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

\AtBeginDocument{{\noindent\small
\emph{Electronic Journal of Differential Equations},
Vol. 2015 (2015), No. 73, pp. 1--19.\newline
ISSN: 1072-6691. URL: http://ejde.math.txstate.edu or http://ejde.math.unt.edu
\newline ftp ejde.math.txstate.edu}
\thanks{\copyright 2015 Texas State University - San Marcos.}
\vspace{9mm}}

\begin{document}
\title[\hfilneg EJDE-2015/73\hfil Existence and multiplicity of solutions]
{Existence and multiplicity of solutions to
operator equations involving duality mappings on Sobolev spaces with variable
exponents}

\author[P. Matei \hfil EJDE-2015/73\hfilneg]
{Pavel Matei}

\address{Pavel Matei \newline
Department of Mathematics and Computer Science\\
Technical University of Civil Engineering\\
124, Lacul Tei Blvd., 020396 Bucharest, Romania}
\email{pavel.matei@gmail.com}

\thanks{Submitted September 18, 2014. Published March 24, 2015.}
\subjclass[2000]{35J60, 35B38, 47J30, 46E30}
\keywords{Mountain Pass Theorem; duality mapping; critical
point; Sobolev space with variable exponent}

\begin{abstract}
 The aim of this article is to study the existence and multiplicity of solutions
 to operator equations involving duality mappings on Sobolev spaces with
 variable exponents. Our main tools are the well known Mountain Pass Theorem and
 its $\mathbb{Z} _2$-symmetric version.
\end{abstract}

\maketitle
\numberwithin{equation}{section}
\newtheorem{theorem}{Theorem}[section]
\newtheorem{lemma}[theorem]{Lemma}
\newtheorem{proposition}[theorem]{Proposition}
\newtheorem{remark}[theorem]{Remark}
\newtheorem{corollary}[theorem]{Corollary}
\allowdisplaybreaks

\section{Introduction}

Our starting point for this article is the references \cite{DJM1,DJM}, where the
existence of the weak solution for Dirichlet's problem with
$p$-Laplacian (when $p$ is a constant  $1<p<\infty $) was obtained using
(among other methods) the Mountain Pass Theorem.
It is well known that the $p$-Laplacian is in fact
the duality mapping on $W_0^{1,p}( \Omega ) $ corresponding to
the gauge function $\varphi ( t) =t^{p-1}$. In \cite{CR} some
results from \cite{DJM} are generalized considering operator equations with
an arbitrary duality mapping on a real reflexive and smooth Banach space,
compactly imbedded in $L^{q}( \Omega ) $, where $1<q<\infty $ and
$\Omega \in \mathbb{R}^{N}$, $N\geq 2$, is a bounded domain with smooth boundary.
In \cite{CGMS} the authors consider more general elliptic equations
than those with $p$-Laplacian and prove the existence of nontrivial
weak solutions of mountain type in an Orlicz-Sobolev space.
Later, by using variational and topological methods, operator equations
involving duality mappings on Orlicz-Sobolev spaces are studied in \cite{DM3}.
In \cite{DM2} the multiplicity of solutions of operator equations involving
duality mappings on a real reflexive and smooth Banach space,
having the Kade\v{c}-Klee property, compactly imbedded in a real Banach
space has studied by using the $\mathbb{Z}_2$-symmetric version of the
Mountain Pass Theorem. Equations of this type in Orlicz-Sobolev spaces
are considered as applications.

In recent years there has been a great interest in the field of
operator equations involving various forms of the
$p( \cdot ) $-Laplacian. The $p( \cdot ) $-Laplacian is the
operator $-\Delta _{p( \cdot ) }:W_0^{1,p( \cdot )
}( \Omega ) \to ( W_0^{1,p( \cdot )}( \Omega ) ) ^{\ast }$,
$\Delta _{p( \cdot ) }u:=\operatorname{div}( | \nabla u| ^{p( \cdot )
-2}\nabla u) $ for $u\in W_0^{1,p( \cdot ) }(\Omega ) $.
Many properties of the classical $p$-Laplacian may be recuperated except
that of being a duality mapping on $W_0^{1,p( \cdot ) }( \Omega ) $.
So, in this article, we will use a natural version of the $p(\cdot ) $-Laplacian
which is appropriate from the standpoint of duality mappings
(see \cite{DM4} or \cite[Section 9.3]{LE}): if $\varphi $ is a gauge function,
the $( \varphi ,p( \cdot ) ) $-Laplacian is
the operator $-\Delta _{( \varphi ,p( \cdot ) )
}:W_0^{1,p( \cdot ) }( \Omega ) \to (
W_0^{1,p( \cdot ) }( \Omega ) ) ^{\ast }$,
$-\Delta _{( \varphi ,p( \cdot ) ) }u:=J_{\varphi }u
\text{ for }u\in W_0^{1,p( \cdot ) }( \Omega )$,
where $J_{\varphi }$ is the duality mapping on $W_0^{1,p( \cdot
) }( \Omega ) $, corresponding to the gauge function $\varphi $.

In particular, if $p( x)$ is constant  and $\varphi ( t)
:=t^{p-1}$, $t\geq 0$, then $\Delta _{( \varphi ,p( \cdot )) }$ coincides
with $\Delta _p$ (see Remark \ref{RR} below).

 The plan of this article is as follows. The main abstract result obtained in
Section \ref{S2} is concerned with the existence of critical points of
functional \eqref{1.1} defined on a real reflexive and smooth Banach space.
The Mountain Pass Theorem and its $\mathbb{Z} _2$-symmetric version (see,
e.g. Rabinowitz \cite{R}) are the basic ingredients which are used.

Section \ref{S3} gathers various definitions and basic properties
related to Lebesgue and Sobolev spaces with variable exponents, needed
through the paper. The standard reference for the basic properties of
variable exponent spaces is \cite{KoR}. Additionally, the reader may also
consult \cite{DHHR,FZ}.  Note that
these spaces occur naturally in connection with various applications
such as the modelling of electrorheological fluids \cite{Ru}.

Let $\Omega$ be a domain in $\mathbb{R} ^{N}$, i.e. a bounded and connected
open subset of $\mathbb{R} ^{N}$ whose boundary $\partial\Omega$ is
Lipschitz-continuous, the set $\Omega$ being locally on the same side of $
\partial\Omega$. Consider the space
\[
U_{\Gamma_0}=\big\{  u\in W^{1,p(\cdot)}(  \Omega) :
u=0\text{ on }\Gamma_0\subset\Gamma=\partial\Omega\big\},
\]
where d$\Gamma-\operatorname{meas}\Gamma_0>0$, with $p(  \cdot)  \in
\mathcal{C}(  \overline{\Omega})  $ and $p(  x)  >1$ for
all $x\in\overline{\Omega}$. For details see \cite[Section 2]{CDM1}.

The main result of this article given in Section \ref{S4} and  concerns
the existence and multiplicity results for operator
equation
\begin{equation}
J_{\varphi}u=N_{g}u, \label{0.1}
\end{equation}
where $J_{\varphi}$ is a duality mapping on $U_{\Gamma_0}$ corresponding to
the gauge function $\varphi$. $N_{g}$ is the Nemytskij operator generated
by a Carath\'{e}odory function $g$ satisfying an appropriate growth condition
ensuring that $N_{g}$ may be viewed as acting from $U_{\Gamma_0}$ into its
dual. In \cite{D}, the author used a topological method to prove the existence
of the weak solution in $W_0^{1,p(\cdot)}(  \Omega)  $ for the
problem $J_{\varphi}u=N_{g}u$. In \cite{CDM2}, the existence of suitable
solutions in $U_{\Gamma_0}$ to equation \eqref{0.1} is proven by three
different methods based, respectively, on reflexivity and smoothness of the
space $U_{\Gamma_0}$, the Schauder fixed point theorem, and the
Leray-Schauder degree.

