\documentclass[reqno]{amsart} \usepackage{hyperref} \AtBeginDocument{{\noindent\small \emph{Electronic Journal of Differential Equations}, Vol. 2013 (2013), No. 253, pp. 1--10.\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-2013/253\hfil Eigenvalue problems] {Eigenvalue problems for $p(x)$-Kirchhoff type equations} \author[G. A. Afrouzi, M. Mirzapour \hfil EJDE-2013/253\hfilneg] {Ghasem A. Afrouzi, Maryam Mirzapour} % in alphabetical order \address{Ghasem Alizadeh Afrouzi \newline Department of Mathematics, Faculty of Mathematical Sciences, University of Mazandaran, Babolsar, Iran} \email{afrouzi@umz.ac.ir} \address{Maryam Mirzapour \newline Department of Mathematics, Faculty of Mathematical Sciences, University of Mazandaran, Babolsar, Iran} \email{mirzapour@stu.umz.ac.ir} \thanks{Submitted August 17, 2013. Published November 20, 2013.} \subjclass[2000]{35J60, 35J35, 35J70} \keywords{$p(x)$-Kirchhoff type equations; variational methods; \hfill\break\indent boundary value problems} \begin{abstract} In this article, we study the nonlocal $p(x)$-Laplacian problem \begin{gather*} -M\Big(\int_{\Omega}\frac{1}{p(x)}|\nabla u|^{p(x)}dx\Big) \operatorname{div}(|\nabla u|^{p(x)-2}\nabla u)= \lambda|u|^{q(x)-2}u \quad \text{ in } \Omega,\\ u=0 \quad \text{on } \partial\Omega, \end{gather*} By means of variational methods and the theory of the variable exponent Sobolev spaces, we establish conditions for the existence of weak solutions. \end{abstract} \maketitle \numberwithin{equation}{section} \newtheorem{theorem}{Theorem}[section] \newtheorem{lemma}[theorem]{Lemma} \newtheorem{proposition}[theorem]{Proposition} \newtheorem{definition}[theorem]{Definition} \allowdisplaybreaks \section{Introduction} The purpose of this article is to show the existence of solutions of the $p(x)$-Kirchhoff type eigenvalue problem $$\label{e1.1} \begin{gathered} -M\Big (\int_{\Omega}\frac{1}{p(x)}|\nabla u|^{p(x)}dx\Big) \operatorname{div}(|\nabla u|^{p(x)-2}\nabla u)= \lambda|u|^{q(x)-2}u \quad \text{in } \Omega,\\ u=0 \quad \text{on } \partial\Omega, \end{gathered}$$ where $\Omega\subset\mathbb{R}^N$, $N\geq3$, is a bounded domain with smooth boundary $\partial\Omega$, $M:\mathbb{R}^+\to \mathbb{R}$ is a continuous function, $p,q$ are continuous functions on $\overline{\Omega}$ such that $11$ for any $x\in \overline{\Omega}$ and $\lambda$ is a positive number. The study of problems involving variable exponent growth conditions has a strong motivation due to the fact that they can model various phenomena which arise in the study of elastic mechanics \cite{Zhikov}, electrorheological fluids \cite{Acerbi} or image restoration \cite{Chen}. Equation \eqref{e1.1} is called a nonlocal problem because of the the term $M$, which implies that the equation in \eqref{e1.1} is no longer a pointwise equation. This causes some mathematical difficulties which make the study of such a problem particularly interesting. Nonlocal differential equations are also called Kirchhoff-type equations because Kirchhoff \cite{K} investigated an equation of the form $$\label{e1.2} \rho\frac{\partial^{2}u}{\partial t^{2}} -\Big(\frac{\rho_0}{h}+\frac{E}{2L}\int_0^L \big|\frac{\partial u}{\partial x}\big|^{2}\,dx\Big) \frac{\partial^{2}u}{\partial x^{2}}=0,$$ which extends the classical D'Alembert's wave equation, by considering the effect of the changing in the length of the string during the vibration. A distinct feature is that the \eqref{e1.2} contains a nonlocal coefficient $\frac{\rho_0}{h}+\frac{E}{2L}\int_0^L |\frac{\partial u}{\partial x}|^{2}\,dx$ which depends on the average $\frac{1}{2L} \int_0^L \big|\frac{\partial u}{\partial x}\big|^{2}\,dx$, and hence the equation is no longer a pointwise equation. The parameters in \eqref{e1.2} have the following meanings: $L$ is the length of the string, $h$ is the area of the cross-section, $E$ is the Young modulus of the material, $\rho$ is the mass density and $P_0$ is the initial tension. Lions \cite{Lions} has proposed an abstract framework for the Kirchhoff-type equations. After the work by Lions \cite{Lions}, various equations of Kirchhoff-type have been studied extensively, see e.g. \cite{Aro,Cava} and \cite{Corr1}-\cite{Dre2}. The study of Kirchhoff type equations has already been extended to the case involving the $p$-Laplacian (for details, see \cite{Dre1,Dre2,Corr1,Corr2}) and $p(x)$-Laplacian (see \cite{Aut,Col,Dai1,Dai2,Fan7}). Motivated by the above papers and the results in \cite{Chung,Mih1}, we consider \eqref{e1.1} to study the existence of weak solutions. \section{Preliminaries} For the reader's convenience, we recall some necessary background knowledge and propositions concerning the generalized Lebesgue-Sobolev spaces. We refer the reader to \cite{Ed1,Ed2,Fan1,Fan2} for details. Let $\Omega$ be a bounded domain of $\mathbb{R}^N$, denote \begin{gather*} C_+(\overline{\Omega})=\{p(x): p(x)\in C(\overline{\Omega}),~p(x)>1, \text{ for all } x\in \overline{\Omega}\};\\ p^+=\max\{p(x): x \in\overline{\Omega}\},\quad p^-=\min\{p(x); x \in\overline{\Omega}\};\\ L^{p(x)}(\Omega)=\big\{u: u\text{ is a measurable real-valued function},\; \int_{\Omega}|u(x)|^{p(x)}dx<\infty\big\}, \end{gather*} with the norm $|u|_{L^{p(x)}(\Omega)}=|u|_{p(x)} =\inf \big\{\lambda>0: \int_{\Omega}|\frac{u(x)}{\lambda}| ^{p(x)}dx\leq 1\big\}$ becomes a Banach space \cite{Kov}. We also define the space $W^{1,p(x)}(\Omega)=\{u\in L^{p(x)}(\Omega): |\nabla u|\in L^{p(x)}(\Omega)\},$ equipped with the norm $\|u\|_{W^{1,p(x)}(\Omega)}=|u(x)|_{L^{p(x)}(\Omega)} +|\nabla u(x)|_{L^{p(x)}(\Omega)}.$ We denote by $W_0^{1,p(x)}(\Omega)$ the closure of $C_0^\infty(\Omega)$ in $W^{1,p(x)}(\Omega)$. Of course the norm $\|u\|=|\nabla u|_{L ^{p(x)}(\Omega)}$ is an equivalent norm in $W_0^{1,p(x)}(\Omega)$. In this paper, we denote by $X=W_0^{1,p(x)}(\Omega)$. \begin{proposition}[\cite{Ed1,Fan2}] \label{prop2.1} $(i)$ The conjugate space of $L^{p(x)}(\Omega)$ is $L^{p'(x)}(\Omega)$, where $\frac{1}{p(x)}+\frac{1}{p'(x)}=1$. For any $u\in L^{p(x)}(\Omega)$ and $v\in L^{p'(x)}(\Omega)$, we have $\int_{\Omega}|uv|dx\leq \Big(\frac{1}{p^{-}} +\frac{1}{p'^{-}}\Big)|u|_{p(x)}|v|_{p'(x)}\leq 2|u|_{p(x)}|v|_{p'(x)}$ $(ii)$ If $p_1(x), p_2(x)\in C_+(\overline{\Omega})$ and $p_1(x)\leq p_2(x)$ for all $x\in \overline{\Omega}$, then $L^{p_2(x)}(\Omega)\hookrightarrow L^{p_1(x)}(\Omega)$ and the embedding is continuous. \end{proposition} \begin{proposition}[\cite{Fan4}] \label{prop2.2} Set $\rho(u)=\int_{\Omega}|\nabla u(x)|^{p(x)}dx$, then for $u\in X$ and $(u_k)\subset X$, we have \begin{itemize} \item[(1)] $\|u\|<1$ (respectively$=1; >1$) if and only if $\rho(u)<1$ (respectively$=1; >1$); \item[(2)] for $u\neq 0$, $\|u\|=\lambda$ if and only if $\rho(\frac{u}{\lambda})=1$; \item[(3)] if $\|u\|>1$, then $\|u\|^{p^{-}}\leq \rho(u)\leq\|u\|^{p^{+}}$; \item[(4)] if $\|u\|<1$, then $\|u\|^{p^{+}}\leq \rho(u)\leq\|u\|^{p^{-}}$; \item[(5)] $\|u_k\|\to 0$ (respectively $\to \infty$) if and only if $\rho(u_k)\to 0$ (respectively $\to \infty$). \end{itemize} \end{proposition} For $x\in \Omega$, let us define $p^*(x)=\begin{cases} \frac{Np(x)}{N-p(x)} & \text{ if } p(x)0 and \beta\geq\alpha>1 such that m_1t^{\alpha-1}\leq M(t)\leq m_2t^{\beta-1}. \item[(M2)] For all t\in \mathbb{R}^+, \widehat{M}(t)\geq M(t)t. \end{itemize} For simplicity, we use c_i, to denote the general nonnegative or positive constant (the exact value may change from line to line). \section{Main results and proofs} \begin{theorem}\label{theo3.1} Assume that M satisfies {\rm (M1)} and {\rm (M2)} and the function q\in C(\overline{\Omega}) satisfies \label{e3.1} \beta p^+0 problem \eqref{e1.1} possesses a nontrivial weak solution. \end{theorem} \begin{lemma}\label{lem3.2} There exist \eta>0 and \alpha>0 such that J_\lambda(u)\geq \alpha>0 for any u\in X with \|u\|=\eta. \end{lemma} \begin{proof} First, we point out that \[ |u(x)|^{q(x)}\leq |u(x)|^{q^-}+|u(x)|^{q^+},\quad \text{for all } x\in \overline{\Omega}.$ Using the above inequality and (M1), we find that \begin{align*} J_\lambda(u) &=\widehat{M}\Big(\int_{\Omega}\frac{1}{p(x)}|\nabla u|^{p(x)}dx\Big) -\lambda\int_{\Omega}\frac{1}{q(x)}|u|^{q(x)}dx\\ &\geq\frac{m_1}{\alpha}\Big(\int_{\Omega}\frac{1}{p(x)} |\nabla u|^{p(x)}dx\Big)^{\alpha}-\frac{\lambda}{q^-}\Big(|u|_{q^-}^{q^-} +|u|_{q^+}^{q^+}\Big). \end{align*} From the assumptions of Theorem \ref{theo3.1}, $X$ is continuously embedded in $L^{q^-}(\Omega)$ and $L^{q^+}(\Omega)$. Then, there exist two positive constants $c_1$ and $c_2$ such that $|u(x)|_{q^-}\leq c_1\|u\|, \quad |u(x)|_{q^+}\leq c_2\|u\|,\quad \text{for all } u\in X.$ Hence, for any $u\in X$ with $\|u\|<1$, we obtain $J_\lambda(u)\geq \frac{m_1}{\alpha (p^+)^\alpha}\|u\|^{\alpha p^+} -\frac{\lambda}{q^-}\Big( c_1^{q^-}\|u\|^{q^-}+c_2^{q^+}\|u\|^{q^+}\Big).$ Since the function $g:[0,1]\to \mathbb{R}$ defined by $g(t)=\frac{m_1}{\alpha (p^+)^\alpha} -\frac{\lambda c_1^{q^-}}{q^-}t^{q^--\alpha p^+} -\frac{\lambda c_2^{q^+}}{q^-}t^{q^+-\alpha p^+},$ is positive in a neighborhood of the origin, the proof is complete. \end{proof} \begin{lemma}\label{lem3.3} There exists $e\in X$ with $\|e\|>\eta$ (where $\eta$ is given in Lemma \ref{lem3.2}) such that $J_\lambda(e)<0$. \end{lemma} \begin{proof} Let $\psi\in C_0^{\infty}(\Omega)$, $\psi\geq0$ and $\psi\neq0$ and $t>1$. By (M1) we have \begin{align*} J_\lambda(t\psi) &=\widehat{M}\Big(\int_{\Omega}\frac{1}{p(x)}|\nabla