\documentclass[reqno]{amsart}
\usepackage{hyperref}

\AtBeginDocument{{\noindent\small
\emph{Electronic Journal of Differential Equations},
Vol. 2009(2009), No. 72, pp. 1--12.\newline
ISSN: 1072-6691. URL: http://ejde.math.txstate.edu or http://ejde.math.unt.edu
\newline ftp ejde.math.txstate.edu}
\thanks{\copyright 2009 Texas State University - San Marcos.}
\vspace{9mm}}

\begin{document}
\title[\hfilneg EJDE-2009/72\hfil Existence of solutions]
{Existence of solutions for a $p(x)$-Laplacian  non-homogeneous
 equations}

\author[I. Andrei \hfil EJDE-2009/72\hfilneg]
{Ionic\u{a} Andrei}

\address{Ionic\u{a} Andrei \newline
Department of Mathematics, High School of Cujmir, 227150 Cujmir, Romania}
\email{andreiionica2003@yahoo.com}

\thanks{Submitted March 9, 2009. Published June 2, 2009.}
\subjclass[2000]{35D05, 35J60, 58E05}
\keywords{$p(x)$-Laplace operator; generalized
Lebesgue-Sobolev space; \hfill\break\indent critical point; weak solution}

\begin{abstract}
 We study the boundary value problem
 \begin{gather*}
 -\mathop{\rm div}(|\nabla u|^{p(x)-2}\nabla  u)=f(x,u)\quad
 \text{in }\Omega, \\
 u=0\quad \text{on }\partial \Omega,
 \end{gather*} 
 where $\Omega$  is a smooth bounded domain  in $\mathbb{R}^N$.
 Our attention is focused  on the cases when
 $$
 f(x,u)=\pm (-\lambda |u|^{p(x)-2}u+|u|^{q(x)-2}u),
 $$
 where $ p(x)<q(x)<N\cdot p(x)/(N-p(x))$ for $x$ in $\Omega$.
\end{abstract}

\maketitle
\numberwithin{equation}{section}
\newtheorem{theorem}{Theorem}[section]
\newtheorem{lemma}[theorem]{Lemma}

\section{Introduction and preliminary results}

In the recent years increasing attention has been paid to the
study of differential and partial differential equations involving
variable exponent conditions. The interest in studying such
problems was stimulated by their applications in elastic
mechanics, fluid dynamics or calculus of variations. For more
information on modelling physical phenomena by equations involving
$p(x)$-growth condition we refer to \cite{ac,di,ha,pf,ru,wi}.
The appearance of such
physical models was facilitated by the development of variable
Lebesgue and Sobolev spaces, $L^{p(x)}$ and $W^{1,p(x)}$, where
$p(x)$ is a real-valued function. Variable exponent Lebesgue
spaces appeared for the first time in literature as early as 1931
in an article by  Orlicz \cite{or}. The spaces $L^{p(x)}$ are
special cases of Orlicz spaces $L^{\varphi}$ originated by Nakano
\cite{na} and developed by Musielak and Orlicz \cite{mu,mu1},
where $f\in L^{\varphi}$ if and only if $\int \varphi
(x,| f(x)|)dx<\infty$ for a suitable $\varphi$. Variable
exponent Lebesque spaces on the real line have been independently
developed by Russian researchers. In that context we refer to the
studies of Tsenov \cite{ts}, Sharapudinov \cite{sh} and Zhikov
\cite{zh1,zh2}.

This paper is motivated by the phenomena that can be modelled by
the equations
\begin{equation}\label{e1}
 \begin{gathered}
  -\mathop{\rm div}(|\nabla u|^{p(x)-2}\nabla u)=f(x,u) \quad
  \text{in } \Omega \\
   u=0 \quad\text{on } \partial\Omega
 \end{gathered}
\end{equation}
where $\Omega\subset \mathbb{R}^N$ $(N\geq 3)$ is a bounded domain
with smooth boundary and $1<p(x)$, $p(x)\in C(\overline{\Omega})$.
Our goal will be to obtain nontrivial weak solutions for
\eqref{e1} in the generalized Sobolev space $W^{1,p(x)}(\Omega)$
for some particular nonlinearities of the type $f(x,u)$. Problems
of type \eqref{e1} have been intensively studied in the past
decades. We refer to
\cite{al,fa1,fa2,gherad,mi,mi0,mi1,mi2,ra1,ra2,zh}, for some
interesting results. We point out the presence in \eqref{e1} of
the $p(x)$-Laplace operator. This is a natural extension of the
$p$-Laplace operator, with $p$ a positive constant. However, such
generalizations are not trivial since the $p(x)$-Laplace operator
possesses a more complicated structure than $p$-Laplace operator,
for example it is inhomogeneous.

We recall some definitions and  properties of the variable
exponent Lebesgue-Sobolev spaces $L^{p(\cdot)}(\Omega)$ and
$W_0^{1,p(\cdot)}(\Omega)$, where $\Omega$ is a bounded domain in
$\mathbb{R}^N$. Roughly speaking, anisotropic Lebesgue and Sobolev spaces
are functional spaces of Lebesgue's and Sobolev's type in which
different space directions have different roles.