All vector and function spaces considered in this paper are real. Given a
normed vector space $X$, the notation $X^{\ast}$ denotes its dual space and
$\langle \cdot,\cdot\rangle _{X,X^{\ast}}$ designates the
associated duality pairing. Often, we shall omit  the spaces in
duality and, simply write $\langle \cdot,\cdot\rangle $.
Strong and weak convergence are denoted by $\to$ and $\rightharpoonup$,
respectively.

\section{An abstract result\label{S2}}

The main result of this article is obtained via the following theorem.

\begin{theorem}\label{T2.1}
Let $X$ be a real reflexive and smooth Banach space, compactly
imbedded in the real Banach space $V$ with the compact injection
$X\overset {i}{\hookrightarrow}V$. Let $H\in\mathcal{C}^{1}(X,\mathbb{R} )$ be a
functional given by
\begin{equation}
H(  u)  :=\Psi(  u)  -G(  iu)  ,\quad u\in X, \label{1.1}
\end{equation}
where:

\noindent\emph{(i)} $\Psi:X\to\mathbb{R} $ satisfies:
\begin{itemize}
\item[(i.1)] at any $u\in X$,
\begin{equation}
\Psi(u):=\Phi(\| u\| _X), \label{1.2}
\end{equation}
with
\begin{equation}
\Phi(t):=\int_0^{t} \varphi(\tau)\mathrm{d}\tau\quad
\text{for any }t\geq0, \label{1.10}
\end{equation}
$\varphi:\mathbb{R} _{+}\to\mathbb{R} _{+}$ being a gauge function
which satisfies
\begin{equation}
\varphi^{\ast}:=\sup_{t>0}\frac{t\varphi(t)}{\Phi(t)}<\infty.
\label{1.3}
\end{equation}

\item[(i.2)] $\Psi'=J_{\varphi}$ satisfies condition $(S)_2$ (see \eqref{2.75});
\end{itemize}

\noindent\emph{(ii)} $G:V\to\mathbb{R} $ satisfies:
\begin{itemize}
\item[(ii.0)] $G(0_V)=0$;

\item[(ii.1)] $G\in\mathcal{C}^{1}(V,\mathbb{R} )$;

\item[(ii.2)]  there is a constant $\theta>\varphi^{\ast}$ such
that, for any $u\in V$,
\begin{equation}
\langle G'(u),u\rangle _{V,V^{\ast}}-\theta G(u)\geq
C=\text{const}.; \label{1.9}
\end{equation}
\end{itemize}

\noindent \emph{(iii)} there exists $c_0>0$ such that for any $u\in X$, with
$\| u\| _X<c_0$, one has
\begin{equation}
H(u)>c_1\| u\| _X^p-c_2 \| i(u)\|_V^{q}, \label{1.4}
\end{equation}
where $i$ stands for the compact injection of $X$ in $V$ while $0<p<q$ and
$c_1>0$, $c_2 >0$;

\noindent\emph{(iv)} for any finite dimensional subspace $X_1\subset X$, there exist
real constants $d_0>0$, $d_1$, $d_2 >0$, $d_{3}$, $s>0$ and $r<s$
(generally depending on $X_1$) such that
\begin{equation}
H(u)\leq d_1\| u\| _X^{r}-d_2 \| u\|
_X^{s}+d_{3}, \label{1.5}
\end{equation}
for any $u\in X_1$ with $\| u\| _X>d_0$.

\noindent Then, the functional $H$ possesses a critical value. Moreover, if the
functional $H$ is even, then $H$ has un unbounded sequence of critical values.
\end{theorem}

Before  proving of Theorem \ref{T2.1}, we list some of the
results to be used.

A function $\varphi:\mathbb{R} _{+}\to\mathbb{R} _{+}$ is said to be a
\textit{gauge} function if $\varphi$ is continuous, strictly increasing,
$\varphi(0)=0$ and $\varphi(t)\to\infty$ as $t\to\infty$.

Firstly, we recall that a real Banach space $X$ is said to be \textit{smooth} if
it has the following property: for any $x\in X$, $x\neq0$, there exists a
unique $u^{\ast}(x)\in X^{\ast}$ such that $\langle u^{\ast
}(x),x\rangle =\| x\| _X$ and $\| u^{\ast
}(x)\| _{X^{\ast}}=1$. It is well known (see, for instance, Diestel
\cite{Di}, Zeidler \cite{Ze}) that the smoothness of $X$ is equivalent to the
G\^{a}teaux differentiability of the norm. Consequently, if $(
X,\| \cdot\| _X)  $ is smooth, then, for any $x\in
X$, $x\neq0$, the only element $u^{\ast}(x)\in X^{\ast}$ with the properties
$\langle u^{\ast}(x),x\rangle =\| x\| _X$ and
$\| u^{\ast}(x)\| _{X^{\ast}}=1$ is $u^{\ast}(x)=\|
\cdot\| _X'(x)$ (where $\| \cdot\|
_X'(x)$ denotes the G\^{a}teaux gradient of the $\|\cdot\| _X$-norm at $x$).

Secondly, if $X$ is a real Banach space, the operator $T:X\to X^{\ast }$ is
said to satisfy \textit{condition }$( S) _2 $ if
\begin{equation}
( S) _2: \quad \text{$x_{n}\rightharpoonup x$, and
$Tx_{n}\to Tx$  imply $x_{n}\to x$ as
$n\to \infty$}.  \label{2.75}
\end{equation}

An operator $T$ is said to satisfy \textit{condition }$(S)_{+}$ if 
\[
( S) _{+}: \quad\text{$x_{n}\rightharpoonup x$ and
$\limsup_{n\to \infty }\langle Tx_{n},x_{n}-x\rangle \leq 0$
 imply $x_{n}\to x$  as $n\to \infty$}.
\]

It is known that if $T$ satisfies condition $(  S)  _{+}$, then $T$
satisfies condition $(  S)  _2 $ (see Zeidler \cite[p. 583]{Ze}).

Let $X$ be a real Banach space and let $H\in\mathcal{C}^{1}(X,\mathbb{R} )$ be
a functional. We say that $H$ satisfies the \textit{Palais-Smale condition} on
$X$ ($(  PS)  $-condition, for short) if any sequence $(
u_{n})  \subset X$ with $(  H(u_{n}))  $ bounded and
$H'(u_{n})\to0$ as $n\to\infty$, possesses a
convergent subsequence. By $(PS)$\textit{-sequence} for $H$ we understand a
sequence $(  u_{n})  \subset X$ which satisfies $(
H(u_{n}))  $ is bounded and $H'(u_{n})\to0$ as
$n\to\infty$.

