\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{equation}
\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}
\end{equation}
and
\begin{equation}
\begin{gathered}
-\Delta _{p(x) }u=f(x,u) \quad \text{in } \Omega ,\\
u=0\quad \text{on } \partial \Omega ,
\end{gathered} \label{E0}
\end{equation}
where $\Omega $ is a bounded smooth domain of $\mathbb{R}^{N}$,
$p\in C(\overline{\Omega }) $ such that $1<p(x) <N$ for any
$x\in \overline{\Omega }$ and $f$ is a Carath\'eodory function.

The study of differential equations and variational problems involving the
$p(x) $-Laplace operator
$-\Delta _{p(x) }u:=-\operatorname{div}(| \nabla u| ^{p(x)-2}\nabla u)$,
which is a natural generalization of the $p$-Laplace operator,
have attracted a special interest in recent years. A lot of researchers
have devoted their work to
this area (see, e.g., \cite{Antonsev,Chabrowski,Hasto,Mihailescu}) since
there are some physical phenomena which can be modelled by such kind of
equations. In particular, we may mention some applications related to the
study of elastic mechanics and electrorheological fluids
\cite{Acerbi,Diening,Halsey,Ruzicka,Zhikov}. The appearance of such physical
models was facilitated by the development of variable exponent Lebesgue
$L^{p(x) }$ and Sobolev spaces $W^{1,p(x) }$.

Generally,  to show the existence of solutions for Dirichlet
problems which is superlinear, it is essential to assume the following
superquadraticity condition, which is known as Ambrosetti-Rabinowitz's type
condition \cite{Ambrosetti}:
\begin{itemize}
\item[(AR)] There exists $M>0$ 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 $1<p^{-}\leq p^{+}<\infty $, then
$L^{p(x) }(\Omega ) $, $W^{1,p(x)}(\Omega) $ and
$W_0^{1,p(x)}(\Omega ) $ are separable and reflexive Banach spaces.
\end{proposition}

\begin{proposition}[\cite{Fan1,Kovacik}] \label{prop4}
Let $q\in C_{+}(\overline{\Omega })$. If
$q(x) <p^{\ast }(x) $ for all $x\in \overline{\Omega }$,
then the embedding $W^{1,p(x)}(\Omega) \hookrightarrow L^{q(x) }(\Omega ) $
is compact and continuous,
where
\[
p^{\ast }(x) =\begin{cases}
\frac{Np(x) }{N-p(x) } & \text{if }p(x)<N, \\
+\infty & \text{if }p(x) \geq N.
\end{cases}
\]
\end{proposition}

Let $E$ be a Banach space with the norm $\|\cdot \|$ and
$E=\overline{\oplus _{j\in\mathbb{N}}X_{j}}$ with
$\dim X_{j}<\infty $ for any $j\in \mathbb{N}$. Set
\begin{gather*}
Y_k=\oplus _{j=0}^{k}X_{j},\quad 
Z_k=\overline{\oplus _{j=k}^{\infty }X_{j}}\,,
\\
B_k=\{ u\in Y_k:\|u\|\leq \rho _k\},\quad
N_k=\{ u\in Z_k:\|u\|=r_k\}\quad
\text{for }\rho _k>r_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
\begin{equation} \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}
\end{equation}
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) <q(x) <p^{\ast}(x) $ for all $x\in \overline{\Omega }$;

\item[(S2)]  $\liminf_{| t|\to \infty }\frac{f(x,t)t}{| t| ^{\theta }}\geq
c>0$ 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{equation}
\begin{gathered}
-\Delta _{p(x) }u=f(x,u) \quad \text{in }\Omega ,\\
u=0\quad \text{on } \partial \Omega ,
\end{gathered} \label{E}
\end{equation}
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) <q(x) <p^{\ast }(x) $
for $x\in \overline{\Omega }$.

\item[(E2)] $\frac{f(x,t)}{t^{p^{-}-1}}$ is increasing in
$t\in\mathbb{R}$ for $t$ large enough.