Set $C_+(\overline\Omega)=\{h\in
C(\overline\Omega):\min_{x\in\overline\Omega}h(x)>1\}.$ For
any $h\in C_+(\overline\Omega)$ we define
$$
h^+=\sup_{x\in\Omega}h(x)\quad\mbox{and}\quad
h^-=\inf_{x\in\Omega}h(x).
$$
For $p\in C_+(\overline\Omega)$, we
introduce {\it the variable exponent Lebesgue space}
\begin{align*}
L^{p(\cdot)}(\Omega)=\big\{&u: u \mbox{ is a
 measurable real-valued function}\\
&\text{such that }\int_\Omega|u(x)|^{p(x)}\,dx<\infty\big\},
\end{align*}
endowed with the so-called {\it Luxemburg norm}
$$
|u|_{p(\cdot)}=\inf\big\{\mu>0;\;\int_\Omega|
\frac{u(x)}{\mu}|^{p(x)}\,dx\leq 1\big\},
$$
which is a
separable and reflexive Banach space. For basic properties of the
variable exponent Lebesgue spaces we refer to \cite{ko}. If $0
<|\Omega|<\infty$ and $p_1$, $p_2$ are variable exponents in
$C_+(\overline\Omega)$ such that $p_1 \leq p_2$  in $\Omega$, then
the  embedding $L^{p_2(\cdot)}(\Omega)\hookrightarrow
L^{p_1(\cdot)}(\Omega)$ is continuous, \cite[Theorem~2.8]{ko}.

Let $L^{p'(\cdot)}(\Omega)$ be the conjugate space of
$L^{p(\cdot)}(\Omega)$, obtained by conjugating the exponent
pointwise that is,  $1/p(x)+1/p'(x)=1$, \cite[Corollary~2.7]{ko}.
For any $u\in L^{p(\cdot)}(\Omega)$ and $v\in
L^{p'(\cdot)}(\Omega)$ the following H\"older type inequality
\begin{equation}\label{Hol}
\big|\int_\Omega uv\,dx\big|\leq\Big(\frac{1}{p^-}+
\frac{1}{{p'}^-}\Big)|u|_{p(\cdot)}|v|_{p'(\cdot)}
\end{equation}
is valid.

An important role in manipulating the generalized
Lebesgue-Sobolev spaces is played by the {\it $p(\cdot)$-modular}
of the $L^{p(\cdot)}(\Omega)$ space, which is the mapping
 $\rho_{p(\cdot)}:L^{p(\cdot)}(\Omega)\to\mathbb{R}$ defined by
$$
\rho_{p(\cdot)}(u)=\int_\Omega|u|^{p(x)}\,dx.
$$
If $(u_n)$, $u\in L^{p(\cdot)}(\Omega)$ then the following
relations hold
\begin{gather}\label{L40}
|u|_{p(\cdot)}<1\;(=1;\,>1)\;\Leftrightarrow\;\rho_{p(\cdot)}(u)
<1\;(=1;\,>1)
\\ \label{L4}
|u|_{p(\cdot)}>1 \;\Rightarrow\; |u|_{p(\cdot)}^{p^-}\leq\rho_{p(\cdot)}(u)
\leq|u|_{p(\cdot)}^{p^+}
\\ \label{L5}
|u|_{p(\cdot)}<1 \;\Rightarrow\; |u|_{p(\cdot)}^{p^+}\leq
\rho_{p(\cdot)}(u)\leq|u|_{p(\cdot)}^{p^-}
\\ \label{L6}
|u_n-u|_{p(\cdot)}\to 0\;\Leftrightarrow\;\rho_{p(\cdot)} (u_n-u)\to 0,
\end{gather}
since $p^+<\infty$. For a proof of these facts see \cite{ko}.
Spaces with $p^{+}=\infty$ have been studied by Edmunds, Lang and
Nekvinda \cite{ed}.

Next, we define $W_0^{1,p(x)}(\Omega)$ as the closure of
$C_0^{\infty}(\Omega)$ under the norm
\[
\| u\|_{p(x)}=|\nabla u|_{p(x)}.
\]
The space $(W_0^{1,p(x)}(\Omega),\| \cdot \|_{p(x)})$
is a separable and reflexive Banach space. We note that if
$q\in C_+(\overline{\Omega})$ and $q(x)<p^*(x)$ for all
$x\in \overline{\Omega}$ then the embedding
$W_0^{1,p(x)}(\Omega)\hookrightarrow L^{q(x)}(\Omega)$ is compact
and continuous, where $p^*(x)=Np(x)/(N-p(x))$ if $p(x)<N$ or
$p^*(x)=+\infty$ if $p(x)\geq N$ \cite[Theorem 3.9 and 3.3]{ko}
(see also \cite [Theorem 1.3 and 1.1]{fa}).

\section{Main results}

In this paper we study \eqref{e1} in the particular cases
when
\[
f(x,t)=\pm (-\lambda| t|^{p(x)-2}t+| t|^{q(x)-2}t)
\]
where $p(x)$, $q(x)\in C_+(\Omega)$ with
$p(x)<q(x)<N\cdot p(x)/(N-p(x))$ for any $x\in \overline{\Omega}$
and $\lambda >0$.

First, we consider the problem
\begin{equation}\label{e6}
 \begin{gathered}
  -\mathop{\rm div}(|\nabla u|^{p(x)-2}\nabla u)
=-\lambda | u | ^{p(x)-2}u +| u | ^{q(x)-2}u \quad\text{in } \Omega \\
   u=0 \quad\text{on }\partial\Omega
 \end{gathered}
\end{equation}
We say that $u\in W_0^{1,p(x)}(\Omega)$ is a weak solution of
 \eqref{e6} if
\[
\int_{\Omega}| \nabla u | ^{p(x)-2}\nabla u \nabla v dx
+\lambda \int_{\Omega} | u | ^{p(x)-2}uv\,dx-
\int_{\Omega}| u | ^{q(x)-2}uv\,dx=0
\]
for all $v\in W_0^{1,p(x)}(\Omega)$.

