\documentclass[reqno]{amsart} \usepackage{hyperref} \AtBeginDocument{{\noindent\small \emph{Electronic Journal of Differential Equations}, Vol. 2013 (2013), No. 14, pp. 1--9.\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-2013/14\hfil Existence and multiplicity of solutions] {Existence and multiplicity of solutions for Dirichlet problems involving the p(x)-Laplace operator} \author[M. Avci \hfil EJDE-2013/14\hfilneg] {Mustafa Avci} % in alphabetical order \address{Mustafa Avci \newline Department of Mathematics, Faculty of Science, Dicle University, 21280-Diyarbakir, Turkey} \email{mavci@dicle.edu.tr} \thanks{Submitted November 11, 2011. Published January 14, 2013.} \subjclass[2000]{35D05, 35J60, 35J70, 58E05} \keywords{$p(x)$-Laplace operator; variable exponent Lebesgue-Sobolev spaces; \hfill\break\indent variational approach; Fountain theorem} \begin{abstract} In this article, we study superlinear Dirichlet problems involving the $p(x)$-Laplace operator without using the Ambrosetti-Rabinowitz's superquadraticity condition. Using a variant Fountain theorem, but not including Palais-Smale type assumptions, we prove the existence and multiplicity of the solutions. \end{abstract} \maketitle \numberwithin{equation}{section} \newtheorem{theorem}{Theorem}[section] \newtheorem{lemma}[theorem]{Lemma} \newtheorem{proposition}[theorem]{Proposition} %\newtheorem{corollary}[theorem]{Corollary} %\newtheorem{definition}[theorem]{Definition} %\newtheorem{example}[theorem]{Example} \newtheorem{remark}[theorem]{Remark} \allowdisplaybreaks \section{Introduction} We study the existence of infinitely many solutions for the Dirichlet boundary problems $$\begin{gathered} -\Delta _{p(x)}u+| u|^{p(x) -2}u=f(x,u) \quad \text{in } \Omega ,\\ u=0\quad \text{on } \partial \Omega , \end{gathered}\label{P0}$$ and $$\begin{gathered} -\Delta _{p(x) }u=f(x,u) \quad \text{in } \Omega ,\\ u=0\quad \text{on } \partial \Omega , \end{gathered} \label{E0}$$ where $\Omega$ is a bounded smooth domain of $\mathbb{R}^{N}$, $p\in C(\overline{\Omega })$ such that $10$ and $\tau >p^{+}$ such that $0<\tau F(x,s) \leq f(x,s) s,\quad |s| \geq M,\; x\in \Omega ,$ where $f$ is the nonlinear term in the equation with $F(x,t)=\int_0^{t}f(x,s)ds$ for $x\in \Omega$ and $t\in\mathbb{R}$. \end{itemize} There are many articles dealing with superlinear Dirichlet problems involving $p(x)$-Laplacian, in which (AR) is the main assumption to get the existence and multiplicity of solutions \cite{Fan3,Fan4}. However, there are many functions which are superlinear but not satisfy (AR). It is well known that the main aim of using (AR) is to ensure the boundedness of the Palais-Smale type sequences of the corresponding functional. In the present paper we do not use (AR) and we know that without (AR) it