\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
\begin{equation}\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}
\end{equation}
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 $1<p(x)<N$ 
and $q(x)>1$ 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
\begin{equation} \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,
\end{equation}
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
\[
p^*(x)=\begin{cases}
\frac{Np(x)}{N-p(x)}      & \text{ if } p(x)<N,\\
\infty       & \text{if } p(x)\geq N.
\end{cases}
\]

\begin{proposition}[\cite{Fan2}] \label{prop2.3}
If $q\in C_+(\overline{\Omega})$ and $q(x)\leq p^*(x)$ $(q(x)<p^*(x))$ 
for $x\in \overline{\Omega}$, then there is a continuous (compact) 
embedding $X\hookrightarrow L^{q(x)}(\Omega)$.
\end{proposition}

\begin{lemma}[\cite{Fan5}] \label{lem2.4}
Denote
\[
I(u)=\int_{\Omega}\frac{1}{p(x)}|\nabla u|^{p(x)}dx,\hspace{0.2cm}\text{ for all }\hspace{0.2cm} u\in X,\hspace{3cm}
\]
then $I(u)\in C^1(X,R)$ and the derivative operator $I'$ of $I$ is
\[
\langle I'(u),v\rangle=\int_{\Omega}|\nabla u|^{p(x)-2}\nabla u\nabla v\,dx,
\quad\text{for all } u,~v\in X,
\]
and we have
\begin{itemize}
\item[(1)] $I$ is a convex functional; 
\item[(2)] $I':X \to X^{*}$ is a bounded homeomorphism and strictly 
monotone operator; 
\item[(3)] $I'$  is a mapping of type $(S_{+})$, namely:
$u_n\rightharpoonup u$ and $\limsup_{n\to+\infty}I'(u_n)
(u_n-u)\leq 0$, imply $u_n\to u$.
\end{itemize}
\end{lemma}