We will prove the following result.

\begin{theorem}\label{thm1}
For every $\lambda >0$, problem \eqref{e6} has infinitely many weak
solutions provided $2\leq p^- $, $p^+<q^-$ and $q^+<N\cdot
p^-/(N-p^-)$.
\end{theorem}

Next, we study the problem
\begin{equation}\label{e7}
 \begin{gathered}
  -\mathop{\rm div}(|\nabla u|^{p(x)-2}\nabla u)
=\lambda | u | ^{p(x)-2}u -| u | ^{q(x)-2}u \quad\text{in } \Omega \\
   u=0 \quad\text{on } \partial\Omega
 \end{gathered}
\end{equation}
We say that $u\in W_0^{1,p(x)}(\Omega)$ is a weak solution of
 \eqref{e7} if
\[
\int_{\Omega}|\nabla u|^{p(x)-2}\nabla u\nabla v\,dx
-\lambda \int_{\Omega}|u|^{p(x)-2}uv dx
+\int_{\Omega}|u|^{q(x)-2}uv\,dx=0
\]
 for all $v\in W_0^{1,p(x)}(\Omega)$.

Next, we prove the following result.

\begin{theorem}\label{thm2}
There exists $\lambda^*>0$ such that for any
$\lambda \geq \lambda^*$ problem \eqref{e7} has a nontrivial
weak solution provided $p^+<q$ and $q^+<N\cdot p^-/(N-p^-)$.
\end{theorem}

We remark that in the particular case corresponding to $p(x)=2$
and $q(x)=q$, $q$ being a constant,  \eqref{e6} becomes
\begin{equation}\label{e8}
 \begin{gathered}
  -\Delta u=-\lambda u +| u | ^{q-2}u \quad\text{in }  \Omega \\
   u=0 \quad\text{on } \partial\Omega
 \end{gathered}
\end{equation}
This problem has been studied by Ambrosetti and Rabinowitz
\cite{am} provided $2<q<2*=2N/(N-2)$. Using the Mountain Pass
Theorem combined with the observation that the operator
$-\Delta +\lambda I$ $(\lambda >0)$ is coercive in
$H_0^1(\Omega)$, Ambrosetti and Rabinowitz showed that problem
\eqref{e8} has a positive solution for any $\lambda >0$.

\section{Proof of Theorem 1}

The key argument in the proof  is the following
version of the Mountain Pass Theorem (see \cite[Theorem 9.12]{ra}):

\subsection*{Mountain Pass Theorem}
 Let $X$ be an infinite dimensional
real Banach space and let $I\in C^1(X,\mathbb{R})$ be even, satisfying
the Palais-Smale condition (i.e., any sequence $\{x_n\}\subset X$
such that $\{I(x_n)\}$ is bounded and $I'(x_n)\to 0$ in
$X^*$ has a convergent subsequence) and $I(0)=0$. Suppose that
\begin{itemize}
\item[(I1)] there exists two constants $\rho$, $a>0$ such that $I(x)\geq
a $ if $\|x\|=\rho$,

\item[(I2)] for each finite dimensional subspace $X_1\subset X$, the set
$\{x\in X_1; I(x)\geq 0\}$ is bounded.
\end{itemize}
Then $I$ has an unbounded sequence of critical values.
\smallskip

Let $E$ denote the generalized Sobolev space
$W_0^{1,p(x)}(\Omega)$ and let $\lambda >0$ be arbitrary but
fixed.

The energy functional corresponding to problem \eqref{e6} is
defined as $J_{\lambda}:E\to \mathbb{R}$,
\[
J_{\lambda}(u)=\int_{\Omega}\frac{1}{p(x)}|\nabla u|^{p(x)}dx
+\lambda \int_{\Omega}\frac{1}{p(x)}|u|^{p(x)}dx
-\int_{\Omega}\frac{1}{q(x)}|u|^{q(x)}dx.
\]
 A simple calculation
based on relations \eqref{L4} and \eqref{L5} and the compact
embedding of $E$ into $L^{r(x)}(\Omega)$ for all $r\in
C_+(\overline{\Omega})$ with $r(x)<p^*(x)$ on $\overline{\Omega}$
shows that $J_{\lambda}$ is well-defined on $E$ and
$J_{\lambda}\in C^1(E,\mathbb{R})$ with the derivative given by
\[
\langle J'_{\lambda}(u),v \rangle
=\int_{\Omega}|\nabla u|^{p(x)-2}\nabla u \nabla v dx
+\lambda \int_{\Omega}|u|^{p(x)-2}uv\,dx
-\int_{\Omega}|u|^{q(x)-2}uv\,dx
\]
for any $u$, $v\in E$. Thus the
weak solutions of \eqref{e6} are exactly the critical points of
$J_{\lambda}$.