becomes a very difficult task to get the boundedness. So, using a weaker assumption (G1) (see main results) instead of (AR), and some variant Fountain theorem, i.e., Theorem \ref{thm5}, we overcome these difficulties. \section{Abstract framework and preliminary results} We state some basic properties of the variable exponent Lebesgue-Sobolev spaces $L^{p(x) }(\Omega )$ and $W^{1,p(x)}(\Omega )$, where $\Omega \subset\mathbb{R}^{N}$ is a bounded domain (for more details, see \cite{Edmunds,Fan1,Fan2,Kovacik}). Set $C_{+}(\overline{\Omega }) =\{ p\in C(\overline{\Omega }): \inf p(x) >1,\forall x\in \overline{\Omega }\} .$ Let $p\in C_{+}(\overline{\Omega })$ and denote $p^{-}:=\inf_{x\in \overline{\Omega }} p(x) \leq p(x) \leq p^{+}:=\sup_{x\in \overline{\Omega }} p(x) <\infty .$ For any $p\in C_{+}(\overline{\Omega })$, we define the variable exponent Lebesgue space by $L^{p(x) }(\Omega ) =\{ u:\Omega \to\mathbb{R}\text{ is measurable, }\int_{\Omega }| u(x)| ^{p(x) }dx<\infty \} .$ Then $L^{p(x) }(\Omega )$ endowed with the norm $| u| _{p(x) }=\inf \{ \mu>0:\int_{\Omega }| \frac{u(x) }{\mu } | ^{p(x) }dx\leq 1\} ,$ becomes a Banach space. The modular of the $L^{p(x) }(\Omega )$ space, which is the mapping $\rho :L^{p(x) }(\Omega ) \to \mathbb{R}$ defined by $\rho (u) =\int_{\Omega }| u(x)| ^{p(x) }dx,\quad \forall u\in L^{p(x) }(\Omega ) .$ \begin{proposition}[\cite{Fan1,Kovacik}] \label{prop1} If $u,u_n\in L^{p(x)}(\Omega )$ ($n=1,2,\dots$), then we have \begin{itemize} \item[(i)] $|u|_{p(x) }<1$ $(=1,>1)$ if and only if $\rho (u) <1$ $(=1,>1)$; \item[(ii)] $|u|_{p(x) }>1$ implies $|u|_{p(x)}^{p^{-}}\leq \rho (u) \leq |u|_{p(x) }^{p^{+}}$, $|u|_{p(x) }<1$ implies $|u|_{p(x) }^{p^{+}}\leq \rho (u) \leq |u|_{p(x) }^{p^{-}}$; \item[(iii)] $\lim_{n\to \infty }|u_n|_{p(x) }=0$ if and only if $\lim_{n\to \infty }\rho (u_n)=0$; $\lim_{n\to \infty }|u_n|_{p(x)}=\infty$ if and only if $\lim_{n\to \infty }\rho(u_n)=\infty$. \end{itemize} \end{proposition} \begin{proposition}[\cite{Fan1,Kovacik}] \label{prop2} If $u,u_n\in L^{p(x) }(\Omega )$ ($n=1,2,\dots$), then the following statements are equivalent: \begin{itemize} \item[(i)] $\lim_{n\to \infty }|u_n-u|_{p(x) }=0$; \item[(ii)] $\lim_{n\to \infty }\rho (u_n-u)=0$; \item[(iii)] $u_n\to u$ in measure in $\Omega$ and $\lim_{n\to \infty}\rho (u_n)=\rho (u)$. \end{itemize} \end{proposition} The variable exponent Sobolev space $W^{1,p(x)}(\Omega)$ is defined by $W^{1,p(x)}(\Omega ) =\{u\in L^{p(x) }(\Omega) : |\nabla u|\in L^{p(x) }(\Omega ) \},$ with the norm $\|u\|_{1,p(x) }=|u|_{p(x)}+|\nabla u|_{p(x)},$ or equivalently $\|u\|_{1,p(x) }=\inf \big\{ \mu>0:\int_{\Omega }(| \frac{\nabla u(x) }{ \mu }| ^{p(x) }+| \frac{u(x) }{ \mu }| ^{p(x) }) dx\leq 1\big\}$ for all\ $u\in W^{1,p(x)}(\Omega )$. The space $W_0^{1,p(x)}(\Omega )$ is defined as the closure of $C_0^{\infty }(\Omega )$ in $W^{1,p(x)}(\Omega )$ with respect to the norm $\|u\|_{1,p(x) }$. For $u\in W_0^{1,p(x)}(\Omega )$, we define an equivalent norm $\|u\|=|\nabla u|_{p(x)},$ since Poincar\'e inequality holds, i.e., there exists a positive constant $c$ such that $|u|_{p(x)}\leq c|\nabla u|_{p(x)},$ for all $u\in W_0^{1,p(x)}(\Omega )$, see \cite{Fan3}. \begin{proposition}[\cite{Fan1,Kovacik}] \label{prop3} If $1r_k>0. \end{gather*} Let us consider the$C^{1}$-functional$I_{\lambda }:E\to\mathbb{R}$defined by $I_{\lambda }(u) :=A(u)-\lambda B(u),\quad \lambda \in [1,2] .$ Now we recall the following variant of the fountain theorem \cite[Theorem 2.1]{Zou}, which is the main tool in the proof of the main results of this article. We will use the following assumptions: \begin{itemize} \item[(F1)]$I_{\lambda }$maps bounded sets to bounded sets uniformly for$\lambda \in [1,2] $. Moreover,$I_{\lambda }(-u)=I_{\lambda }(u)$for all$(\lambda ,u)\in [1,2] \times E$; \item[(F2)]$B(u)\geq 0$for all$u\in E$, and$A(u)\to \infty $or$B(u)\to \infty $as$\|u\|\to \infty, $\item[(F3)]$B(u)\leq 0$for all$u\in E$;$B(u)\to -\infty$as$\|u\|\to \infty$. \end{itemize} \begin{theorem} \label{thm5} Assume the functional$I_{\lambda }$satisfies {\rm (F1)}, and either {\rm(F2)} or {\rm (F3)}. For$k\geq 2$, let \begin{gather*} \Gamma _k:=\{ \psi \in C(B_k,E):\psi \text{ is odd, } \psi \big| _{\partial B_k} ={\rm id}\}, \\ c_k(\lambda ):=\inf_{\psi \in \Gamma _k}\max_{u\in B_k}I_{\lambda }(\gamma (u)),\\ b_k(\lambda ):=\inf_{u\in Z_k,\|u\|=r_k}I_{\lambda }(u),\\ a_k(\lambda ):=\max_{u\in Y_k,\|u\|=\rho _k}I_{\lambda}(u). \end{gather*} If$b_k(\lambda )>a_k(\lambda )$for all$\lambda \in [1,2] $, then$c_k(\lambda )\geq b_k(\lambda )$for all$\lambda \in [1,2] $. Moreover, for a.e$\lambda \in [1,2] $, there exists a sequence$\{ u_n^{k}(\lambda )\} _{n=1}^{\infty }$such that$\sup_n\|u_n^{k}(\lambda )\|<\infty$,$ I_{\lambda }'(u_n^{k}(\lambda ))\to 0$and$I_{\lambda }(u_n^{k}(\lambda ))\to c_k(\lambda )$as$n\to \infty$. \end{theorem} \section{Main results} First, we study the Dirichlet boundary-value problem $$\label{P} \begin{gathered} -\Delta _{p(x) }u+| u|^{p(x) -2}u=f(x,u) \quad \text{in } \Omega ,\\ u=0\quad \text{on } \partial \Omega , \end{gathered}$$ where$\Omega $is a bounded smooth domain of$\mathbb{R}^{N}$. We assume the following conditions: \begin{itemize} \item[(S1)]$f:\overline{\Omega }\times\mathbb{R}\to\mathbb{R}$is a Carath\'eodory function and$| f(x,t)| \leq c(1+| t| ^{q(x) -1}) $for a.e.