t\psi|^{p(x)}dx\Big) -\lambda\int_{\Omega}\frac{1}{q(x)}|t\psi|^{q(x)}dx\\ &\leq\frac{m_2}{\beta}\Big(\int_{\Omega}\frac{1}{p(x)} |\nabla t\psi|^{p(x)}dx\Big)^{\beta}-\lambda\frac{t^{q^-}}{q^+} \int_{\Omega}|\psi|^{q(x)}dx\\ &\leq\frac{m_2}{\beta (p^-)^{\beta}}t^{\beta p^+} \Big(\int_{\Omega}|\nabla \psi|^{p(x)}dx\Big)^{\beta} -\lambda\frac{t^{q^-}}{q^+}\int_{\Omega}|\psi|^{q(x)}dx. \end{align*} Since $\beta p^+1$ large enough, we can take $e=t\psi$ such that $\|e\|>\eta$ and $J_\lambda(e)<0$. \end{proof} \begin{proof}[Proof of Theorem \ref{theo3.1}] By Lemmas \ref{lem3.2}--\ref{lem3.3} and the mountain pass theorem of Ambrosetti and Rabinowitz \cite{AmbRab}, we deduce the existence of a sequence $(u_n)\subset X$ such that $$\label{e3.2} J_\lambda(u_n)\to c_3>0, \quad J'_\lambda(u_n)\to 0\quad \text{as }n\to\infty.$$ We prove that $(u_n)$ is bounded in $X$. Arguing by contradiction. We assume that, passing eventually to a subsequence, still denote by $(u_n)$, $\|u_n\|\to\infty$ and $\|u_n\|>1$ for all $n$. By \eqref{e3.2} and (M1)-(M2), for $n$ large enough, we have \begin{align*} &1+c_3+\|u_n\|\\ & \geq J_\lambda(u_n)-\frac{1}{q^-}\langle J'_\lambda(u_n),u_n\rangle\\ &\geq M\Big(\int_{\Omega}\frac{1}{p(x)}|\nabla u_n|^{p(x)}dx\Big) \int_{\Omega}\frac{1}{p(x)}|\nabla u_n|^{p(x)}dx -\lambda\int_{\Omega}\frac{1}{q(x)}|u_n|^{q(x)}dx\\ &\hspace{0.3cm}-\frac{1}{q^-}M\Big(\int_{\Omega}\frac{1}{p(x)} |\nabla u_n|^{p(x)}dx\Big)\int_{\Omega}|\nabla u_n|^{p(x)}dx +\frac{\lambda}{q^-}\int_{\Omega}|u_n|^{q(x)}dx\\ &\geq\frac{m_1}{\alpha (p^+)^{\alpha-1}}\Big(\frac{1}{p^+} -\frac{1}{q^-}\Big)\|u_n\|^{\alpha p^-}+\lambda \Big(\frac{1}{q^-} -\frac{1}{q(x)}\Big)\int_{\Omega}|u_n|^{q(x)}dx\\ &\geq\frac{m_1}{\alpha (p^+)^{\alpha-1}}\Big(\frac{1}{p^+} -\frac{1}{q^-}\Big)\|u_n\|^{\alpha p^-}+\lambda \Big(\frac{1}{q^-} -\frac{1}{q(x)}\Big)\Big(c_1\|u_n\|^{q^-}+c_2\|u_n\|^{q^+}\Big). \end{align*} Dividing the above inequality by $\|u_n\|^{\alpha p^-}$, taking into account \eqref{e3.1} holds true and passing to the limit as $n\to \infty$, we obtain a contradiction. It follows that $(u_n)$ is bounded in $X$. This information, combined with the fact that $X$ is reflexive, implies that there exists a subsequence, still denote by $(u_n)$ and $u_1\in X$ such that $(u_n)$ converges weakly to $u_1$ in $X$. Note that Proposition \ref{prop2.3} yields that $X$ is compactly embedded in $L^{q(x)}(\Omega)$, it follows that $(u_n)$ converges strongly to $u_1$ in $L^{q(x)}(\Omega)$. Then by H\"{o}lder inequality we deduce $$\label{e3.} \lim_{n\to\infty}\int_{\Omega}|u_n|^{q(x)-2}u_n(u_n-u_1)dx=0.$$ Using \eqref{e3.2}, we infer that $$\label{e3..} \lim_{n\to\infty}\langle J'_\lambda (u_n),u_n-u_1\rangle=0.$$ Since $(u_n)$ is bounded in $X$, passing to a subsequence, if necessary, we may assume that $\int_{\Omega}\frac{1}{p(x)}|\nabla u_n|^{p(x)}dx\to t_0\geq 0 \quad \text{ as } n\to\infty.$ If $t_0=0$ then $(u_n)$ converges strongly to $u_1=0$ in $X$ and the proof is complete. If $t_0>0$ then since the function $M$ is continuous, we obtain $M\Big(\int_{\Omega}\frac{1}{p(x)}|\nabla u_n|^{p(x)}dx\Big)\to M(t_0)\geq 0 \quad \text{as } n\to\infty.