The main tools used in proving Theorem \ref{T2.1} are the well known Mountain
Pass Theorem and its $\mathbb{Z} _2$-symmetric version.

\begin{theorem}[{\cite[Theorem 2.2]{R}}] \label{T2.2}
 Let $X$ be a real Banach space and let
$H$ belong to $\mathcal{C}^{1}(  X,\mathbb{R} ) $ satisfying the
$(PS)$-condition. Suppose that $H(0)=0$ and that the following
conditions hold:
\begin{itemize}
\item[(G1)] There exist $\rho>0$ and $r>0$ such that $H(u)\geq r$
for $\| u\| =\rho$;

\item[(G2)] There exists $e\in X$ with $\| e\|
>\rho$ such that $H(e)\leq0$.
\end{itemize}
Let
\begin{gather}
\Gamma=\{\gamma\in\mathcal{C}([0,1];X):\gamma(0)=0,\gamma(1)=e\},\nonumber\\
c=\inf_{\gamma\in\Gamma}\max_{0\leq t\leq1}H(\gamma(t)).
\label{1.6}
\end{gather}
Then, $H$ possesses a critical value $c>r$.
\end{theorem}

\begin{theorem}[{\cite[Theorem 9.12]{R}}] \label{T2.3}
 Let $X$ be an infinite dimensional real Banach space and let
 $H\in\mathcal{C}^{1}(  X,\mathbb{R})  $ be even, satisfying the $(PS)$-condition,
  and $H(0)=0$. Assume {\rm (G1)} and 
\begin{itemize}
\item[(G2')] for each finite dimensional subspace $X_1$
of $X$ the set $\{  u\in X_1\mid H(u)\geq0\}  $ is bounded.
\end{itemize}
Then $H$ possesses an unbounded sequence of critical values.
\end{theorem}

Now, we  show that under the assumptions of Theorem \ref{T2.1}, the
functional $H$ has a mountain pass geometry. More precisely:

\begin{proposition} \label{P2.1}
Let $X$ be a real Banach space, imbedded in the real Banach space
$V$, with the injection $X\overset{i}{\hookrightarrow}V$.
Let $H\in \mathcal{C}^{1}(  X,\mathbb{R} )  $ be given with $H(0)=0$. Suppose
that $H$ satisfies the hypotheses
{\rm (iii)} and {\rm (iv)} in Theorem \ref{T2.1}.
Then, the functional $H$ satisfies the conditions
{\rm (G1), (G2)}, and {\rm (G2')}
 in Theorems \ref{T2.2} and \ref{T2.3}.
\end{proposition}

\begin{proof}
Indeed, let $C$ be such that$ \| i(u)\| _V\leq C\|
u\| _X$, for any $u\in X$. According to \cite[Theorem 1, p. 422]{DM2},
 from \eqref{1.4}  it follows that  (G1) is satisfied with
\begin{equation}
0<\rho<\min \Big( c_0,\big(  \frac{c_1}{2C^{q}c_2 }\big)^{1/(  q-p)}\Big)  \label{1.7}
\end{equation}
and $r=c_1\rho^p/2$.

Next we show that (G2) is also satisfied. Let $X_1$ be a finite dimensional subspace of $X$ and let
$e_0\in X_1$ with $\| e_0\| _X>d_0$. Since for any
$\lambda>1$, one has $\| \lambda e_0\| _X>d_0$, it
follows from \eqref{1.5} that,
\begin{equation}
H(\lambda e_0)\leq d_1\lambda^{r}\| e_0\| _X
^{r}-d_2 \lambda^{s}\| e_0\| _X^{s}+d_{3}.
\label{1.8}
\end{equation}


Since, in general $s>r$, from \eqref{1.8} we deduce that
$H(\lambda e_0)\to-\infty$ as $\lambda\to\infty$. Consequently, 
there exists a $\lambda_0$ such that, for $\lambda\geq\lambda_0$,
$H(\lambda e_0)<0$. Let $e:=\lambda e_0$ with
$\lambda>\max( 1,\lambda_0 ,\rho/\| e_0\| _X)  $, $\rho$ being given by
\eqref{1.7}. Clearly with such a choice one has $\| e\|_X>\rho$ and
 $H(  e)  <0$.

Finally, according to \cite[Theorem 1, p. 422]{DM2},  from
\eqref{1.5} it follows that (G2') is fulfilled.
The proof is complete.
\end{proof}

To prove that the functional $H$ satisfies the $(PS)$-condition,
the following result will be useful.

\begin{proposition}[{\cite[Corollary 1]{DM1}}] \label{P2.4}
Let $X$ be a real reflexive Banach space, compactly imbedded in the real Banach
space $V$ and $H\in\mathcal{C}^{1}(X,\mathbb{R} )$ be such that
\[
H'(u)=Su-Nu,
\]
where $S:X\to X^{\ast}$ is monotone, hemicontinuous, satisfies
condition $(  S)  _2 $ and $N:V\to V^{\ast}$ is
demicontinuous. Assume that any Palais-Smale sequence for $H$ is bounded. Then
$H$ satisfies the $(  PS)  $-condition.
\end{proposition}

To apply Proposition \ref{P2.4}, we recall that, if $X$ is a real
smooth Banach space and $\varphi:\mathbb{R} _{+}\to\mathbb{R} _{+}$ is
a gauge function, the duality mapping on $X$ corresponding to $\varphi$ is the
mapping $J_{\varphi}:X\to X^{\ast}$ defined by
\[
J_{\varphi}0:=0,\quad J_{\varphi}x:=\varphi(  \| x\|
_X)  \| \cdot\| _X'(x),\quad \text{if } x\neq0.
\]


The following result is standard in the theory of monotone operators (see,
e.g. Browder \cite{Br}, Zeidler \cite{Ze}).

\begin{proposition}\label{P2.5}
Let $X$ be a real reflexive and smooth Banach space. Then, any
duality mapping $J_{\varphi}:X\to X^{\ast}$ is:
\begin{itemize}
\item[(a)] monotone ($\langle J_{\varphi}u-J_{\varphi}v,u-v\rangle
\geq0$, $u,v\in X$);

\item[(b)] demicontinuous ($x_{n}\to x\Rightarrow J_{\varphi}
x_{n}\rightharpoonup J_{\varphi}x$).
\end{itemize}
\end{proposition}

Since, generally, demicontinuity implies hemicontinuity, it follows that any
duality mapping $J_{\varphi}:X\to X^{\ast}$ is hemicontinuous
($\langle J_{\varphi}(u+\lambda v),w\rangle \to\langle
J_{\varphi}u,w\rangle $ as $\lambda\searrow0$ for all $u,v,w\in X$).
Consequently, from Proposition \ref{P2.4}, we obtain the following result.

\begin{corollary}\label{C1.1}
Let $X$ be a real reflexive Banach space, compactly imbedded in
the real Banach space $V$ and $H\in\mathcal{C}^{1}(X,\mathbb{R} )$ such that
\[
H'(u)=J_{\varphi}u-Nu,
\]
where $J_{\varphi}\ $is a duality mapping corresponding to the gauge function
$\varphi$, satisfying condition $(  S)  _2 $ and
$N:V\to V^{\ast}$ is demicontinuous. Assume that any Palais-Smale
sequence for $H$ is bounded. Then $H$ satisfies the $(  PS)  $-condition.
\end{corollary}

Taking into account \cite[Corollary 2, p. 897]{DM1}, we obtain

\begin{corollary} \label{C2.2}
Let $X$ be a real reflexive and smooth Banach space, compactly
imbedded in the real Banach space $V$ with the compact injection $X\overset
{i}{\hookrightarrow}V$. Let $H\in\mathcal{C}^{1}(X,\mathbb{R} )$ be a
functional given by
\[
H(  u)  =\Psi(  u)  -G( iu) ,\quad u\in X,
\]
where:
\begin{itemize}
\item[(i.1)] at any $u\in X$, $\Psi(u)=\Phi(\| u\|_X)$ with
$\Phi$ given by \eqref{1.10}, where $\varphi:\mathbb{R}_{+}\to\mathbb{R} _{+}$
 is a gauge function which satisfies
\eqref{1.3};

\item[(i.2)]  $\Psi'$ satisfies condition \emph{(S)}$_2 $;