$x\in \overline{\Omega }$and all$t\in\mathbb{R}$,$f(x,t)t\geq 0$for all$t>0$, where$p,q\in C_{+}(\overline{\Omega }) $such that$p(x) 0$uniformly for$x\in \overline{\Omega }$, where$p^{+}<\theta \leq q^{-}$; \item[(S3)]$\lim_{t\to 0}\frac{f(x,t)}{t^{p^{-}-1}}=0$uniformly for$x\in \overline{\Omega }$,$\frac{f(x,u)}{u^{p^{-}-1}}$is an increasing function of$t\in\mathbb{R}$for all$x\in \overline{\Omega }$. \item[(S4)]$f(x,-t)=-f(x,t)$for all$x\in \overline{\Omega }$,$t\in\mathbb{R}$. \item[(G1)] There exists a constant$\xi \geq 1$, such that for any$s\in [0,1] $,$t\in\mathbb{R}$, and for each$G_{\gamma }\in \mathcal{F} $, and all$\eta \in [p^{-},p^{+}] $, the inequality$\xi G_{\gamma }(x,t)\geq G_{\eta }(x,st) $hold for a.e.$x\in \overline{\Omega }$, where $\mathcal{F} =\{ G_{\gamma }:G_{\gamma }(x,t) =f(x,t) t-\gamma F(x,t),\gamma \in [p^{-},p^{+}] \}.$ Note that when$p(x) \equiv p$a constant,$\mathcal{F} =\{ f(x,t) t-pF(x,t)\} $is consist of only one element. \end{itemize} \begin{remark} \label{rmk6} \rm It is not difficult to show that if$f(x,t) $is increasing in$ t $, then (AR) implies (G1) when$t$is large enough. However, in general, (AR) does not imply (G1); see \cite[Remark 3.3]{Zang}. \end{remark} \begin{theorem} \label{thm7} Assume that {\rm (S1)--(S4), (G1)} hold. Then problem \eqref{P} has infinitely many solutions$\{ u_k\} $satisfying $J(u_k) =\int_{\Omega }\frac{1}{p(x) }(| \nabla u_k| ^{p(x) }+| u_k| ^{p(x) })dx-\int_{\Omega}F(x,u_k)dx\to \infty \quad \text{as } k\to \infty ,$ where$J:W^{1,p(x)}(\Omega ) \to\mathbb{R}$is the functional corresponding to problem \eqref{P} and$F(x,t)=\int_0^{t}f(x,s)ds$. \end{theorem} \begin{remark} \label{rmk8} \rm Condition (S1) implies that the functional$J$is well defined and of class$C^{1}$. It is well known that the critical points of$J$are weak solutions of \eqref{P}. Moreover, the derivative of$J$is given by $\langle J'(u) ,\upsilon \rangle =\int_{\Omega }(| \nabla u| ^{p(x) -2}\nabla u\nabla \upsilon +| u| ^{p(x) -2}u\upsilon )dx-\int_{\Omega }f(x,u)\upsilon dx,$ for any$u,\upsilon \in W^{1,p(x)}(\Omega ) $. \end{remark} Second, we consider the Dirichlet boundary problem $$\begin{gathered} -\Delta _{p(x) }u=f(x,u) \quad \text{in }\Omega ,\\ u=0\quad \text{on } \partial \Omega , \end{gathered} \label{E}$$ where$\Omega $is a bounded smooth domain of$\mathbb{R}^{N}$. We will use the following assumptions: \begin{itemize} \item[(E1)]$f:\overline{\Omega }\times\mathbb{R}\to\mathbb{R}$is a Carath\'eodory function and$| f(x,t)| \leq c(1+| t| ^{q(x) -1})$a.e.$x\in \overline{\Omega }$and all$t\in\mathbb{R}$, where$p,q\in C_{+}(\overline{\Omega }) $such that$p(x) \overline{b}_k>0$, and$\{ z_n\} _{n=1}^{\infty }\subset W^{1,p(x)}(\Omega ) $, such that $J_{\lambda }'(z_n)=0,\quad J_{\lambda }(z_n)\in [\overline{b}_k,\overline{c}_k] .