$ Thus, by (M1), for sufficiently large $n$, we have \label{e3.3} 00, \quad J'_\lambda(u_1)= 0; \] that is, $u_1$ ia a nontrivial weak solution of \eqref{e1.1}. \end{proof} \begin{theorem}\label{theo3.4} If we assume that {\rm (M1)--(M2)} hold and $q\in C_+(\overline{\Omega})$ satisfies \label{e3.6} 10$such that for any$\lambda>\lambda^*$, problem \eqref{e1.1} possesses a nontrivial weak solution. \end{theorem} Under the theorem's conditions, we want to construct a global minimizer of the functional. We start with the following auxiliary result. \begin{lemma}\label{lem3.5} The functional$J_\lambda$is coercive on$X$. \end{lemma} \begin{proof} By Theorem \ref{theo3.1} and Proposition \ref{prop2.2}, we deduce that for all$u\in X$, $J_\lambda(u)\geq \frac{m_1}{\alpha (p^+)^{\alpha}} \Big(\int_{\Omega}|\nabla u|^{p(x)}dx\Big))^{\alpha} -\frac{\lambda}{q^-}\Big( c_1\|u\|^{q-}+c_2\|u\|^{q^+}\Big).$ Now we set$\|u\|>1$, then $J_\lambda(u)\geq \frac{m_1}{\alpha (p^+)^{\alpha}}\|u\|^{\alpha p^-} -\frac{\lambda}{q^-}\Big( c_1\|u\|^{q-}+c_2\|u\|^{q^+}\Big).$ Since by relation \eqref{e3.6} we have$\alpha p^->q^+\geq q^-$, we infer that$J_\lambda(u)\to\infty$as$\|u\|\to\infty$. In other words,$J_\lambda$is coercive in$X$. \end{proof} \begin{proof}[Proof of Theorem \ref{theo3.4}]$J_\lambda(u)$is a coercive functional and weakly lower semi-con\-tinuous on$X$. These two facts enable us to apply \cite[Theorem 1.2]{Struw} in order to find that there exists$u_\lambda\in X$a global minimizer of$J_\lambda$and thus a weak solution of problem \eqref{e1.1}. We show$u_\lambda$is not trivial for$\lambda$large enough. Letting$t_0>1$be a constant and$\Omega_1$be an open subset of$\Omega$with$|\Omega_1|>0$, we assume that$v_0\in C_0^\infty(\overline{\Omega})$is such that$v_0(x)=t_0$for any$x\in \overline{\Omega_1}$and$0\leq v_0(x)\leq t_0$in$\Omega\backslash\Omega_1. We have \begin{align*} J_\lambda(v_0) &=\widehat{M}\Big(\int_{\Omega}\frac{1}{p(x)}|\nabla v_0|^{p(x)}dx\Big) -\lambda\int_{\Omega}\frac{1}{q(x)}|v_0|^{q(x)}dx\\ &\leq c_6-\frac{\lambda}{q^+}\int_{\Omega}|v_0|^{q(x)}dx \leq c_6-\frac{\lambda}{q^+}t_0^{q^-}|\Omega_1|. \end{align*} So there exists\lambda^*>0$such that$J_\lambda(v_0)<0$for any$\lambda\in [\lambda^*,+\infty)$. It follows that for any$\lambda\geq \lambda^*$,$u_\lambda$is a nontrivial weak solution of problem \eqref{e1.1} for$\lambda$large enough. \end{proof} \begin{theorem}\label{theo3.6} If$q\in C_+(\overline{\Omega})$with \label{e3.7} 10$ such that for any $\lambda\in (0,\lambda^{**})$, problem \eqref{e1.1} possesses a nontrivial weak solution. \end{theorem} We plan to apply Ekeland variational principle \cite{Ek} to get a nontrivial solution to problem \eqref{e1.1}. We start with two auxiliary results. \begin{lemma}\label{lem3.7} There exists $\lambda^{**}>0$ such that for any $\lambda \in (0,\lambda^{**})$ there are $\rho,a>0$ such that $J_\lambda(u)\geq a>0$ for any $u\in X$ with $\|u\|=\rho$. \end{lemma} \begin{proof} Under the assumption of Theorem \ref{theo3.6}, $X$ is continuously embedded in $L^{q(x)}(\Omega)$. Thus, there exists a positive constant $c_7$ such that $$\label{e3.8} |u|_{q(x)}\leq c_7\|u\| \quad \text{for all } u\in X.