\item[(ii)] $G:V\to\mathbb{R} $ is $\mathcal{C}^{1}$ \ on $V$ and
satisfies: there is a constant $\theta>\varphi^{\ast}$ such that, at any
$u\in V$,
\[
\langle G'(u),u\rangle _{V,V^{\ast}}-\theta G(u)\geq
C=\text{const.};
\]
\end{itemize}
Then, the functional $H$ satisfies the $(  PS)  $-condition.
\end{corollary}

\begin{proof}
The hypotheses of Corollary \ref{C1.1} are fulfilled with $N=G'$.
Indeed, by Asplund's Theorem \cite{As}, $\Psi'=J_{\varphi
}$ and, by hypothesis (i.2) $J_{\varphi}$ satisfies condition $(
S)  _2 $. The demicontinuity of $G'$ is assumed by (ii.2).
According to \cite[Corollary 2, p. 897]{DM1} we obtain that any
$(  PS)  $ sequence for $H$ is bounded.
\end{proof}

\begin{proof}[Proof of Theorem \ref{T2.1}]
The assumptions of Theorem \ref{T2.1} entail the fulfillment of
those of Corollary \ref{C2.2}, therefore
the functional $H$ satisfies the $(PS)$-condition. According to
Proposition \ref{P2.1}, the functional $H$ satisfies the conditions
(G1), (G2), and (G2') from Theorems \ref{T2.2} and
\ref{T2.3}. Applying these theorems, the conclusions of Theorem \ref{T2.1} follow.
\end{proof}

\section{Lebesgue and Sobolev spaces with variable exponent\label{S3}}


The Lebesgue measure in $\mathbb{R} ^{N}$ is denoted d$x$. No distinction
will be made between d$x$-measurable functions and their equivalence classes
modulo the relation of d$x$-almost everywhere equality. The notation
$\mathcal{D}(  \Omega)  $ denotes the space of functions that are
infinitely differentiable in $\Omega$ and whose support is a compact subset
of $\Omega$.

The usual Lebesgue and Sobolev spaces, i.e., \emph{with constant exponent}
$p\geq1$, are denoted $L^p(\Omega)$ and $W^{1,p}(\Omega)$.

Given a function $p(  \cdot)  \in L^{\infty}(  \Omega)
$ that satisfies
\[
1\leq p^{-}:=\operatorname{ess\,inf}_{x\in\Omega}p(  x)  \leq
p^{+}:=\operatorname{ess\,sup}_{x\in\Omega}p(  x)  ,
\]
the Lebesgue space $L^{p(\cdot)}(  \Omega)  $ with variable
exponent $p(  \cdot)  $ is defined as
\[
L^{p(\cdot)}(  \Omega)  :=\{v:\Omega\to\mathbb{R};
v\text{ is d}x\text{-measurable and }\rho_{0,p(\cdot)}(v)
:=\int_{\Omega}| v(x)| ^{p(x)}\mathrm{d}x<\infty\},
\]
where $\rho_{0,p(\cdot)}(v)$ is called the \textit{convex modular} of $v$.

\begin{theorem} \label{thm4}
Let $\Omega$ be a domain in $\mathbb{R} ^{N}$.

\noindent\emph{(a)} Let $p(  \cdot)  \in L^{\infty}(  \Omega)  $
be such that $p^{-}\geq1$. Equipped with the norm
\[
v\in L^{p(\cdot)}(  \Omega)  \to\| v\|
_{0,p(\cdot)}:=\inf\{\lambda>0;\text{ }\int_{\Omega}|
\frac{v(x)}{\lambda}| ^{p(x)}\mathrm{d}x\leq1\},
\]
the space $L^{p(\cdot)}(  \Omega)  $ is a separable Banach space.
If $p^{-}>1$, the space $L^{p(\cdot)}(  \Omega)  $ is uniformly
convex, hence reflexive.

\noindent\emph{(b)} Let $p_1(  \cdot)  \in L^{\infty}(
\Omega)  $ and $p_2 (  \cdot)  \in L^{\infty}(
\Omega)  $ be such that $p_1^{-}\geq1$ and $p_2 ^{-}\geq1$. Then
\[
L^{p_2 (\cdot)}(  \Omega)  \hookrightarrow L^{p_1(\cdot)}(
\Omega)
\]
if and only if
\[
p_1(x)\leq p_2 (x)\quad \text{for almost all }x\in\Omega.
\]


\noindent\emph{(c)} For any $u\in L^{p(\cdot)}(  \Omega)  $ with
$p(\cdot)  \in L^{\infty}(  \Omega)  $ satisfying $p^{-}>1$ and
$v\in L^{p'(\cdot)}(  \Omega)  $,
\begin{equation}
\int_{\Omega}| u(x)v(x)| dx\leq\Big(  \frac
{1}{p^{-}}+\frac{1}{(  p')  ^{-}}\Big)  \|
u\| _{0,p(\cdot)}\| v\| _{0,p'(\cdot)}\,. \label{3.1}
\end{equation}
\end{theorem}

\begin{remark}[{\cite[p. 430]{FZ}}] \label{R2}
If $p(x)$ is constant, then the
space $L^{p(\cdot )}( \Omega ) $ coincides with the
classical Lebesgue space $L^p( \Omega ) $ and the norms
on these spaces are equal.
\end{remark}

The next theorem sums up the relations between the norm
$\|\cdot\| _{0,p(\cdot)}$ and the convex modular $\rho_{0,p(\cdot)}$.
Its proof can be found in \cite{FZ}.

\begin{theorem}\label{T3.2}
Let $p(  \cdot)  \in L^{\infty}(  \Omega)$ be such that
 $p^{-}\geq1$ and let $u\in L^{p(\cdot)}(  \Omega)$.
The following properties hold:
\begin{itemize}
\item[(a)] If $u\neq0$, then $\| u\| _{0,p(\cdot)}=a$ if and
only if $\rho_{0,p(\cdot)}(  a^{-1}u)  =1$.

\item[(b)] $\| u\| _{0,p(\cdot)}<1$ (resp. $=1$ or $>1$)
 if and only if $\rho_{0,p(\cdot)}(u)<1$ (resp. $=1$, or $>1$).

\item[(c)] $\| u\| _{0,p(\cdot)}>1$ implies
$\|u\| _{0,p(\cdot)}^{p^{-}}\leq\rho_{0,p(\cdot)}(u)
\leq\|u\| _{0,p(\cdot)}^{P^{+}}$.

\item[(d)] $\| u\| _{0,p(\cdot)}<1$ implies
$\|u\| _{0,p(\cdot)}^{p^{+}}\leq\rho_{0,p(\cdot)}(u)\leq\|u\| _{0,p(\cdot)}^{p^{-}}$.
\end{itemize}
\end{theorem}

The Sobolev space $W^{1,p(  \cdot)  }(  \Omega)  $
with variable exponent $p(  \cdot)  $ is defined as
\[
W^{1,p(  \cdot)  }(  \Omega)  :=\{  v\in
L^{p(\cdot)}(  \Omega)  : \partial_{i}v\in L^{p(\cdot
)}(  \Omega)  ,1\leq i\leq N\}  ,
\]
where, for each $1\leq i\leq N$, $\partial_{i}$ denotes the distributional
derivative operator with respect to the\ $i$-th variable.

\begin{theorem}\label{T3.3}
Let $\Omega$ be a domain in $\mathbb{R} ^{N}$.

\noindent\emph{(a)} Let $p(  \cdot)  \in L^{\infty}(  \Omega)  $
be such that $p^{-}\geq1$. Equipped with the norm
\[
v\in W^{1,p(  \cdot)  }(  \Omega)  \to\|
v\| _{1,p(\cdot)}:=\| v\| _{0,p(\cdot)}
+{\textstyle\sum_{i=1}^{N}} \| \partial_{i}v\|
_{0,p(\cdot)},
\]
the space $W^{1,p(  \cdot)  }(  \Omega)  $ is a
separable Banach space. If $p^{-}>1$, the space $W^{1,p(  \cdot)
}(  \Omega)  $ is reflexive.