$ \end{lemma} \begin{proof} It is easy to prove that, for some$\rho _k>0$large enough, we have$a_k(\lambda ):=\max_{u\in Y_{k,}\|u\|=p_k}J_{\lambda }(u) \leq 0$uniformly for$\lambda \in [1,2] $. Indeed, by the conditions (S1)--(S3), for any$\varepsilon >0$there exists$C_{\varepsilon }>0$such that$f(x,u)u\geq C_{\varepsilon}| u| ^{\theta }-\varepsilon | u|^{p^{-}}$. Further, on the finite dimensional subspace$Y_k$, we can find some constants$c>0such that $| u| _{\theta }\geq c\|u\|_{1,p(x) }, \quad | u| _{p^{-}}\leq c\| u\|_{1,p(x) },\quad \forall u\in Y_k.$ By Propositions \ref{prop1} and \ref{prop4}, we have \begin{align*} J_{\lambda }(u) &\leq \frac{1}{p^{-}}\int_{\Omega}(| \nabla u| ^{p(x) }+| u| ^{p(x) }) dx-\frac{\lambda C_{\varepsilon }}{ \theta }\int_{\Omega }| u| ^{\theta }dx+\frac{ \lambda \varepsilon }{p^{-}}\int_{\Omega }| u|^{p^{-}}dx \\ &\leq \frac{1}{p^{-}}\|u\|_{1,p(x) }^{p^{+}}- \frac{\lambda C_{\varepsilon }c^{\theta }}{\theta }\|u\| _{1,p(x) }^{\theta }+\frac{\lambda c^{p^{-}}}{p^{-}}\| u\|_{1,p(x) }^{p^{-}}\text{\ }. \end{align*} Since\theta >p^{+}$, it follows that $a_k(\lambda ):=\max_{u\in Y_{k,}\|u\|_{1,p(x) } =\rho _k}J_{\lambda }(u) \to -\infty \quad \text{as }\|u\|_{1,p(x) }\to +\infty$ uniformly for$\lambda \in [1,2] $and for all$u\in Y_k$. On the other hand, by conditions (S1) and (S3), for any$\varepsilon >0$there exists$C_{\varepsilon }>0$such that$| f(x,u)| \leq \varepsilon| u| ^{p^{-}-1}+C_{\varepsilon }| u|^{q(x) -1}$. Let $\beta _k:=\sup_{u\in Z_k,\|u\|_{1,p(x) }=1}| u| _{q(x) },\quad \vartheta _k:=\sup_{u\in Z_k,\|u\|_{1,p(x) }=1}| u| _{p^{-}}.$ Then$\beta _k\to 0$and$\vartheta _k\to 0$as$k\to \infty $(see \cite{Fan4}). Therefore, when$u\in Z_k $and$\|u\|_{1,p(x) }>1, we have \begin{align*} J_{\lambda }(u) &\geq \frac{1}{p^{+}}\int_{\Omega}(| \nabla u| ^{p(x) }+|u| ^{p(x) }) dx -\lambda \varepsilon \int_{\Omega }| u| ^{p^{-}}dx -\lambda C_{\varepsilon }\int_{\Omega }| u| ^{q(x)}dx \\ &\geq \frac{1}{p^{+}}\|u\|_{1,p(x)}^{p^{-}}-c| u| _{p^{-}}^{p^{-}}-c| u| _{q(x) }^{q^{+}} \\ &\geq \frac{1}{p^{+}}\|u\|_{1,p(x)}^{p^{-}} -c\vartheta _k^{p^{-}}\|u\|_{1,p(x) }^{p^{-}} -c\beta _k^{q^{+}}\|u\|_{1,p(x) }^{q^{+}}, \end{align*} wherec=\max \{ 2\varepsilon ,2C_{\varepsilon }\} $. For sufficiently large$k$, we have$c\vartheta _k^{p^{-}}<\frac{1}{2p^{+}}$. Then, it follows $J_{\lambda }(u) \geq \frac{1}{2p^{+}}\|u\| _{1,p(x) }^{p^{-}}-c\beta _k^{q^{+}}\|u\|_{1,p(x) }^{q^{+}}.$ If we choose$r_k:=(2cq^{+}\beta _k^{q^{+}}) ^{\frac{1}{ p^{-}-q^{+}}}$, then for$u\in Z_k$with$\|u\|_{1,p(x) }=r_k, we obtain \begin{align*} J_{\lambda }(u) &\geq \frac{1}{2p^{+}}(2cq^{+}\beta_k^{q^{+}}) ^{\frac{p^{-}}{p^{-}-q^{+}}} -c\beta _k^{q^{+}}(2cq^{+}\beta _k^{q^{+}}) ^{\frac{q^{+}}{p^{-}-q^{+}}} \\ &\geq \frac{q^{+}-p^{+}}{2p^{+}q^{+}}(2cq^{+}\beta _k^{q^{+}}) ^{\frac{p^{-}}{p^{-}-q^{+}}}:=\overline{b}_k, \end{align*} which implies $b_k(\lambda ):=\inf_{u\in Z_k,\|u\|_{1,p(x) }=r_k} J_{\lambda }(u) \to \infty \quad \text{as }k\to \infty$ uniformly for\lambda $. So, by Theorem \ref{thm5}, for a.e.