*} Hence, by \eqref{phi2}, Lemma \ref{F1} and $m1]}\widetilde{\Phi}_*(|u_n|^{m-1})+C_3|\Omega|\\ &\leq C_1\int_\Omega\Phi_*(u_n)+C_2\int_{[|u_n|>1]} \widetilde{\Phi}_*(|u_n|^{l^*-1})+C_3|\Omega|. \end{align*} Now, by Lemmas \ref{F3} and \ref{F0}, $$ \int_\Omega\widetilde{\Phi}_*( f(x,u_n)) \leq K_1\int_\Omega \Phi_*(u_n)+K_2|\Omega|<+\infty. $$ From Claim 1, there is a subsequence $\{u_n\}$ such that $$ \phi_*(|u_n|)u_n+f(x,u_n)\rightharpoonup \widetilde{w}_2 \quad \text{ weakly in }L_{\widetilde{\Phi}_*}(\Omega), $$ for some $\widetilde{w}_2\in L_{\widetilde{\Phi}_*}(\Omega)$. Since \[ J'_\lambda(u_n)v=\int_\Omega\phi(|\nabla u_n|)\nabla u_n\nabla v -\int_\Omega\left(\phi_*(|u_n|)u_n+f(x,u_n)\right)v \to 0, \] as $n\to \infty$ for any $v\in W_0^{1,\Phi}(\Omega)$, $$ \int_\Omega(\widetilde{w}_1\nabla v-\widetilde{w}_2v)=0, $$ for any $v\in W_0^{1,\Phi}(\Omega)$. Substituting $v=u\psi_\epsilon$ we have $$ \int_\Omega(\widetilde{w}_1\nabla(u\psi_\epsilon)-\widetilde{w}_2u\psi_\epsilon)=0. $$ Namely, $$ \int_\Omega(\widetilde{w}_1\nabla\psi_\epsilon)u =-\int_\Omega(\widetilde{w}_1\nabla u-\widetilde{w}_2u)\psi_\epsilon. $$ Noting $\widetilde{w}_1\nabla u-\widetilde{w}_2u\in L^1(\Omega)$, we see that right-hand side tends to 0 as $\epsilon\to 0$. Hence we have $$ \int_\Omega (\widetilde{w}_1\nabla \psi_\epsilon)u\to 0, $$ as $\epsilon\to 0$. Letting $\epsilon\to 0$ in \eqref{eq est02}, we obtain $l\mu_{j} \leq \lambda m^*\nu_{j}$. Hence, \[ S_N^{-\alpha}\nu_j \leq \mu_{j}^\beta \leq \big(\frac{l\lambda}{m^*}\big)^{\beta} \nu_{j}^\beta, \] for some $\alpha\in\{l^*,m^*\}$, $\beta\in\{\frac{l^*}{l},\frac{m^*}{l},\frac{l^*}{m},\frac{m^*}{m}\}$, and so $$ \nu_{j} \geq \big(\frac{l}{\lambda m^*}\big)^{\frac{\beta}{\beta-1}} S_{N}^{-\frac{\alpha}{\beta-1}} \quad \text{ or} \quad \nu_{j} = 0 . $$ \end{proof} \begin{lemma} Assume that {\rm (F1)--(F2)}. Then, $ J_{\lambda} $ satisfies $(PS)_{d} $ for $d\in(0,d_\lambda)$ where $$ d_\lambda= \min\Big\{\frac{l^*-\theta}{\theta S_N^{\frac{\alpha}{\beta-1}} \lambda^{\frac{1}{\beta-1}}}(\frac{l}{m^*})^{\frac{\beta}{\beta-1}}; \alpha\in\{l^*,m^*\},\; \beta\in\{\frac{l^*}{l},\frac{m^*}{l},\frac{l^*}{m}, \frac{m^*}{m}\}\Big\}. $$ \end{lemma} \begin{proof} Using that $J_\lambda(u_n) = d + o_n(1)$ and $J_\lambda'(u_n)=o_n(1)$, we have \begin{align*} d = \lim_{n \to \infty} I(u_n) &= \lim_{n \to \infty} \big( J_\lambda(u_n) - \frac{1}{\theta} J_\lambda'(u_n)u_n \big) \\ &\geq \lim_{n \to \infty} \Big[( 1 - \frac{m}{\theta})\int_{\Omega} \Phi( |\nabla u_n|)+ \lambda(\frac{l^*}{\theta}-1) \int_{\Omega} \Phi_*( u_n) \\ &\quad - \int_{\Omega}\big( F(x, u_n) - \frac{1}{\theta} f(x,u_n)u_n \big) \Big] \\ &\geq\lambda (\frac{l^*}{\theta}-1 )\int_{\Omega} \Phi_*( u_n). \end{align*} Recalling that $$ \lim_{n \to \infty} \int_{\Omega} \Phi_*(u_n)dx = \Big[ \int_{\Omega} \Phi_*(u) + \sum_{j \in \mathcal{J}}\nu_{j} \Big] \geq \nu_{j}, $$ we derive that \begin{align*} d &\geq \lambda \big(\frac{l^*}{\theta}-1\big) \big(\frac{l}{\lambda m^*}\big)^\frac{\beta}{\beta-1}S_N^{-\frac{\alpha}{\beta-1}} \\ &=\big(\frac{l^*-\theta}{\theta}\big) \big(\frac{l}{m^*}\big)^{\frac{\beta}{\beta-1}} S_N^{-\frac{\alpha}{\beta-1}}\lambda^{\frac{1}{1-\beta}}, \end{align*} for some $\alpha\in\{l^*,m^*\}$, $\beta\in\{\frac{l^*}{l},\frac{m^*}{l},\frac{l^*}{m},\frac{m^*}{m}\}$, which is an absurd. From this, we must have $ \nu_{j} = 0 $ for any $ j \in \mathcal{J}$, leading to \begin{align} \int_{\Omega}\Phi_*(u_n) \to \int_{\Omega}\Phi_*(u) \label{conv_critica}. \end{align} Combining the last limit with