\noindent\emph{(b)} Let $p_1(  \cdot)  \in L^{\infty}(
\Omega)  $ with $p_1^{-}\geq1$ and $p_2 (  \cdot)  \in
L^{\infty}(  \Omega)  $ with $p_2 ^{-}\geq1$ be such that
\[
p_1(x)\leq p_2 (x)\text{ for almost all }x\in\Omega.
\]
Then
\[
W^{1,p_2 (  \cdot)  }(  \Omega)  \hookrightarrow
W^{1,p_1(  \cdot)  }(  \Omega)  .
\]

\noindent\emph{(c)} Let $p(  \cdot)  \in\mathcal{C}(  \overline{\Omega
})  $ be such that $p^{-}\geq1$. Given any $x\in\overline{\Omega}$, let
\begin{equation}
p^{\ast}(x):=\frac{Np(x)}{N-p(x)}\text{ if }p(x)<N,\quad\text{and}\quad
p^{\ast}(x):=\infty\text{\ if }p(x)\geq N, \label{3.3}
\end{equation}
and let $q(  \cdot)  \in\mathcal{C}(  \overline{\Omega})  $ be a function
that satisfies
\begin{equation}
1\leq q(  x)  <p^{\ast}(  x)  \quad \text{for each }
x\in\overline{\Omega}. \label{3.4}
\end{equation}
Then the following compact injection holds:
\[
W^{1,p(\cdot)}(  \Omega)  \Subset L^{q(\cdot)}(
\Omega)  ,
\]
so that, in particular,
$W^{1,p(\cdot)}(  \Omega)  \Subset L^{p(\cdot)}(\Omega)$.



\noindent\emph{(d)} The function defined by
\[
v\in W^{1,p(\cdot)}(  \Omega)  \to\| v\|
_{1,p(\cdot),\nabla}:=\| v\| _{0,p(\cdot)}+\|
| \nabla v| \| _{0,p(\cdot)},
\]
is a norm on $W^{1,p(\cdot)}(  \Omega)  $, equivalent with the norm
$\| \cdot\| _{1,p(\cdot)}$.
\end{theorem}

The following theorem  concerns  the definition of the space
$U_{\Gamma_0}$ (\cite[Theorem 6]{CDM1}).

\begin{theorem}\label{T3.4}
Let $\Omega$ be a domain in $\mathbb{R} ^{N}$, $N\geq2$, let
$\Gamma_0$ be a d$\Gamma$-measurable subset of $\Gamma=\partial\Omega$ that
satisfies d$\Gamma-\operatorname{meas}$ $\Gamma_0>0$, let $p(  \cdot)
\in\mathcal{C}(  \overline{\Omega})  $ be such that $p(x)  >1$ for all
$x\in\overline{\Omega}$ and let
\[
U_{\Gamma_0}:=\{  u\in(  W^{1,p(  \cdot)  }(
\Omega)  ,\| \cdot\| _{1,p(\cdot),\nabla}):
\operatorname{ tr }u=0\text{ on }\Gamma_0\}  .
\]
Then:

\noindent\emph{(a)} The space $U_{\Gamma_0}$ is closed in $(  W^{1,p(
\cdot)  }(  \Omega)  ,\| \cdot\|
_{1,p(\cdot),\nabla})  $; hence $(  U_{\Gamma_0},\|
\cdot\| _{1,p(\cdot),\nabla})  $ is a separable reflexive
Banach space.

\noindent\emph{(b)} The map
\begin{equation}
u\in U_{\Gamma_0}\to\| u\| _{0,p(\cdot),\nabla
}:=\| | \nabla u| \| _{0,p(
\cdot)  } \label{3.2}
\end{equation}
is a norm on $U_{\Gamma_0}$ equivalent with the norm $\|
\cdot\| _{1,p(\cdot),\nabla}$.

\noindent\emph{(c)} The norm $\| u\| _{0,p(  \cdot)
,\nabla}$ is Fr\'{e}chet-differentiable at any nonzero $u\in U_{\Gamma_0}$
and the Fr\'{e}chet-differential of this norm at any nonzero $u\in
U_{\Gamma_0}$ is given for any $h\in U_{\Gamma_0}$ by
\[
\langle \| \cdot\| _{0,p(\cdot),\nabla}'(u),h\rangle
=\frac{\int_{\Omega\backslash\Omega_{0,u}
}p(x)\frac{| \nabla u(  x)  | ^{p(x)-2}\text{
}\langle \nabla u(  x)  ,\nabla h(  x)
\rangle }{\| u\| _{0,p(\cdot),\nabla}^{p(x)-1}}
\mathrm{d}x}{\int_{\Omega}p(x)\frac{| \nabla u(
x)  | ^{p(x)}}{\| u\| _{0,p(\cdot),\nabla
}^{p(x)}}\mathrm{d}x},
\]
where $\Omega_{0,u}:=\{  x\in\Omega;| \nabla u(  x)| =0\}  $.
\end{theorem}

By Theorem \ref{T3.3} (c) and Theorem \ref{T3.4} (a)--(b) we derive the following
result.

\begin{lemma}\label{L1}
Let $p(  \cdot)  \in\mathcal{C}(  \overline{\Omega
})  $ be such that $p^{-}\geq1$. Given any $x\in\overline{\Omega}$, let
$p^{\ast}$ be given by \eqref{3.3} and let $q(  \cdot)
\in\mathcal{C}(  \overline{\Omega})  $ be a function that satisfies
\eqref{3.4}. Then the following compact inclusion holds:
\[
\big(  U_{\Gamma_0},\| \cdot\| _{1,p(\cdot),\nabla
}\big)  \Subset \big(  L^{q(\cdot)}(  \Omega)  ,\|
\cdot\| _{0,q(  \cdot)  }\big)  .
\]
\end{lemma}

\begin{remark} \label{R1}\rm
If $\varphi ^{\ast }<q^{-}$, then
$L^{q(\cdot )}( \Omega ) \hookrightarrow L^{\varphi ^{\ast }}( \Omega
) $, therefore $U_{\Gamma _0}$ is compactly imbedded
in $L^{\varphi ^{\ast }}( \Omega ) $.
\end{remark}

The above remark will be useful in the upcoming section.

\begin{proposition}[{\cite[Proposition 4]{DJ}}] \label{P4.1}
Let $X$ be a real reflexive Banach space, compactly embedded in
the real Banach space $Z$. Denote by $i$ the compact injection of $X$ into $Z
$ and, for any $r\in\lbrack1,\infty)$, define
\[
\lambda_{1,r}=\inf\{  \frac{\| u\| _X^{r}}{\|
i(u)\| _{Z}^{r}}\mid u\in X\backslash\{0_X\}\}  .
\]
Then, $\lambda_{1,r}$ is attained and $\lambda_{1,r}^{-1/r}$ is the best
constant $c_{Z}$ in the writing of the imbedding of $X$ into $Z$:
\[
\| i(u)\| _{Z}\leq c_{Z}\| u\| _X,\quad\text{for all }u\in X.
\]
\end{proposition}

Taking into account Remark \ref{R1}, we obtain the following result.

\begin{corollary} \label{coro3}
Let $\Omega$ be a domain in $\mathbb{R} ^{N}$ $(N\geq2)$, let
$p\in\mathcal{C}(  \overline{\Omega})  $ and
$q\in\mathcal{C} (  \overline{\Omega})  $ be two functions such that $p^{-}>1$,
$q^{-}>1$ and \eqref{3.4} holds. For $\varphi^{\ast}<q^{-}$ define
\begin{equation}
\lambda_{1,\varphi^{\ast}}:=\inf\big\{  \frac{\| u\|
_{0,p(  \cdot)  ,\nabla}^{\varphi^{\ast}}}{\| u\|
_{L^{\varphi^{\ast}}(  \Omega)  }^{\varphi^{\ast}}}:u\in
U_{\Gamma_0}\backslash\{  0\}  \big\}  , \label{4.9}
\end{equation}
where $i$ is the compact injection $i:U_{\Gamma_0}\to L^{\varphi
^{\ast}}(  \Omega)  $. Then $\lambda_{1,\varphi^{\ast}}$ is
attained and $\lambda_{1,\varphi^{\ast}}^{-1/\varphi^{\ast}}$ is the best
constant $c$ in the imbedding of $U_{\Gamma_0}$ in $L^{\varphi^{\ast}
}(  \Omega)  $, namely,
\[
\| i(  u)  \| _{L^{\varphi^{\ast}}(
\Omega)  }\leq c\| u\| _{0,p(  \cdot),\nabla}\quad \text{for all }u\in U_{\Gamma_0}.
\]
\end{corollary}