$$ Now, Let us assume that $\|u\|<\min \{1,\frac{1}{c_7}\}$, where $c_7$ is the positive constant from above. Then we have $|u|_{q(x)}<1$. Using Proposition \ref{prop2.2} we obtain $$\label{e3.9} \int_{\Omega}|u|^{q(x)}dx\leq|u|_{q(x)}^{q^-} \quad \text{for all u\in X with \|u\|=\rho \in (0,1)}.$$ Relations \eqref{e3.8} and \eqref{e3.9} imply $$\label{e3.10} \int_{\Omega}|u|^{q(x)}dx\leq c_7^{q^-}\|u\|^{q^-} \quad \text{for all u\in X with \|u\|=\rho}.$$ Using the hypotheses (M1) and \eqref{e3.10}, we deduce that for any $u\in X$ with $\|u\|=\rho$, the following hold \begin{align*} J_\lambda(u) &=\widehat{M}\Big(\int_{\Omega}\frac{1}{p(x)}|\nabla u|^{p(x)}dx\Big) -\lambda\int_{\Omega}\frac{1}{q(x)}|u|^{q(x)}dx\\ &\geq \frac{m_1}{\alpha (p^+)^{\alpha}}\|u\|^{\alpha p^+} -\frac{\lambda}{q^-}c_7^{q^-}\|u\|^{q^-}\\ &=\rho^{q^-}\Big(\frac{m_1}{\alpha (p^+)^{\alpha}}\rho^{\alpha p^+-q^-} -\frac{\lambda}{q^-}c_7^{q^-}\Big). \end{align*} By \eqref{e3.7} we have $q^-\leq q^+0$. \end{proof} \begin{lemma}\label{lem3.8} For any $\lambda\in (0,\lambda^{**})$ given by \eqref{e3.11}, there exists $\varphi\in X$ such that $\varphi\geq 0$, $\varphi\neq 0$ and $J_\lambda(t\varphi)<0$ for all $t>0$ small enough. \end{lemma} \begin{proof} Assumption \eqref{e3.7} implies that $q(x)<\beta p(x)$. Let $\epsilon_0>0$ such that $q^-+\epsilon_0<\beta p^-$. Since $q\in C(\overline{\Omega})$, there exists an open set $\Omega_0\subset \Omega$ such that $|q(x)-q^-|<\epsilon_0$ for all $x\in \Omega_0$. It follows that $q(x)0. \] On the other hand, by Lemma \ref{lem3.8}, there exists$\varphi\in X$such that $J_\lambda(t\varphi)<0 \quad \text{for t>0 small enough}.$ Moreover, for$u\in B_{\rho}(0)$, $J_\lambda(u)\geq \frac{m_1}{\alpha (p^+)^{\alpha}}\|u\|^{\alpha p^+} -\frac{\lambda}{q^-}c_7^{q^-}\|u\|^{q^-}.$ It follows that $-\infty0 and v\in B_{\rho}(0). The above relation yields \[ \frac{J_\lambda(u_{\varepsilon}+tv)-J_\lambda(u_{\varepsilon})}{t} +\varepsilon \|v\|\geq0.$ Letting$t\to0$it follows that$\langle J_\lambda'(u_{\varepsilon}),v\rangle+\varepsilon \|v\|>0$and we infer that$\|J_\lambda'(u_{\varepsilon})\|\leq \varepsilon$. We deduce that there exists a sequence$(v_n)\subset B_{1}(0)$such that $$\label{e3.12} J_\lambda(v_n)\to c_8,\quad J_\lambda'(v_n)\to 0.$$ It is clear that$(v_n)$is bounded in$X$. Thus, there exists$u_2\in X$such that, up to a subsequence,$(v_n)$converges weakly to$u_2$in$X$. Actually, with similar arguments as those used in the proof Theorem \ref{theo3.1}, we can show that$v_n\to u_2$in$X$. Thus, by relation \eqref{e3.12}, $J_\lambda(u_2)=c_8<0, \quad J_\lambda'(u_2)= 0;$ i.e.,$u_2$is a nontrivial weak solution for problem \eqref{e1.1}. \end{proof} \begin{thebibliography}{99} \bibitem{Acerbi} E. Acerbi, G. Mingione; {Gradient estimate for the$p(x)$-Laplacian system}, \emph{J. Reine Angew. Math}, \textbf{584} (2005), 117-148. \bibitem{AmbRab} A. Ambrosetti, P. H. Rabinowitz; {Dual variational methods in critical points theory and applications}, \emph{J. Funct. Anal.}, \textbf{14} (1973), 349-381. \bibitem{Aro} A. Arosio, S. Panizzi; {On the well-posedness of the Kirchhoff string}, \emph{Trans. Amer. Math. Soc.