We show now that the Mountain Pass Theorem can be applied in this
case.

\begin{lemma}\label{lem1}
There exist $\eta >0$ and $\alpha >0$ such that
$J_{\lambda}(u)\geq \alpha >0$ for any $u\in E$ with
$\|u\|_{p(x)}=\eta$
\end{lemma}

\begin{proof}
We first point out that we have
\begin{equation}\label{e11}
|u(x)|^{q^-}+|u(x)|^{q^+}\geq |u(x)|^{q(x)}, \quad \forall x\in
\overline{\Omega}
\end{equation}
Using \eqref{e11} we deduce that
\begin{equation}\label{e12}
J_{\lambda}(u)\geq \frac{1}{p^+}\cdot\int_{\Omega}|\nabla
u|^{p(x)}dx-\frac{1}{q^-}\cdot
\Big(\int_{\Omega}|u|^{q^-}dx+\int_{\Omega}|u|^{q^+}dx\Big)
\end{equation}

Since $p^+<q^-\leq q^+<p^{*}(x)$ for any $x\in \overline{\Omega}$
and $E$ is continuously embedded in $L^{q^-}(\Omega)$ and in
$L^{q^+}(\Omega)$, it follows that there exist two positive
constant $C_1$ and $C_2$ such that
\begin{equation}\label{e13}
\|u\|_{p(x)}\geq C_1 \cdot |u|_{q^+}, \quad
\|u\|_{p(x)}\geq C_2\cdot |u|_{q^-}, \quad \forall u\in E.
\end{equation}
Next, we focus our attention on the case when $u\in E$ with
$\|u\|_{p(x)}<1$. For such a $u$ by relation \eqref{L5} we obtain
\begin{equation}\label{e14}
\int_{\Omega}|\nabla u|^{p(x)}dx\geq \|u\|_{p(x)}^{p^+}.
\end{equation}
Relations \eqref{e12}, \eqref{e13} and \eqref{e14} imply
\begin{align*}
J_{\lambda}(u)
&\geq \frac{1}{p^+}\cdot \|u\|_{p(x)}^{p^+}
 -\frac{1}{q^-}\cdot \Big[\Big(\frac{1}{C_1}\cdot \|u\|_{p(x)}\Big)^{q^+}
+\Big( \frac{1}{C_2}\cdot \|u\|_{p(x)}\Big)^{q^-}\Big]\\
&=(\beta-\gamma\cdot \|u\|_{p(x)}^{q^+-p^+}-\delta \cdot
\|u\|_{p(x)}^{q^--p^+})\cdot \|u\|_{p(x)}^{p^+}
\end{align*}
for any $u\in E$ with $\|u\|_{p(x)}<1$, where $\beta$, $\gamma$
and $\delta$ are positive constants.

We remark that the function $g:[0,1]\to \mathbb{R}$ defined by
\[
g(t)=\beta -\gamma \cdot t^{q^+-p^+}-\delta \cdot t^{q^--p^+}
\]
is positive in a neighborhood of the origin. We conclude that
Lemma \ref{lem1} holds.
\end{proof}

\begin{lemma}\label{lem2}
If $E_1\subset E$ is a finite dimensional subspace, the set
$S=\{u\in E_1; J_{\lambda}\geq 0\}$ is bounded in $E$.
\end{lemma}

\begin{proof}
To prove this lemma, we first show that
\begin{equation}\label{e15}
\int_{\Omega}\frac{1}{p(x)}|\nabla u|^{p(x)}dx\leq K_1 \cdot
\Big( \|u\|_{p(x)}^{p^-}+\|u\|_{p(x)}^{p^+}\Big), \quad \forall
u\in E
\end{equation}
where $K_1$ is a positive constant.
Indeed, using relations \eqref{L4} and \eqref{L5} we obtain
\begin{equation}\label{e16}
\int_{\Omega}|\nabla u|^{p(x)}dx \leq |\nabla
u|_{p(x)}^{p^-}+|\nabla
u|_{p(x)}^{p^+}=\|u\|^{p^-}_{p(x)}+\|u\|^{p^+}_{p(x)}, \quad
\forall u\in E.
\end{equation}
On the other hand
\[
\int_{\Omega}\frac{1}{p(x)}|\nabla u|^{p(x)}dx
\leq \frac{1}{p^+}\int_{\Omega}|\nabla u|^{p(x)}dx
\]
and thus \eqref{e15} holds.
 Also, for each $\lambda >0$ there exists a positive
constant $K_2(\lambda)$ such that
\begin{equation}\label{e19}
\lambda \cdot \int_{\Omega}\frac{1}{p(x)}|u|^{p(x)}dx\leq
K_2(\lambda)\cdot\left(\|u\|_{p(x)}^{p^-}+\|u\|_{p(x)}^{p^+}\right),
\quad \forall u\in E.
\end{equation}
By inequalities \eqref{e15} and \eqref{e19}, we
get
\[J_{\lambda}(u)\leq K_1 \cdot \left( \|u\|_{p(x)}^{p-}+\|u\|_{p(x)}^{p^+}\right)+K_2(\lambda)\cdot
\left(\|u\|_{p(x)}^{p^-}+\|u\|_{p(x)}^{p^+}\right)
-\frac{1}{q^+}\int_{\Omega}|u|^{q(x)}dx\]
for all $u\in E$.