\section{Main result} \label{S4}

In this section we study the existence and multiplicity of weak solutions for
the boundary value problem
\begin{gather}
J_{\varphi}u=g(x,u)\quad \text{in }\Omega, \label{4.1} \\
u=0\quad \text{on }\Gamma_0\subset\partial\Omega, \label{4.2}
\end{gather}
in the following framework:

\noindent$\bullet$ $J_{\varphi}:\big(  U_{\Gamma_0},\| \cdot\|
_{0,p(  \cdot)  ,\nabla}\big)  \to\big(  U_{\Gamma_0
},\| \cdot\| _{0,p(  \cdot)  ,\nabla}\big)^{\ast}$ is the duality
 mapping on \\
$(  U_{\Gamma_0},\|\cdot\| _{0,p(  \cdot)  ,\nabla})  $ subordinated to
the gauge function $\varphi$: such that
$J_{\varphi }0=0$, and
\[
\langle J_{\varphi }u,h\rangle
=\varphi (\| u\|_{0,p( \cdot ) ,\nabla }) \frac{\int_{\Omega
\backslash \Omega _{0,u}}p(x)\frac{| \nabla u( x)
| ^{p(x)-2}\langle \nabla u( x) ,\nabla
h( x) \rangle }{\| u\| _{0,p(\cdot
),\nabla }^{p(x)-1}}\mathrm{d}x}{\int_{\Omega }p(x)\frac{|
\nabla u( x) | ^{p(x)}}{\| u\|
_{0,p(\cdot ),\nabla }^{p(x)}}\mathrm{d}x},
\]
at any nonzero $u\in U_{\Gamma _0}$, for any $h\in U_{\Gamma _0}$
(here $\Omega _{0,u}:=\{ x\in \Omega : | \nabla u( x)| =0\} $).

\noindent $\bullet $ $g:\Omega \times \mathbb{R}\to \mathbb{R}$ is a
 Carath\'{e}odory function.

\begin{remark}\label{RR} \rm
By Remark \ref{R2}, if $p( x)$ is constant on $\Omega $, then 
$\| u\| _{0,p(\cdot ) }=\| u\| _{L^p( \Omega ) }$, and
\[
\int_{\Omega }p(x)\frac{| \nabla u( x)
| ^{p(x)}}{\| u\| _{0,p(\cdot ),\nabla }^{p(x)}}\mathrm{d}x=p;
\]
therefore,
\[
\langle J_{\varphi }u,h\rangle =\varphi (\| u\|
_{0,p( \cdot ) ,\nabla })\frac{\int_{\Omega \backslash
\Omega _{0,u}}| \nabla u( x) | ^{p-2}\text{ }
\langle \nabla u( x) ,\nabla h( x) \rangle
\mathrm{d}x}{\| u\| _{L^p( \Omega )}^{p-1}}.
\]
Moreover, if $\varphi ( t) =t^{p-1}$, $t\geq 0$, we obtain that
\[
\langle J_{\varphi }u,h\rangle
=\int_{\Omega \backslash \Omega _{0,u}}| \nabla u( x) | ^{p-2}
\langle \nabla u( x) ,\nabla h( x) \rangle \mathrm{d}x;
\]
that is,
\[
\langle J_{\varphi }u,h\rangle =\langle -\Delta
_pu,h\rangle .
\]
Consequently, in this case equation \eqref{4.1} can be rewritten as
\[
-\Delta _pu=g(x,u)\quad \text{in }\Omega .
\]
\end{remark}

By a (weak) solution to the problem \eqref{4.1}, \eqref{4.2} we understand a
solution to the equation
\begin{equation}
J_{\varphi}u=N_{g}u, \label{4.7}
\end{equation}
$N_{g}$ being the Nemytskij operator generated by $g$.

Our goal is to prove the main result of this paper.

\begin{theorem} \label{T4.1}
Let $\Omega$ be a domain in $\mathbb{R} ^{N}$
$(N\geq 2)$, let $p\in\mathcal{C}(  \overline{\Omega})  $ be a
function such that $p^{-}>1$, and let $p^{\ast}(\cdot)$ be given by
\eqref{3.3}. Let $\varphi:\mathbb{R} _{+}\to\mathbb{R} _{+}
$ be a gauge function which satisfies \eqref{1.3}, where $\Phi$ is
given by \eqref{1.10}. Let there be given a Carath\'{e}odory function
$g:\Omega\times\mathbb{R} \to\mathbb{R} $ satisfying the hypotheses:
\begin{itemize}
\item[(H1)] there exists a function $q(  \cdot)
\in\mathcal{C}(  \overline{\Omega})  $ that satisfies
\eqref{3.4} such that
\begin{equation}
| g(  x,s)  | \leq C_1| s|
^{q(  x)  /q'(  x)  }+a(  x)  \quad\text{
for almost all $x\in\Omega$  and all $s\in\mathbb{R}$} , \label{4.4}
\end{equation}
where $\frac{1}{q(x)}+\frac{1}{q'(x)}=1$, a is a bounded function,
$a(  x)  \geq0$ for almost
all $x\in\Omega$, and $C_1$ is a constant, $C_1>0$;

\item[(H2)] there exist $s_0>0$ and $\theta>\varphi^{\ast}
:=\sup_{t>0}\frac{t\varphi(t)}{\Phi(t)}$ such that
\begin{equation}
0<\theta G(x,s)\leq sg(x,s), \label{4.5}
\end{equation}
for almost every $x\in\Omega$ and all $s$ with $| s| \geq
s_0$, where
\begin{equation}
G(  x,s)  :={\textstyle\int_0^{s}} g(
x,\tau)  \mathrm{d}\tau. \label{4.6}
\end{equation}
\end{itemize}
Also assume that
\begin{itemize}
\item[(H3)]
\begin{equation}
\limsup_{s\to0}\frac{g(  x,s)  }{|
s| ^{\varphi^{\ast}-2}s}<\frac{\varphi^{\ast}\Phi(  1)
}{2}\lambda_{1,\varphi^{\ast}} \label{4.17}
\end{equation}
uniformly with respect to almost all $x\in\Omega$, where $\lambda
_{1,\varphi^{\ast}}$ is given by \eqref{4.9}.

\item[(H4)]  $\varphi^{\ast}<q^{-}$.
\end{itemize}
Let
$N_{g}:L^{q(  \cdot)  }(  \Omega)  \to L^{q'(  \cdot)  }(  \Omega)$,
with $(N_{g}u)  (  x)  =g(  x,u(  x)  )$  for almost all $x\in\Omega$,
denote the Nemytskij operator generated by $g$.