\begin{definition}\label{def2.5} \rm
A function $u\in X$ is said to be a weak solution of  \eqref{e1.1} if 
\[
M \Big(\int_{\Omega}\frac{1}{p(x)}|\nabla u|^{p(x)}dx\Big)
\int_{\Omega}|\nabla u|^{p(x)-2}\nabla u \nabla v\,dx
-\lambda \int_{\Omega}|u|^{q(x)-2}uv\,dx=0,
\]
for all $v\in X$.
\end{definition}
The Euler-Lagrange functional associated to \eqref{e1.1} is 
\[
J_\lambda(u)=\widehat{M}\Big(\int_{\Omega}
\frac{1}{p(x)}|\nabla u|^{p(x)}dx\Big)
-\lambda\int_{\Omega}\frac{1}{q(x)}|u|^{q(x)}dx,
\]
where $\widehat{M}(t)=\int_0^t M(\tau)d\tau$. Then
\[
\langle J_\lambda'(u),v\rangle
=M\Big(\int_{\Omega}\frac{1}{p(x)}|\nabla u|^{p(x)}dx\Big)
\int_{\Omega}|\nabla u|^{p(x)-2}\nabla u\nabla v\,dx
-\lambda \int_{\Omega} |u|^{q(x)-2}uv\,dx,
\]
for all $u,v\in X$, then we know that the weak solution of \eqref{e1.1} 
corresponds to the critical point of the functional $J_\lambda$.
Hereafter $M(t)$ is supposed to verify the following assumptions:
\begin{itemize}
\item[(M1)] There exists $m_2\geq m_1>0$ 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
\begin{equation}\label{e3.1}
\beta p^+<q^-\leq q^+<p^*(x).
\end{equation}
Then for any $\lambda>0$ 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 $\beta p^+<q^-$, we obtain $\lim_{t\to\infty}J_\lambda(t\psi)=-\infty$. 
Then for $t>1$ 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
\begin{equation}\label{e3.2}
J_\lambda(u_n)\to c_3>0, \quad
 J'_\lambda(u_n)\to 0\quad \text{as }n\to\infty.
\end{equation}
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
\begin{equation}\label{e3.}
\lim_{n\to\infty}\int_{\Omega}|u_n|^{q(x)-2}u_n(u_n-u_1)dx=0.
\end{equation}
Using \eqref{e3.2}, we infer that
\begin{equation}\label{e3..}
\lim_{n\to\infty}\langle J'_\lambda (u_n),u_n-u_1\rangle=0.
\end{equation}
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
\begin{equation}\label{e3.3}
0<c_4\leq M\Big(\int_{\Omega}\frac{1}{p(x)}|\nabla u_n|^{p(x)}dx\Big)\leq c_5.
\end{equation}
From \eqref{e3.}-\eqref{e3.3}, we deduce that
\begin{equation} \label{e3.4}
\lim_{n\to\infty}\int_{\Omega}|\nabla u_n|^{p(x)-2}\nabla u_n
 (\nabla u_n-\nabla u_1)dx=0.
\end{equation}
Using Lemma \ref{lem2.4}, we deduce that actually $(u_n)$ converges strongly to $u_1$ in $X$. Then by relation \eqref{e3.2} we have
\[
J_\lambda(u_1)= c_3>0, \quad  J'_\lambda(u_1)= 0;
\]
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 
\begin{equation}\label{e3.6}
1<q^-\leq q^+<\alpha p^-,
\end{equation}
then there exists $\lambda^*>0$ 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
\begin{equation}\label{e3.7}
1<q(x)<p(x)<p^*(x),
\end{equation}
then there exists $\lambda^{**}>0$ 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
\begin{equation} \label{e3.8}
|u|_{q(x)}\leq c_7\|u\| \quad \text{for all } u\in X.
\end{equation}
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
\begin{equation} \label{e3.9}
\int_{\Omega}|u|^{q(x)}dx\leq|u|_{q(x)}^{q^-} \quad
\text{for all $u\in X$ with $\|u\|=\rho \in (0,1)$}.
\end{equation}
Relations \eqref{e3.8} and \eqref{e3.9} imply
\begin{equation} \label{e3.10}
\int_{\Omega}|u|^{q(x)}dx\leq c_7^{q^-}\|u\|^{q^-} \quad
\text{for all $u\in X$ with $\|u\|=\rho$}.
\end{equation}
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 $q^-\leq q^+<p^-\leq p^+<\alpha p^+$. So, if we take
\begin{equation}\label{e3.11}
\lambda^{**}=\frac{m_1q^-}{2\alpha (p^+)^{\alpha}}\rho^{\alpha p^+-q^-},
\end{equation}
then for any $\lambda \in (0,\lambda^{**})$ and $u\in X$ with $\|u\|=\rho$,
there exists $a=\frac{\rho^{\alpha p^+}}{2\alpha (p^+)^{\alpha}}$
such that $J_\lambda(u)\geq a>0$.
\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 
$q(x)<q^-+\epsilon_0<\beta p^-$ for all $x\in \Omega_0$.