Let $u\in E$ be arbitrary but fixed. We define
\[
\Omega_1=\{x\in \Omega; |u(x)|<1\}, \quad \Omega_2
=\Omega\setminus \Omega_1.
\]
Then we have
\begin{align*}
J_{\lambda}(u)
&\leq K_1 \cdot \left( \|u\|_{p(x)}^{p-}+\|u\|_{p(x)}^{p^+}\right)+K_2(\lambda)\cdot
 \left(\|u\|_{p(x)}^{p^-}+\|u\|_{p(x)}^{p^+}\right)
 -\frac{1}{q^+}\int_{\Omega}|u|^{q(x)}dx\\
& \leq K_1 \cdot \left( \|u\|_{p(x)}^{p-}+\|u\|_{p(x)}^{p^+}\right)+K_2(\lambda)\cdot
 \left(\|u\|_{p(x)}^{p^-}+\|u\|_{p(x)}^{p^+}\right)
 -\frac{1}{q^+}\int_{\Omega_2}|u|^{q(x)}dx\\
&\leq K_1 \cdot \left( \|u\|_{p(x)}^{p-}+\|u\|_{p(x)}^{p^+}\right)
  +K_2(\lambda)\cdot \left(\|u\|_{p(x)}^{p^-}+\|u\|_{p(x)}^{p^+}\right)
  -\frac{1}{q^+}\int_{\Omega_2}|u|^{q^-}dx\\
&\leq K_1 \cdot \left( \|u\|_{p(x)}^{p-}+\|u\|_{p(x)}^{p^+}\right)
  +K_2(\lambda)\cdot \left(\|u\|_{p(x)}^{p^-}+\|u\|_{p(x)}^{p^+}\right)\\
&\quad -\frac{1}{q^+}\int_{\Omega}|u|^{q^-}dx+\frac{1}{q^+}
 \int_{\Omega_1}|u|^{q^-}dx.
\end{align*}
But there exists a positive constant $K_3$ such that
\[
\frac{1}{q^+}\int_{\Omega_1}|u|^{q^-}dx\leq K_3,\quad \forall u\in E.
\]
Thus we deduce that
\[
J_{\lambda}(u)\leq K_1 \cdot
\big( \|u\|_{p(x)}^{p-}+\|u\|_{p(x)}^{p^+}\big)
+K_2(\lambda)\cdot  \big(\|u\|_{p(x)}^{p^-}+\|u\|_{p(x)}^{p^+}\big)
-\frac{1}{q^+}\int_{\Omega}|u|^{q^-}dx+K_3,
\]
for all $u\in E$.
The functional $|\cdot|_{q^-}:E\to \mathbb{R}$ defined
by
\[
|u|_{q^-}=\Big( \int_{\Omega}|u|^{q^-}dx\Big)^{1/q^-}
\]
is a norm in $E$. In the finite dimensional subspace $E_1$ the
norms $|\cdot|_{q^-}$ and $\|\cdot\|_{p(x)}$ are equivalent, so
there exists a positive constant $K=K(E_1)$ such that
\[
\|u\|_{p(x)}\leq K\cdot |u|_{q^-}, \quad \forall u\in E_1.
\]
As a consequence we have that there exists a positive constant
$K_4$ such that
\[
J_{\lambda}(u)\leq K_1 \cdot \left( \|u\|_{p(x)}^{p-}
 +\|u\|_{p(x)}^{p^+}\right)+K_2(\lambda)\cdot
 \left(\|u\|_{p(x)}^{p^-}+\|u\|_{p(x)}^{p^+}\right)
-K_4\cdot \|u\|^{q^-}+K_3,
\]
for all $u\in E_1$. Hence
 \[
K_1 \cdot \left(\|u\|_{p(x)}^{p-}+\|u\|_{p(x)}^{p^+}\right)
  +K_2(\lambda)\cdot \left(\|u\|_{p(x)}^{p^-}+\|u\|_{p(x)}^{p^+}\right)
-K_4\cdot \|u\|^{q^-}_{p(x)}+K_3\geq 0,
\]
for all $u\in S$.
 and since $q^->p^+$ we conclude that $S$ is bounded in $E$.
The proof  is complete.
\end{proof}

\begin{lemma}\label{lem3}
If $\{u_n\}\subset E$ is a sequence which satisfies the conditions
\begin{gather}\label{e20}
|J_{\lambda}(u_n)|<M, \\
\label{e21}
J_{\lambda}'(u_n)\to 0 \quad as \quad n\to \infty
\end{gather}
where $M$ is a positive constant, then $\{u_n\}$ possesses a
convergent subsequence.
\end{lemma}

\begin{proof}
First, we show  that $\{u_n\}$ is bounded in $E$. Assume the
contrary. Then, passing if necessary to a subsequence, still
denoted by $\{u_n\}$, we may assume that
$\|u_n\|_{p(x)}\to \infty$ as $n\to \infty$. Thus, we may assume that
$\|u_n\|_{p(x)}>1$ for any integer $n$.

By \eqref{e21} we deduce that there exists $N_1>0$ such that for
any $n>N_1$, we have
\[
\|J_{\lambda}'(u_n)\|\leq 1.
\]
On the other hand, for any $n>N_1$ fixed, the application
\[
E\ni v\to \langle J'_{\lambda}(u_n),v  \rangle
\]
is linear and continuous. The above information implies
\[
|\langle J'_{\lambda}(u_n), v \rangle|
\leq \|J_{\lambda}'(u_n)\|\cdot \|v\|_{p(x)}\leq \|v\|_{p(x)},
\quad \forall v\in E, n>N_1.
\]
Setting $v=u_n$ we have
\[
-\|u_n\|_{p(x)}\leq \int_{\Omega}|\nabla u_n|^{p(x)}dx
+\lambda \int_{\Omega}|u_n|^{p(x)}dx-\int_{\Omega}|u_n|^{q(x)}dx
\leq \|u_n\|_{p(x)}
\]
for all $n>N_1$. We obtain
\begin{equation}\label{e22}
-\|u_n\|_{p(x)}- \int_{\Omega}|\nabla u_n|^{p(x)}dx-\lambda
\int_{\Omega}|u_n|^{p(x)}dx\leq -\int_{\Omega}|u_n|^{q(x)}dx
\end{equation}
for any $n>N_1$.