\end{itemize}

\begin{theorem} \label{thm9}
Assume that {\rm (S2), (S4), (G1), (E1)-(E2)} hold.
Then problem \eqref{E} has infinitely
many solutions $\{ u_k\}$ satisfying
\[
\Psi (u_k) =\int_{\Omega }\frac{1}{p(x) }| \nabla u_k| ^{p(x)
}dx-\int_{\Omega }F(x,u_k)dx\to \infty \quad \text{as }
k\to \infty ,
\]
where $\Psi :W_0^{1,p(x)}(\Omega ) \to\mathbb{R}$
 is the functional corresponding to problem \eqref{E}.
\end{theorem}

Since the proof of Theorem \ref{thm9} is very similar to the proof of 
Theorem \ref{thm7}, we
only prove Theorem \ref{thm7} and omit the other proof. 

We say that $u\in W^{1,p(x)}(\Omega ) $ is a weak solution of
\eqref{P} if
\[
\int_{\Omega }(| \nabla u| ^{p(x)
-2}\nabla u\nabla \upsilon +| u| ^{p(x)
-2}u\upsilon )dx=\int_{\Omega }f(x,u)\upsilon dx,
\]
for any $\upsilon \in W^{1,p(x)}(\Omega )$.

Let us choose an orthonormal basis $\{ e_{j}\} \subset
W^{1,p(x)}(\Omega ) $ and define $X_{j}:=\mathbb{R}e_{j}$.
Then the spaces $Y_k$ and $Z_k$ can be defined as in  Section 2.
Let us consider the $C^{1}$-functional
$J_{\lambda }:W^{1,p(x)}(\Omega ) \to\mathbb{R}$ defined by
\[
J_{\lambda }(u)
=\int_{\Omega }\frac{1}{p(x)}(| \nabla u| ^{p(x) }+|u| ^{p(x) })dx
-\lambda \int_{\Omega }F(x,u)dx
:=A(u) -\lambda B(u) ,\quad \lambda \in[1,2] .
\]
Then $B(u) \geq 0$, $A(u) \to \infty $ as $\|u\|_{1,p(x) }\to \infty $, and
$J_{\lambda }(-u) =J_{\lambda }(u) $ for all
$\lambda\in $ $[1,2] $, $u\in W^{1,p(x)}(\Omega ) $.

In the view of Theorem \ref{thm5}, we can get the proof of 
Theorem \ref{thm7} by help of the
following two lemmas.

\begin{lemma} \label{lem10} 
Under the assumptions of Theorem \ref{thm7} there exist a sequence $\lambda_n\to 1$, 
as $n\to \infty $, $\overline{c}_k>\overline{b}_k>0$, 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 $c>0$ such 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*}
where $c=\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
\begin{equation}
J_{\lambda _n}(t_nz_{n)}:=\max_{t\in [0,1]}J_{\lambda _n}(tz_n) .  \label{3.1}
\end{equation}
Let $\overline{\omega }_n:=(2p^{+}c)^{\frac{1}{p^{-}}}\omega _n$ with
 $c>0$. Then for $n$ is large enough, we have
\begin{equation}
\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}
\end{equation}
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 $0<t_n<1$, when $n $ is large enough. Hence, we obtain
\begin{equation}
\langle J_{\lambda _n}'(t_nz_n),t_nz_n\rangle
=A'(t_nz_n) -\lambda_nB'(t_nz_n) =0.  \label{e3.3}
\end{equation}
Thus, from \eqref{e3.2} and \eqref{e3.3}, we can write
\[
\lambda _n(\frac{1}{\overline{p}_{t_n}}B'(
t_nz_n) -B(t_nz_n) ) =\frac{1}{\overline{p}
_{t_n}}A'(t_nz_n) -\lambda _nB(
t_nz_n) =J_{\lambda _n}(t_nz_n) \to
\infty
\]
as $n\to \infty $, where $\overline{p}_{t_n}=\frac{A'(t_nz_n) }{A(t_nz_n) }$.

Let $\gamma _{z_n}=\overline{p}_n=\frac{A'(z_n) }{A(z_n) }$, 
$\gamma _{t_nz_n}=\overline{p}_{t_n}$, then 
$\gamma _{z_n},\gamma _{t_nz_n}\in [p^{-},p^{+}] $. Thus, 
$G_{\gamma _{z_n}}$, $G_{\gamma _{t_nz_n}}\in \mathcal{F} $. Using 
condition (G1)  and the fact 
$\inf_n \frac{\overline{p}_{t_n}}{\overline{p}_n\xi }>0$, 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{equation}
\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}
\end{equation}
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
\begin{equation}
\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}
\end{equation}
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
\begin{equation}
\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}
\end{equation}
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}

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\end{document}