Let $\varphi \in C_0^{\infty}(\Omega)$ be such that 
$\operatorname{supp}(\varphi)\supset \overline{\Omega_0}$, 
$\varphi(x)=1$ for all 
$x\in \overline{\Omega_0}$ and $0\leq \varphi\leq1$ in $\Omega$. 
Then for any $t\in(0,1)$, we have
\begin{align*}
J_\lambda(t\varphi)
&=\widehat{M}\Big(\int_{\Omega}\frac{1}{p(x)}|\nabla t\varphi|^{p(x)}dx\Big)
-\lambda\int_{\Omega}\frac{1}{q(x)}|t\varphi|^{q(x)}dx\\
&\leq \frac{m_2}{\beta}\Big(\int_{\Omega}\frac{1}{p(x)}|\nabla t\varphi|^{p(x)}
dx\Big)^{\beta}-\lambda\int_{\Omega}\frac{1}{q(x)}t^{q(x)}|\varphi|^{q(x)}dx\\
&\leq \frac{m_2}{\beta (p^-)^{\beta}}t^{\beta p^-}
\Big(\int_{\Omega}|\nabla \varphi|^{p(x)}dx\Big)^{\beta}
-\frac{\lambda}{q^+}t^{q^-+\epsilon_0}\int_{\Omega_0}|\varphi|^{q(x)}dx<0,
\end{align*}
for all $t<\delta^{\frac{1}{\beta p^--q^--\epsilon_0}}$ with
\[
0<\delta<\min\Bigl\{1,\frac{\lambda \beta (p^-)^{\beta}}{m_2q^+}
\frac{\int_{\Omega_0}|\varphi|^{q(x)}dx}
{\big(\int_{\Omega}|\nabla \varphi|^{p(x)}dx\big)^{\beta}}\Bigl\}.
\]
\end{proof}

\begin{proof}[Proof of Theorem \ref{theo3.6}]
Let $\lambda^{**}$ be defined as in \eqref{e3.11} and
 $\lambda\in (0,\lambda^{**})$. By Lemma \ref{lem3.7}, it follows 
that on the boundary of the ball centered at the origin and of 
radius $\rho$ in $X$, we have
\[
\inf_{\partial B_{\rho}(0)}J_\lambda(u)>0.
\]
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
\[
-\infty<c_8=\inf_{\overline{B_{\rho}(0)}}J_\lambda(u)<0.
\]
We let now $0<\varepsilon<\inf_{\partial B_{\rho}(0)}J_\lambda
-\inf_{B_{\rho}(0)}J_\lambda$. Applying Ekeland variational
principle \cite{Ek} to the functional $J_\lambda:\overline{B_{\rho}(0)}\to \mathbb{R}$,
we find $u_{\varepsilon}\in \overline{B_{\rho}(0)}$ such that
\begin{gather*}
J_\lambda(u_{\varepsilon}) <\inf_{\overline{B_{\rho}(0)}}J_\lambda
+\varepsilon\\
J_\lambda(u_{\varepsilon}) <J_\lambda(u)+\varepsilon\|u-u_{\varepsilon}\|,
\quad u\neq u_{\varepsilon}.
\end{gather*}
Since
\[
J_\lambda(u_{\varepsilon})\leq\inf_{\overline{B_{\rho}(0)}}J_\lambda
+\varepsilon\leq \inf_{B_{\rho}(0)}J_\lambda
+\varepsilon<\inf_{\partial B_{\rho}(0)}J_\lambda,
\]
we deduce that $u_{\varepsilon}\in B_{\rho}(0)$. Now, we define 
$K_\lambda:\overline{B_{\rho}(0)}\to \mathbb{R}$
by $K_\lambda(u)=J_\lambda(u)+\varepsilon \|u-u_{\varepsilon}\|$. 
It is clear that $u_{\varepsilon}$ is a minimum point of $K_\lambda$ and thus
\[
\frac{K_\lambda(u_{\varepsilon}+tv)-K_\lambda(u_{\varepsilon})}{t}\geq0,
\]
for small $t>0$ 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
\begin{equation}\label{e3.12}
J_\lambda(v_n)\to c_8,\quad J_\lambda'(v_n)\to 0.
\end{equation}
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}

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\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
 \begin{equation}\label{e4.1}
m_1q^-(p^-)^{\alpha}>m_2\alpha p^-(p^+)^{\alpha}.
\end{equation}
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 $n$ large 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 
$\alpha p^+<q^-\leq q^+<p^*(x)$. The proofs of Theorems and Lemmas are 
similar to the original proofs, but replacing $\beta$ by $\alpha$.
\smallskip

The authors would like to thank anonymous reader and the editor for
allowing us to correct our mistake.

\end{document}