Provided that $\|u_n\|_{p(x)}>1$ relations \eqref{e20}, \eqref{e22}
and \eqref{L4} imply
\begin{align*}
M&>J_{\lambda}(u_n)\\
&\geq \Big( \frac{1}{p^+}-\frac{1}{q^-}\Big)\cdot
\int_{\Omega}(|\nabla u_n|^{p(x)})dx\\
&\quad +\lambda \cdot \Big( \frac{1}{p^+}-\frac{1}{q^-}\Big)\cdot
  \int_{\Omega}|u_n|^{p(x)}dx-\frac{1}{q^-}\cdot \|u_n\|_{p(x)}\\
&\geq \Big( \frac{1}{p^+}-\frac{1}{q^-}\Big)\cdot
 \int_{\Omega} |\nabla u_n|^{p(x)}dx-\frac{1}{q^-}\cdot \|u_n\|_{p(x)}\\
&\geq \Big( \frac{1}{p^+}-\frac{1}{q^-}\Big)
\cdot \|u_n\|^{p^-}_{p(x)}-\frac{1}{q^-}\cdot \|u_n\|_{p(x)}.
\end{align*}
Letting $n\to \infty$ we obtain a contradiction. It follows that
$\{u_n\}$ is bounded in $E$.

Since $\{u_n\}$ is bounded in $E$ we deduce that there exists a
subsequence, again denoted by $\{u_n\}$, and $u_0\in E$ such that
$\{u_n\}$ converges weakly to $u_0$ in $E$. Using Theorem 1.3 in
\cite{fa}, $E$ is compactly embedded in $L^{p(x)}(\Omega)$ and in
$L^{q(x)}(\Omega)$ it follows that $\{u_n\}$ converges strongly to
$u_0$ in $L^{p(x)}(\Omega)$ and $L^{q(x)}(\Omega)$. The above
information and relation \eqref{e21} imply
\[
\langle J'_{\lambda}(u_n)-J'_{\lambda}(u_0), u_n-u_0\rangle \to 0 \quad
\text{as } n\to \infty.
\]
On the other hand, we have
\begin{align*}
&\int_{\Omega}(|\nabla u_n|^{p(x)-2}\nabla u_n-|\nabla u_0|^{p(x)-2}
 \nabla u_0)\cdot (\nabla u_n-\nabla u_0)dx\\
&=\langle J'_{\lambda}(u_n)-J'_{\lambda}(u_0), u_n-u_0 \rangle
 -\lambda \cdot \int_{\Omega}(|u_n|^{p(x)-2}u_n-|u_0|^{p(x)-2}u_0)
 (u_n-u_0)dx\\
&\quad +\int_{\Omega}(|u_n|^{q(x)-2}u_n-|u_0|^{q(x)-2}u_0)(u_n-u_0)dx.
\end{align*}
Using the fact that $\{u_n\}$ converges strongly to $u_0$ in
$L^{q(x)}(\Omega)$ and inequality \eqref{Hol},
 we have
\begin{align*}
&\big|\int_{\Omega}(|u_n|^{q(x)-2}u_n-|u_0|^{q(x)-2}u_0)(u_n-u_0)dx\big|\\
&\leq\big|\int_{\Omega}|u_n|^{q(x)-2}u_n(u_n-u_0)dx\big|
 +\big|\int_{\Omega}|u_0|^{q(x)-2}u_0(u_n-u_0)dx\big|\\
&\leq C_3\cdot \|u_n|^{q(x)-1}|_{\frac{q(x)}{q(x)-1}}\cdot |u_n-u_0|_{q(x)}
 +C_4\cdot \|u_0|^{q(x)-1}|_{\frac{q(x)}{q(x)-1}}\cdot |u_n-u_0|_{q(x)}
\end{align*}
where $C_3$ and $C_4$ are positive constants. Since
$|u_n-u_0|_{q(x)}\to 0$ as $n\to \infty$ we deduce that
\begin{gather}\label{e24}
\lim_{n\to \infty}\int_{\Omega}(|u_n|^{q(x)-2}u_n-|u_0|^{q(x)-2}u_0)
(u_n-u_0)dx=0,\\
\label{e25}
\lim_{n\to \infty}\int_{\Omega}(|u_n|^{p(x)-2}u_n-|u_0|^{p(x)-2}u_0)
 (u_n-u_0)dx=0.
\end{gather}
By \eqref{e24} and \eqref{e25}, we obtain
\begin{equation}\label{e26}
\lim_{n\to \infty}\int_{\Omega}(|\nabla u_n|^{p(x)-2}\nabla u_n
-|\nabla u_0|^{p(x)-2}\nabla u_0)\cdot (\nabla u_n -\nabla
u_0)dx=0.
\end{equation}
It is known that
\begin{equation}\label{e27}
(|z|^{r-2}z-|t|^{r-2}t)\cdot (z-t)\geq
\big(\frac{1}{2}\big)^r|z-t|^r, \quad \forall r\geq 2,\;
z,t\in \mathbb{R}^N.
\end{equation}
Relations \eqref{e26} and \eqref{e27} yield
\[
\lim_{n\to \infty}\int_{\Omega}|\nabla u_n-\nabla u_0|^{p(x)}dx=0\,.
\]
This fact and relation \eqref{L6} imply
$\|u_n-u_0\|_{p(x)}\to \infty$ as $n\to \infty$.
The proof  is complete.
\end{proof}