$\lambda \in [1,2] $, there exists a sequence$\{ u_n^{k}(\lambda )\} _{n=1}^{\infty }$such that \begin{gather*} \sup_n \|u_n^{k}(\lambda )\|_{1,p(x) }<\infty ,\quad J_{\lambda }'(u_n^{k}(\lambda ))\to 0,\\ J_{\lambda }(u_n^{k}(\lambda )) \to c_k(\lambda )\geq b_k(\lambda )\geq \overline{b}_k\quad \text{as }n\to \infty . \end{gather*} Moreover, since$c_k(\lambda )\leq \sup_{u\in B_k}J_{\lambda }(u):=\overline{c}_k$and$W^{1,p(x)}(\Omega ) $is embedded compactly to$L^{q(x) }(\Omega ) $, and thanks to the standard arguments,$\{ u_n^{k}(\lambda )\} _{n=1}^{\infty} $has a convergent subsequence. Hence, there exists$z^{k}(\lambda )$such that$J_{\lambda }'(z^{k}(\lambda )) =0$and$J_{\lambda }(z^{k}(\lambda )) \in [\overline{b}_k,\overline{c}_k] $. Consequently, we can find$\lambda _n\to 1$and$\{ z_n\} $desired as the claim. \end{proof} \begin{lemma} \label{lem11}$\{ z_n\} _{n=1}^{\infty }$is bounded in$W^{1,p(x)}(\Omega ) $. \end{lemma} \begin{proof} We argue by contradiction. Passing to a subsequence if necessary, still denoted by$\{ z_n\} $, we may assume that$\| z_n\|_{1,p(x) }\to \infty $as$n\to \infty $. Let$\{ \omega _n\} \subset W^{1,p(x)}(\Omega) $and put$\omega _n:=\frac{z_n}{\|z_n\|_{1,p(x) }}$. Since$\|\omega _n\|_{1,p(x) }=1$, up to subsequences, we obtain \begin{gather*} \omega _n \rightharpoonup \omega \quad \text{in }W^{1,p(x)}(\Omega) , \\ \omega _n \to \omega \quad \text{in }L^{\gamma (x)}(\Omega ) , \\ \omega _n(x) \to \omega (x) \quad \text{a.e. }x\in \Omega . \end{gather*} Then, the main concern is whether$\{ \omega _n\} \subset W^{1,p(x)}(\Omega ) $vanish or not. We shall prove that none of these alternatives can occur and this contradiction will prove that$\{\omega _n\} \subset W^{1,p(x)}(\Omega ) $is bounded. If$\omega =0$, we can define a sequence$\{ t_n\} \subset\mathbb{R}$, as argued in \cite{Zang}, such that $$J_{\lambda _n}(t_nz_{n)}:=\max_{t\in [0,1]}J_{\lambda _n}(tz_n) . \label{3.1}$$ Let$\overline{\omega }_n:=(2p^{+}c)^{\frac{1}{p^{-}}}\omega _n$with$c>0$. Then for$n$is large enough, we have $$\begin{split} J_{\lambda _n}(t_nz_n) &\geq J_{\lambda _n}(\overline{\omega }_n) \geq A((2p^{+}c)^{\frac{1}{p^{-}}}\omega_n) -\lambda _nB(\overline{\omega }_n) \\ &\geq \frac{1}{p^{+}}(2p^{+}c)A(\omega _n) -\lambda _nB(\overline{\omega }_n) \geq 2c-\lambda _nB( \overline{\omega }_n) \geq c, \end{split} \label{e3.2}$$ which implies that$\lim_{n\to \infty } J_{\lambda_n}(t_nz_n) =\infty $by the fact$c>0$can be large arbitrarily. Noting that$J_{\lambda _n}(0) =0$and$J_{\lambda _n}(z_n) \to c$, then$00, we have \begin{align*} (\frac{1}{\overline{p}_n}B'(z_n) -B(z_n) ) &= \frac{1}{\overline{p}_n}\int_{\Omega}G_{\gamma _{z_n}}(x,z_n)dx \geq \frac{1}{\overline{p}_n\xi } \int_{\Omega }G_{\gamma _{t_nz_n}}(x,t_nz_n)dx \\ &= \frac{\overline{p}_{t_n}}{\overline{p}_n\xi } \big(\frac{1}{\overline{p}_{t_n}}B'(t_nz_n) -B( t_nz_n) \big) \to +\infty . \end{align*} This contradicts the following result of Lemma \ref{lem10}, $\lambda _n(\frac{1}{\ \overline{p}_n}B'( z_n) -B(z_n) ) =J_{\lambda _n}(z_n)-\frac{1}{ \overline{p}_n}\langle J_{\lambda _n}'(z_n),z_n\rangle =J_{\lambda _n}(z_n)\in [\overline{b}_k,\overline{c}_k] .$ If\omega \neq 0$, since$J_{\lambda _n}'(z_n)=0$, we have, by Proposition \ref{prop1}, $$\begin{split} 1-o(1) &=\int_{\Omega }\frac{f(x,z_n)z_n}{\varphi (z_n) }dx\text{ }\geq \int_{\Omega }\frac{ f(x,z_n)z_n}{\|z_n\|_{1,p(x) }^{p^{+}}}dx \\ &\geq \int_{\Omega }\frac{f(x,z_n)z_n}{\| z_n\|_{1,p(x) }^{\theta }}dx=\int_{\Omega } \frac{f(x,z_n)z_n}{| z_n| ^{\theta }}| \omega _n| ^{\theta }dx, \end{split} \label{e3.4}$$ where$\varphi (z_n) :=\int_{\Omega }(| \nabla z_n| ^{p(x) }+| z_n| ^{p(x) }) dx$. Define the set$\Omega _0=\{ x\in \Omega :\omega (x)=0\} $. Then for$x\in \Omega \backslash \Omega _0=\{ x\in \Omega :\omega (x) \neq 0\} $, we have$|z_n(x) | \to +\infty $as$n\to \infty $. Hence, by$(\mathbf{S}_{1}) $and$(\mathbf{S}_2) $, we have $\frac{f(x,z_n)z_n}{| z_n| ^{\theta }}| \omega _n| ^{\theta }dx\to +\infty \quad \text{as }n\to \infty .$ Using Fatou's lemma and that$| \Omega \backslash \Omega_0| >0$, we obtain $$\int_{\Omega \backslash \Omega _0}\frac{f(x,z_n)z_n}{ | z_n| ^{\theta }}| \omega _n| ^{\theta }dx\to +\infty \quad \text{as }n\to \infty . \label{e3.5}$$ On the other hand, by condition (S2), there exists$c>-\infty $such that$\frac{f(x,t)t}{t^{\theta }}\geq c$for$t\in\mathbb{R}$and a.e.$x\in \overline{\Omega }$. Moreover, we have$\int_{\Omega_0}| \omega _n| ^{\theta }dx\to 0\ $as$n\to \infty $. Thus, there exists$\Lambda >-\infty $such that $$\int_{\Omega _0}\frac{f(x,z_n)z_n}{| z_n| ^{\theta }}| \omega _n| ^{\theta }dx\geq c\int_{\Omega _0}| \omega _n| ^{\theta }dx\geq \Lambda >-\infty . \label{e3.6}$$ Combining \eqref{e3.4}, \eqref{e3.5} and \eqref{e3.6}, we obtain a contradiction. Therefore,$\{ z_n\}_{n=1}^{\infty }$is bounded, and the proof is complete. \end{proof} \begin{thebibliography}{99} \bibitem{Acerbi} E. Acerbi, G. Mingione; \emph{Regularity results for stationary electrorheological fluids}, Arch. Ration. Mech. Anal. 164 (2002), 213-259. \bibitem{Ambrosetti} A. Ambrosetti, P. H. Rabinowitz; \emph{Dual variational methods in critical point theory and applications}, J. Funct. Anal. 173\textbf{\ }(14), 349-381. \bibitem{Antonsev} S. N. Antontsev, S. I. Shmarev; \emph{Elliptic equations and systems with nonstandard growth conditions: Existence, uniqueness and localization properties of solutions}, Nonlinear Anal. 65 (2006), 728--761. \bibitem{Chabrowski} J. 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