Then under these assumptions, problem \eqref{4.1}, \eqref{4.2} has a
weak non-trivial solution in the space $U_{\Gamma_0}$ (endowed with the norm
\eqref{3.2}). Moreover, if $g$ is odd in the second argument:
$g(  x,-s)  =-g(x,s)$, $s\in\mathbb{R} $, then the problem
\eqref{4.1}, \eqref{4.2} has a sequence of weak solutions.
\end{theorem}

To prove this theorem, we apply Theorem \ref{T2.1} to the functional
$H:U_{\Gamma_0}\to\mathbb{R} $,
\begin{equation}
H(  u)  :=\Phi(  \| u\| _{0,p(
\cdot)  ,\nabla})  -\mathcal{G}(  u)  ,
\label{4.10}
\end{equation}
where
\begin{equation}
\mathcal{G}(  u)  :=\int_{\Omega} G(
x,u(  x)  )  \mathrm{d}x. \label{4.8}
\end{equation}


\begin{proposition}\label{P4.2}
Under the hypotheses of Theorem \ref{T4.1}, the functional
$H$ given by \eqref{4.10}, is well-defined and $\mathcal{C}^{1}$ on
$U_{\Gamma_0}$, with
\[
H'(  u)  =J_{\varphi}(  u)  -g(
x,u)
\]
\end{proposition}

\begin{proof}
The well-definedness of functional $H$ is  reduced to proving that
for any $u\in U_{\Gamma_0}$, $\int_{\Omega} G(x,u(x))\mathrm{d}x$ makes sense.
Indeed, by using \eqref{4.4} it follows that 
\begin{equation}
| G(  x,s)  | \leq\frac{C_1}{q^{-}}|
s| ^{q(  x)  }+a(  x)  | s|. \label{4.29}
\end{equation}
Thus
\[
\int_{\Omega} G(  x,u(x))  \mathrm{d}x\leq
\frac{C_1}{q^{-}}\int_{\Omega} |
u(x)| ^{q(  x)  }\mathrm{d}x+\int_{\Omega} a(  x)  | u(x)|
\mathrm{d}x.
\]
Since, for any $u\in U_{\Gamma_0}$, we have $u\in L^{q(  \cdot)  }
(\Omega) $ and $a\in L^{q'(  \cdot)  }(\Omega)$, it follows
that $\int_{\Omega} a(  x)  |u(x)| \mathrm{d}x$ makes sense.
Consequently $\int _{\Omega} G(  x,u(x))  \mathrm{d}x<\infty$.

Now, we  show that $H\in\mathcal{C}^{1}$ over $U_{\Gamma_0}$. First, we
will prove that $\Psi:U_{\Gamma_0}\to\mathbb{R} $,
$\Psi( u)  :=\Phi(  \| u\| _{1,p(  \cdot),\nabla})  $, is
$\mathcal{C}^{1}$ over $U_{\Gamma_0}$. Indeed,
according to \cite[Theorem 6]{CDM1}, $\Psi$ is continuously Fr\'{e}chet
differentiable at any nonzero $u\in U_{\Gamma_0}$ and, for any $h\in
U_{\Gamma_0}$ one has
\[
\langle \Psi'(u),h\rangle =\varphi(\| u\|_{0,p(  \cdot)  ,\nabla})
\frac{\int_{\Omega\backslash\Omega_{0,u}}p(x)\frac{| \nabla u(  x)  |
^{p(x)-2} \langle \nabla u(  x)  ,\nabla h(
x)  \rangle }{\| u\| _{0,p(  \cdot)
,\nabla}^{p(x)-1}}\mathrm{d}x}{\int_{\Omega}p(x)\frac{|
\nabla u(  x)  | ^{p(x)}}{\| u\|
_{0,p(  \cdot)  ,\nabla}^{p(x)}}\mathrm{d}x},
\]
where $\Omega_{0,u}:=\{  x\in\Omega;| \nabla u(  x)| =0\}  $.

If $u=0$, then a direct calculus shows that $\Psi$ is G\^{a}teaux
differentiable at zero and
\[
\langle \Psi'(0),h\rangle
=\lim_{t\to0}t^{-1}\Phi(  | t| \| h\|_{0,p(  \cdot)  ,\nabla})  =\lim_{t\to
0}\varphi(| t| \| h\| _{0,p(
\cdot)  ,\nabla})sgn\text{ }t\| h\| _{0,p(\cdot)  ,\nabla}=0.
\]
Moreover, $u\to\Psi'(u)$ is continuous at zero. Indeed, from
Theorem \ref{T3.2} (b), we obtain
\begin{equation}
\int_{\Omega}p(x)\frac{| \nabla u(  x)
| ^{p(x)}}{\| u\| _{0,p(  \cdot)
,\nabla}^{p(x)}}\mathrm{d}x\geq p^{-}\rho_{p(  \cdot)  }\Big(
\frac{| \nabla u| }{\| u\| _{0,p(
\cdot)  ,\nabla}}\Big)  =p^{-}. \label{4.13}
\end{equation}
On the other hand, by using Schwarz's inequality for nonnegative bilinear
symmetric forms and inequality \eqref{3.1}, it follows that
\begin{equation} \label{4.12}
\begin{aligned}
&\big| \int_{\Omega\backslash\Omega_{0,u}}p(x)\frac{|
\nabla u(  x)  | ^{p(x)-2}\langle \nabla
u(  x)  ,\nabla h(  x)  \rangle }{\|
u\| _{0,p(  \cdot)  ,\nabla}^{p(x)-1}}\mathrm{d}x\big|
\\
&\leq p^{+}{\int_{\Omega}}\Big(  \frac{|
\nabla u(  x)  | }{\| u\| _{0,p(
\cdot)  ,\nabla}}\Big)  ^{p(  x)  -1}| \nabla h(  x)  | \mathrm{d}x
\\
&\leq M\| | \nabla h| \| _{0,p(\cdot
)}\big\| \Big(  \frac{| \nabla u| }{\|
u\| _{0,p(  \cdot)  ,\nabla}}\Big)  ^{p(\cdot)  -1}\big\| _{0,p'(\cdot)}
\\
&=M\| h\| _{0,p(  \cdot)  ,\nabla} \big\|
\Big( \frac{| \nabla u| }{\| u\|
_{0,p(  \cdot)  ,\nabla}}\Big)  ^{p(  \cdot)
-1}\big\| _{0,p'(\cdot)},
\end{aligned}
\end{equation}
where $M=p^{+}\cdot(  \frac{1}{p^{-}}+\frac{1}{p^{\prime-}})  $.
Since
\[
\rho_{p'(  \cdot)  }\Big(  \Big(  \frac{|
\nabla u| }{\| u\| _{0,p(  \cdot)
,\nabla}}\Big)  ^{p(  \cdot)  -1}\Big)  =\rho_{p(
\cdot)  }\Big(  \frac{| \nabla u| }{\|
u\| _{0,p(  \cdot)  ,\nabla}}\Big)  =1,
\]
by Theorem \ref{T3.2} (b) we have
\[
\big\| \Big(  \frac{| \nabla u| }{\|
u\| _{0,p(\cdot),\nabla}}\Big)  ^{p(  \cdot)
-1}\big\| _{0,p'(\cdot)}=1\,,
\]
therefore, from \eqref{4.12} we obtain
\[
\big| \int_{\Omega\backslash\Omega_{0,u}}p(x)\frac{|
\nabla u(  x)  | ^{p(x)-2}\text{ }\nabla u(
x)  \cdot\nabla h(  x)  }{\| u\|
_{0,p(  \cdot)  ,\nabla}^{p(x)-1}}\mathrm{d}x\big| \leq
M \| h\| _{0,p(  \cdot)  ,\nabla}\,.
\]
From \eqref{4.13} and \eqref{4.12} we infer that
\[
| \langle \Psi'(u),h\rangle | \leq
\frac{M}{p^{-}}\cdot\varphi(\| u\| _{0,p(  \cdot)
,\nabla})\cdot\| h\| _{0,p(  \cdot)  ,\nabla}\,,
\]
for any nonzero $u\in U_{\Gamma_0}$ and for any $h\in U_{\Gamma_0}$.
Thus
\[
\| \Psi'(u)\| \leq\frac{M}{p^{-}}\varphi
(\| u\| _{0,p(  \cdot)  ,\nabla})\to
0\quad \text{as }\| u\| _{0,p(  \cdot)  ,\nabla
}\to0\,;
\]
therefore $\Psi$ is $\mathcal{C}^{1}$. To conclude that $H$ is
$\mathcal{C}^{1}$, the $\mathcal{C}^{1}$-property of the functional $\mathcal{G}$
given by \eqref{4.8}, has to be proven.