\begin{proof}[Completed proof of Theorem \ref{thm1}]
 It is clear that the functional $J_{\lambda}$ is even and verifies
$J_{\lambda}(0)=0$. Lemma \ref{lem3} implies that $J_{\lambda}$
satisfies the Palais-Smale condition. On the other hand,
Lemmas \ref{lem1} and \ref{lem2} show that conditions (I1) and (I2)
are satisfied. The
Mountain Pass Theorem can be applied to the functional
$J_{\lambda}$. We conclude that equation \eqref{e6} has infinitely
many weak solutions in $E$. The proof  is
complete.
\end{proof}

\section{Proof of Theorem \ref{thm2}}

Let $E$ denote the generalized Sobolev space
$W_0^{1,p(x)}(\Omega)$ and let $\lambda >0$ be arbitrary but
fixed.

We start by introducing the energy functional corresponding to
problem \eqref{e6} as $I_{\lambda}:E\to \mathbb{R}$,
\[
I_{\lambda}(u)=\int_{\Omega}\frac{1}{p(x)}|\nabla u|^{p(x)}dx
-\lambda \int_{\Omega}\frac{1}{p(x)}|u|^{p(x)}dx
+\int_{\Omega}\frac{1}{q(x)}|u|^{q(x)}dx.
\]
The same arguments as those used in the case of the functional
$J_{\lambda}$ show that $I_{\lambda}$ is well-defined on $E$ and
$I_{\lambda}\in C^1(E,\mathbb{R})$ with the derivative given by
\[
\langle I'_{\lambda}(u), v\rangle
=\int_{\Omega}|\nabla u|^{p(x)-2}\nabla u \nabla v dx
-\lambda \int_{\Omega}|u|^{p(x)-2}uv\,dx
+\int_{\Omega}|u|^{q(x)-2}uv dx
\]
for any $u$, $v\in E$. We obtain that the weak solutions of
\eqref{e6} are the critical points of $I_{\lambda}$.

This time our idea is to show that $I_{\lambda}$ possesses a
nontrivial global minimum point in $E$. With this end in view we
start by proving two auxiliary results.
\begin{lemma}\label{lem4}
The functional $I_{\lambda}$ is coercive on $E$.
\end{lemma}

\begin{proof}
To prove this lemma, we first show that for any $a$,
$b>0$ and $0<k<l$ the following inequality holds:
\begin{equation}\label{e28}
a\cdot t^k-b\cdot t^l\leq a\cdot
\big(\frac{a}{b}\big)^{k/(l-k)},\quad \forall t\geq 0.
\end{equation}
Indeed, since the function
$[0,+\infty ) \ni t\to t^{\theta}$
is increasing for any $\theta >0$ it follows that
\[
a-b\cdot t^{l-k}<0, \quad \forall t>\big(\frac{a}{b}\big)^{1/(l-k)}
\]
and
\[
t^k \cdot(a-b\cdot t^{l-k})\leq a\cdot t^k<a\cdot
\big(\frac{a}{b}\big)^{k/(l-k)}, \forall t\in
[0, (\frac{a}{b})^{1/(l-k)}].
\]
The above two inequalities show that \eqref{e28} holds.
Using \eqref{e28} we deduce that for any $x\in \Omega$ and $u\in E$,
we have
\begin{align*}
\frac{\lambda}{p^-}|u(x)|^{p(x)}-\frac{1}{q^+}|u(x)|^{q(x)}
&\leq \frac{\lambda}{p^-}
 \big[ \frac{\lambda \cdot q^+}{p^-}\big]^{p(x)/(q(x)-p(x))}\\
&\leq \frac{\lambda}{p^-}\big[\big( \frac{\lambda \cdot q^+}{p^-}
 \big)^{p^+/(q^{-}-p^+)}+\big( \frac{\lambda \cdot q^+}{p^-}
 \big)^{p^-/(q^{+}-p^-)} \big]\\
&=C
\end{align*}
where $C$ is a positive constant independent of $u$ and $x$.
Integrating the above inequality over $\Omega$ we obtain
\begin{equation}\label{e29}
\frac{\lambda}{p^-}\int_{\Omega}|u|^{p(x)}dx
 -\frac{1}{q^+}\int_{\Omega}|u|^{q(x)}dx\leq D
\end{equation}
where $D$ is a positive constant independent of $u$.

Using inequalities \eqref{e11} and \eqref{e29} we obtain that for any
$u\in E$ with $\|u\|_{p(x)}>1$,
\begin{align*}
I_{\lambda}(u)
&\geq \frac{1}{p^+}\int_{\Omega}|\nabla u|^{p(x)}dx
 -\frac{\lambda}{p^-}\int_{\Omega}|u|^{p(x)}dx
 +\frac{1}{q^+}\int_{\Omega}|u|^{q(x)}dx\\
&\geq \frac{1}{p^+}\|u\|^{p^-}_{p(x)}
-\Big( \frac{\lambda}{p^-}\int_{\Omega}|u|^{p(x)}dx
  -\frac{1}{q^+}\int_{\Omega}|u|^{q(x)}dx\Big)\\
&\geq \frac{1}{p^+}\|u\|_{p(x)}^{p^-}-D.
\end{align*}
Thus $I_{\lambda}$ is coercive and the proof of  is
complete.
\end{proof}