}, \textbf{348} (1996), 305-330. \bibitem{Aut} G. Autuori, P. Pucci, M. C. Salvatori; {Global nonexistence for nonlinear Kirchhoff systems}, \emph{Arch. Rat. Mech. Anal.}, \textbf{196} (2010), 489-516. \bibitem{Cava} M. M. Cavalcanti, V. N. Domingos Cavalcanti, J. A. Soriano; {Global existence and uniform decay rates for the Kirchhoff-Carrier equation with nonlinear dissipation}, \emph{Adv. Differential Equations}, \textbf{6} (2001), 701-730. \bibitem{Chen} Y. Chen, S. Levine, M. Rao; {Variable exponent, linear growth functionals in image processing}, \emph{SIAM J. Appl. Math.}, \textbf{66} (2006), 1383-1406. \bibitem{Chung} N. T. Chung; {Multiplicity results for a class of$p(x)$-Kirchhoff type equations with combined nonlinearities}, \emph{Electron. J. Qual. Theory Differ. Equ.}, \textbf{42} (2012), 1-13. \bibitem{Col} F. Colasuonno, P. Pucci; {Multiplicity of solutions for$p(x)$-polyharmonic Kirchhoff equations}, \emph{Nonlinear Anal.}, \textbf{74} (2011), 5962-5974. \bibitem{Corr1} F. J. S. A. Corr\^{e}a, G. M. Figueiredo; {On an elliptic equation of$p$-Kirchhoff type via variational methods}, \emph{Bull. Aust. Math. Soc.}, \textbf{74} (2006), 263-277. \bibitem{Corr2} F. J. S. A. Corr\^{e}a, G. M. Figueiredo; {On a$p$-Kirchhoff equation via Krasnoselskii's genus}, \emph{Appl. Math. Letters}, \textbf{22} (2009), 819-822. \bibitem{Dai1} G. Dai, R. Hao; {Existence of solutions for a$p(x)$-Kirchhoff-type equation}, \emph{J. Math. Anal. Appl.}, \textbf{359} (2009), 275-284. \bibitem{Dai2} G. Dai, R. Ma; {Solutions for a$p(x)$-Kirchhoff-type equation with Neumann boundary data}, \emph{Nonlinear Anal.}, \textbf{12} (2011), 2666-2680. \bibitem{Dre1} M. Dreher; {The Kirchhoff equation for the p-Laplacian}, \emph{Rend. Semin. Mat. Univ. Politec. Torino}, \textbf{64} (2006), 217-238. \bibitem{Dre2} M. Dreher; {The wave equation for the p-Laplacian}, \emph{Hokkaido Math. J.}, \textbf{36} (2007), 21-52. \bibitem{Ed1} D. E. Edmunds, J. R\'akosn\'{\i}k; {Density of smooth functions in$W^{k,p(x)}(\Omega)$}, \emph{Proc. R. Soc. A}, \textbf{437} (1992), 229-236. \bibitem{Ed2} D. E. Edmunds, J. R\'akosn\'{\i}k; {Sobolev embedding with variable exponent}, \emph{Studia Math.}, \textbf{143} (2000), 267-293. \bibitem{Ek} I. Ekeland; {On the variational principle}, \emph{J. Math. Anal. Appl.}, \textbf{47} (1974), 324-353. \bibitem{Fan1} X. L. Fan, J. S. Shen, D. Zhao; {Sobolev embedding theorems for spaces$W^{k,p(x)}(\Omega)$}, \emph{J. Math. Anal. Appl.}, \textbf{262} (2001), 749-760. \bibitem{Fan2} X. L. Fan, D. Zhao; {On the spaces$L^{p(x)}$and$W^{m,p(x)}$}, \emph{J. Math. Anal. Appl.}, \textbf{263} (2001), 424-446. \bibitem{Fan4} X. L. Fan, X. Y. Han; {Existence and multiplicity of solutions for$p(x)$-Laplacian equations in$\mathbb{R}^{N}$}, \emph{Nonlinear Anal.}, \textbf{59} (2004), 173-188. \bibitem{Fan5} X. L. Fan, Q. H. Zhang; {Existence of solutions for$p(x)$-Laplacian Dirichlet problems}, \emph{Nonlinear Anal.}, \textbf{52} (2003), 1843-1852. \bibitem{Fan7} X. L. Fan; {On nonlocal$p(x)$-Laplacian Dirichlet problems}, \emph{Nonlinear Anal.