Br\'ezis and Lieb \cite{Brezis_Lieb}, we obtain \[ \int_{\Omega} \Phi_*(u_n - u) \to 0 \text{ as} \quad n \to \infty, \] from where it follows by Lemma \ref{F2} \[ u_n \to u \text{ in } L_{\Phi_*}(\Omega). \] Now, as $ J_\lambda'(u_n)u_n = o_n(1) $, the last limit gives \[ \int_{\Omega}\phi(|\nabla u_n|)| u_n|^2 = \lambda \int_{\Omega}\phi_*( |u_n|)u_n^2 + \int_{\Omega} f(x, u_n)u_n+ o_n(1). \] In what follows, let us denote by $\{P_n\}$ the sequence $$ P_n(x) = \langle \phi(|\nabla u_n(x)|) \nabla u_n(x) - \phi(|\nabla u(x)|) \nabla u(x), \nabla u_n(x) - \nabla u (x) \rangle. $$ Since $\Phi$ is convex in $\mathbb{R}$ and $\Phi(|.|)$ is $C^1$ class in $\mathbb{R}^N$, has $P_n(x) \geq 0$. From definition of $\{P_n\}$, \[ \int_{\Omega} P_n = \int_{\Omega} \phi(|\nabla u_n|)|\nabla u_n|^2 - \int_{\Omega}\phi( |\nabla u_n|) \nabla u_n \nabla u - \int_{\Omega} \phi(|\nabla u|) \nabla u \nabla(u_n - u). \] Recalling that $u_n \rightharpoonup u$ in $W_0^{1,\Phi}(\Omega)$, we have \begin{align} \int_{\Omega} \phi(|\nabla u|) \nabla u \nabla(u_n - u) \to 0 \quad \text{as } n \to \infty, \end{align} which implies that \[ \int_{\Omega} P_n = \int_{\Omega} \phi(|\nabla u_n|) |\nabla u_n|^2 - \int_{\Omega} \phi(|\nabla u_n|)\nabla u_n \nabla u + o_n(1). \] On the other hand, from $ J_\lambda'(u_n)u_n = o_n(1)$ and $ J_\lambda'(u_n)u = o_n(1)$, we derive \begin{align*} 0\leq\int_{\Omega} P_n &= \lambda \int_{\Omega} \phi_*(|u_n|)|u_n|^2 - \lambda \int_{\Omega}\phi_*(|u_n|)u_n u\\ & \quad +\int_{\Omega} f(x, u_n)u_n - \int_{\Omega}f(x, u_n) u + o_n(1). \end{align*} Combining \eqref{conv_critica} with the compactness Lemma of Strauss \cite{chabro}, we deduce that \[ \int_{\Omega} P_n \to 0 \quad \text{as } n \to \infty. \] Using that $\Phi$ is convex, from a result due to Dal Maso and Murat \cite{Maso}, it follows that $$ \nabla u_n(x)\to \nabla u(x) \text{ a.e. }\Omega. $$ Now, using Lebesgue's Theorem, $$ \int_\Omega\Phi(|\nabla u_n-\nabla u|)dx \to 0, $$ which shows that \begin{equation} \label{E2} u_n \to u \text{ in } W^{1,\Phi}_{0}(\Omega) . \end{equation} \end{proof} The next lemma is similar to \cite[Lemma~5]{xinmin} and its proof will be omitted. \begin{lemma} Under conditions {\rm (F1)--(F2)}, there is sequence $ \{M_{m}\} \subset (0,+\infty) $ independent of $ \lambda $ with $ M_{m} \leq M_{m+1}$, such that for any $ \lambda > 0 $, \begin{eqnarray} c_{m}^{\lambda} = \inf_{K \in \Gamma_{m}} \max_{u \in K} J_{\lambda}(u) < M_{m}. \end{eqnarray} \end{lemma} \begin{proof}[Proof of Theorem \ref{T1}] For each $ k \in \mathbb{N} $, choose $ \lambda_{k} $ such that $M_{k} < d_{\lambda_{k}}$. Thus, for $ \lambda \in (\lambda_{k+1}, \lambda_{k}) $, \[ 0 < c^{\lambda}_{1} \leq c^{\lambda}_{2} \leq \dots \leq {c}^{\lambda}_{k} < M_{k} \leq d_\lambda. \] By Theorem~\ref{genus}, the levels $ c^{\lambda}_{1} \leq c^{\lambda}_{2} \leq \dots \leq c^{\lambda}_{k} $ are critical values of $ J_{\lambda}$. Thus, if $$ c^{\lambda}_{1} < c^{\lambda}_{2} < \dots < c^{\lambda}_{k} , $$ the functional $J_{\lambda}$ has at least $k$ critical points. Now, if $ c^{\lambda}_{j} = c^{\lambda}_{j+ 1} $ for some $ j = 1, 2, \dots , k $, it follows from Theorem~\ref{genus} that $K_{c^{\lambda}_{j}} $ is an infinite set \cite[Cap. 7]{Rabinowitz1}. Then, in this case, problem \eqref{prob1} has infinitely many solutions. \end{proof} \begin{thebibliography}{00} \bibitem{adams} A. Adams, J. F. Fournier; \emph{Sobolev spaces}, 2nd ed., Academic Press, (2003). \bibitem{adams2} A. Adams, L. I. 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