\begin{lemma}\label{lem5}
The functional $I_{\lambda}$ is weakly lower semicontinuous.
\end{lemma}

\begin{proof}
First we prove that the functional $A:E\to \mathbb{R}$,
\[
A(u)=\int_{\Omega}\frac{1}{p^(x)}|\nabla u|^{p(x)}dx,
\]
is convex. Indeed, since the function
$[0,\infty)\ni t\to t^s$
is convex for any $s>1$, we deduce that for each $x\in \Omega$
fixed it the inequality
\[
\big|\frac{z+t}{2}\big|^{p(x)}\leq \big| \frac{|z|+|t|}{2}\big|^{p(x)}
\leq \frac{1}{2}|z|^{p(x)}+\frac{1}{2}|t|^{p(x)},\quad
\forall z,\; t \in \mathbb{R}^N
\]
holds. Using the above inequality we deduce that
\[
\big| \frac{\nabla u +\nabla v}{2}\big|^{p(x)}
\leq \frac{1}{2}|\nabla u|^{p(x)}+\frac{1}{2}|\nabla v|^{p(x)},
\quad \forall u,v\in E, \; x\in \Omega .
\]
Multiplying with $1/p(x)$ and integrating over $\Omega$ we obtain
\[
A\big(\frac{u+v}{2}\big)\leq \frac{1}{2}A(u)+\frac{1}{2}A(v),
\quad \forall u,v\in E.
\]
Thus $A$ are convex.

Next, we show that the functional $A$ is weakly lower
semicontinuous on $E$. Taking into account that $A$ is convex, by
\cite[Corollary III.8]{br} it is sufficient to show that $A$ is
strongly lower semicontinuous on $E$. We fix $u\in E$ and $\varepsilon>0$.
Let $v\in E$ be arbitrary. Since $A$ is convex and inequality
\eqref{Hol} holds; we have
\begin{align*}
A(u)&\geq A(u)+\langle A'(u),v-u\rangle\\
&\geq A(u)-\int_{\Omega}|\nabla u|^{p(x)-1}|\nabla (v-u)|dx\\
&\geq A(u)-D_1\cdot \|\nabla u|^{p(x)-1}|_{\frac{p(x)}{p(x)-1}}
 \cdot |\nabla (u-v)|_{p(x)}\\
&\geq A(u)-D_2\cdot \|u-v\|_{p(x)}\\
&\geq A(u)-\varepsilon
\end{align*}
for all $v\in E$ with $\|u-v\|_{p(x)}<\varepsilon /[\|\nabla
u|^{p(x)-1}|_{\frac{p(x)}{p(x)-1}}]$. We have denoted by $D_1$ and
$D_2$ two positive constants. It follows that $A$ is strongly
lower semicontinuous and since it is convex we obtain  that $A$ is
weakly lower semicontinuous.

Finally, we remark that if $\{u_n\}\subset E$ is a sequence which
converges weakly to $u$ in $E$ then $\{u_n\}$ converges strongly
to $u$ in $L^{p(x)}(\Omega)$ and $L^{q(x)}(\Omega)$. Thus,
$I_{\lambda}$ is weakly lower semicontinuous. The proof is complete.
\end{proof}

\begin{proof}[Proof of Theorem \ref{thm2}]
By Lemmas \ref{lem4} and \ref{lem5},
we deduce that $I_{\lambda}$ is coercive and weakly lower
semicontinuous on $E$. Then  \cite[Theorem 1.2]{st} implies that
there exist a global minimizer $u_{\lambda}\in E$ of $I_{\lambda}$
and thus a weak solution of problem \eqref{e7}.

We show that $u_{\lambda}$ is not trivial for $\lambda$ large
enough. Indeed, letting $t_0>1$ be a fixed real and $\Omega_1$ an
open subset of $\Omega$ with $|\Omega_1|>0$ we deduce that there
exists $u_0\in C^{\infty}_0(\Omega)\subset E$ such that
$u_0(x)=t_0$ for any $x\in \overline{\Omega}_1$ and $0\leq
u_0(x)\leq t_0$ in $\Omega \setminus \Omega_1$. We have
\begin{align*}
I_{\lambda}(u_0)
&=\int_{\Omega}\frac{1}{p(x)}|\nabla u_0|^{p(x)}dx
 -\lambda \int_{\Omega}\frac{1}{p(x)}|u_0|^{p(x)}dx
 +\int_{\Omega}\frac{1}{q(x)}|u_0|^{q(x)}dx\\
&\leq L-\frac{\lambda}{p^+}\int_{\Omega_1}|u_0|^{p(x)}dx\\
&\leq L-\frac{\lambda}{p^+}\cdot t_0^{p^-}\cdot |\Omega_1|
\end{align*}
where $L$ is a positive constant. Thus, there exists $\lambda^*>0$
such that $I_{\lambda}(u_0)<0$ for any
$\lambda \in [\lambda^*,\infty)$. It follows that
$I_{\lambda}(u_{\lambda})<0$
for any $\lambda\geq \lambda^*$ and thus $u_{\lambda}$ is a
nontrivial weak solution of problem \eqref{e7} for $\lambda$ large
enough. The proof of  is complete.
\end{proof}

\subsection*{Remark}
After this article was accepted, the author learned that the results here
are a particular case of the results in \cite{mii}.

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