}, \textbf{72} (2010), 3314-3323. \bibitem{K} G. Kirchhoff; \emph{Mechanik}, Teubner, Leipzig, 1883. \bibitem{Kov} O. Kov\'{a}\v{c}ik, J. R\'{a}kosn\'{\i}k; {On the spaces$L^{p(x)}(\Omega)$and$W^{1,p(x)}(\Omega)$}, \emph{Czechoslovak Math. J.}, \textbf{41} (1991), 592-618. \bibitem{Lions} J. L. Lions; {On some questions in boundary value problems of mathematical physics}, \emph{in: Proceedings of international Symposium on Continuum Mechanics and Partial Differential Equations, Rio de Janeiro 1977, in: de la Penha, Medeiros (Eds.), Math. Stud., North-Holland}, \textbf{30} (1978), 284-346. \bibitem{Mih1} M. Mih\u{a}ilescu, P. Pucci, V. R\u{a}dulescu; {Eigenvalue problems for anisotropic quasilinear elliptic equations with variable exponent}, \emph{J. Math. Anal. Appl.}, \textbf{340} (2008), 687-698. \bibitem{Struw} M. Struwe; {Variational methods: Applications to Nonlinear Partial Differential Equatios and Hamiltonian systems}, Springer-Verlag, Berlin, 1996. \bibitem{Zhikov} V. V. Zhikov; {Averaging of functionals of the calculus of variations and elasticity theory}, \emph{Math. USSR. Izv.}, \textbf{9} (1987), 33-66. \end{thebibliography} \section{Corrigendum posted in August 27, 2014} A reader pointed out that no function$M(t)$can satisfy both hypotheses (M1) and (M2). In response, we present a proof of our results by adding the following assumption $$\label{e4.1} m_1q^-(p^-)^{\alpha}>m_2\alpha p^-(p^+)^{\alpha}.$$ and without assumption (M2). \subsection*{Modified assumptions} We delete the assumption (M2) and re-state the following: \begin{itemize} \item [(M1)] There exist$m_2\geq m_1 > 0$and$\alpha> 1$such that $$m_1t^{\alpha-1} \leq M(t) \leq m_2t^{\alpha-1}, \quad \forall t \in \mathbb{R}^+$$ (The original (M1) implies$\alpha=\beta$, so we rename constant$\alpha$.); \end{itemize} In the proof of Theorem \ref{theo3.1}, By \eqref{e3.2} and (M1), for$nlarge enough, we can write \begin{align*} &1+c_3+\|u_n\|\\ & \geq J_\lambda(u_n)-\frac{1}{q^-}\langle J'_\lambda(u_n), u_n\rangle \\ &= \widehat{M}\Big(\int_{\Omega}\frac{1}{p(x)}|\nabla u_n|^{p(x)}\,dx\Big) -\lambda\int_{\Omega}\frac{1}{q(x)}|u_n|^{q(x)}\,dx\\ &\hspace{0.3cm}-\frac{1}{q^-}M\Big(\int_{\Omega}\frac{1}{p(x)}|\nabla u_n|^{p(x)} \,dx\Big)+\frac{\lambda}{q^-}\int_{\Omega}|u_n|^{q(x)\,}dx\\ &\geq\frac{m_1}{\alpha}\Big(\int_{\Omega}\frac{1}{p(x)}|\nabla u_n|^{p(x)} \,dx\Big)^{\alpha}-\frac{m_2}{q^-}\Big(\int_{\Omega}\frac{1}{p(x)} |\nabla u_n|^{p(x)}\,dx\Big)^{\alpha-1}\int_{\Omega}|\nabla u_n|^{p(x)}\,dx\\ &\hspace{0.3cm}+\lambda \int_{\Omega}\Big(\frac{1}{q^-}-\frac{1}{q(x)}\Big) |u_n|^{q(x)}\,dx\\ &\geq \Big(\frac{m_1}{\alpha (p^+)^{\alpha}}-\frac{m_2}{q^-(p^-)^{\alpha-1}}\Big) \|u_n\|^{\alpha p^-}+\lambda \Big(c_1\|u_n\|^{q^-}+c_2\|u_n\|^{q^+}\Big). \end{align*} Dividing the above inequality by\|u_n\|^{\alpha p^-}$, taking into account \eqref{e3.1} and \eqref{e4.1} hold true and passing to the limit as$n\to \infty$, we obtain a contradiction. It follows that$(u_n)$is bounded in$X$. Theorem \ref{theo3.6} remains unchanged. However, Theorems \ref{theo3.1} and \ref{theo3.4} need to be stated without assumption (M2). Relation \eqref{e3.1} need to be changed by$\alpha p^+