As far as the $\mathcal{C}^{1}$-regularity of $\mathcal{G}$ is concerned, for
a later use, we shall prove more: $\mathcal{G}$ is $\mathcal{C}^{1}$ on
$L^{q(  \cdot)  }(  \Omega)  $ and
\begin{equation}
\langle \mathcal{G}'(u),h\rangle ={\textstyle\int
_{\Omega}} g(  x,u(x))  h(x)\mathrm{d}x,u,h\in
L^{q(  \cdot)  }(  \Omega)  . \label{4.14}
\end{equation}

Indeed, let $u,h\in L^{q(  \cdot)  }(  \Omega)  $.
According to \cite[p. 178]{KR} and by using H\"{o}lder's type inequality
\eqref{3.1},
\begin{align*}
&| \mathcal{G}(u+h)-\mathcal{G}(u)-\langle \mathcal{G}'(u),h\rangle |\\
&=\big| \int_{\Omega} [  g(  x,u(x)+\theta
(x)h(x))  h(x)-g(x,u(x))h(x)]  \mathrm{d}x\big| \\
&\leq M\| g(  x,u(x)+\theta(  x)  h(x))
-g(x,u(x))\| _{0,q'(  \cdot)  }\|
h\| _{0,q(  \cdot)  },
\end{align*}
where $0\leq\theta(x)\leq1$. Consequently,
\[
\frac{| \mathcal{G}(u+h)-\mathcal{G}(u)-\langle \mathcal{G}
'(u),h\rangle | }{\| h\| _{0,q(\cdot)  }}
\leq M\| g(  x,u(x)+\theta(  x)  h(x))
-g(x,u(x))\| _{0,q'(  \cdot)  }.
\]
Suppose $\| h\| _{0,q(  \cdot)  }\to0$.
Taking into account the continuity of Nemytskij operators
 \cite[Theorem 1.16]{FZ}, it follows that $\mathcal{G}$ is
Fr\'{e}chet differentiable on
$L^{q(  \cdot)  }(  \Omega)  $ and $\mathcal{G}'$ is given by \eqref{4.14}.

Moreover, the operator $\mathcal{G}':L^{q(  \cdot)
}(  \Omega)  \to(  L^{q(  \cdot)  }(\Omega)  )  ^{\ast}$
given by \eqref{4.14} is continuous \cite[Theorem 1.16]{FZ}.

Now, since $U_{\Gamma_0}$ is continuously imbedded in
$L^{q(\cdot)  }(  \Omega)  $ and $\mathcal{G}$ is $\mathcal{C}^{1}$
on $L^{q(  \cdot)  }(  \Omega)  $, it follows that
$\mathcal{G}$ is $\mathcal{C}^{1}$ on $U_{\Gamma_0}$.
\end{proof}

\begin{proposition} \label{P4.3}
Let $q\in\mathcal{C}_{+}(  \overline{\Omega})  $ and
$g:\Omega\times\mathbb{R} \to\mathbb{R} $ be a Carath\'{e}odory
function which satisfies the growth condition \eqref{4.4} and the
hypothesis {\rm (H2)} modified as follows: there exist $s_0>0$ and
$\theta>0$ such that \eqref{4.5} holds for almost all
$x\in\Omega$ and all $s$ with $| s| \geq s_0$, where $\mathcal{G}$
is given by \eqref{4.8}.
Then, the functional $\mathcal{G}:L^{q(  \cdot)  }(
\Omega)  \to\mathbb{R} $ given by \eqref{4.8} satisfies
the inequality \eqref{1.9}.
\end{proposition}

\begin{proof}
One has
\[
\langle \mathcal{G}'(u),u\rangle -\theta\mathcal{G}
(u)={\int_{\Omega}} [  g(  x,u(
x)  )  u(  x)  -\theta G(x,u(  x)  )]
\mathrm{d}x.
\]

Now, we shall give an estimation for the right term of this equality.
Define $\overline{\Omega}=\{x\in\Omega:| u(x)| >s_0\}$. Taking into
account \eqref{4.5}, one has
\begin{equation}
\int_{\overline{\Omega}} [  g(  x,u(
x)  )  u(  x)  -\theta G(x,u(  x)  )]
\mathrm{d}x\geq0. \label{4.15}
\end{equation}
Also, considering \eqref{4.29}, one has
\begin{align*}
\big| {\int_{\Omega\backslash\overline{\Omega}}}
G(  x,u(x))  \mathrm{d}x\big|
&\leq{\int _{\Omega\backslash\overline{\Omega}}} [  c| u(
x)  | ^{q(  x)  }+| u(  x) | a(  x)  ]  \mathrm{d}x\\
& \leq c s_0^{q^{+}} \operatorname{vol}(\Omega)
+s_0{\int_{\Omega}} a(  x)  \mathrm{d}x=K,
\end{align*}
where $c:=C_1/q^{-}$.

On the other hand, from \eqref{4.4}, it follows that
\begin{align*}
\big| {\int_{\Omega\backslash\overline{\Omega}}}
g(  x,u(  x)  )  u(  x)  \mathrm{d}x\big|
&\leq{\int_{\Omega\backslash\overline{\Omega}}} [
c| u(  x)  | ^{q(  x)  }+|u(  x)  | a(  x)  ]  \mathrm{d}x \\
&\leq c s_0^{q^{+}} \operatorname{vol}(\Omega)+s_0{\int_{\Omega}} a(  x)
  \mathrm{d}x=K.
\end{align*}
Thus
\begin{equation}
\big| {\int_{\Omega\backslash\overline{\Omega}}}
[  g(  x,u(  x)  )  u(  x)  -\theta
G(x,u(  x)  )]  \mathrm{d}x\big| \leq C,
\label{4.16}
\end{equation}
with $C:=K(  1+\theta) $.
From \eqref{4.15} and \eqref{4.16}, we infer that
\[
{\int_{\Omega}} [  g(  x,u(  x)
)  u(  x)  -\theta G(x,u(  x)  )]
\mathrm{d}x\geq-C,
\]
that is \eqref{1.9}.
\end{proof}

Using the same arguments as in \cite[Remark 7.2, p. 26]{DM3},
we obtain the following result.

\begin{lemma} \label{L2}
Let $\varphi:\mathbb{R} _{+}\to\mathbb{R} _{+}$ be a gauge
function which satisfies \eqref{1.3}, where $\Phi$ is given by
\eqref{1.10}. Then, for all $u\in U_{\Gamma_0}$ with
$\| u\| _{0,p(  \cdot)  ,\nabla}<1$ one has
\begin{equation}
\Phi(\| u\| _{0,p(  \cdot)  ,\nabla})\geq
\Phi(  1)  \| u\| _{0,p(  \cdot)
,\nabla}^{\varphi^{\ast}}\,. \label{4.24}
\end{equation}
Also for all $u\in U_{\Gamma_0}$ with $\| u\| _{0,p(
\cdot)  ,\nabla}>1$ one has
\[
\Phi(\| u\| _{0,p(  \cdot)  ,\nabla})\leq
\Phi(  1)  \| u\| _{0,p(  \cdot)
,\nabla}^{\varphi^{\ast}}\,.
\]
\end{lemma}

\begin{proof}[Proof of Theorem \ref{T4.1}]
 We  use Theorem \ref{T2.1} with
$X=U_{\Gamma_0}$ and $V=L^{q(\cdot)}(  \Omega)  $. Indeed, $X$ is
reflexive (Theorem \ref{T3.4}, (a)) and smooth (Theorem \ref{T3.4} (c)).
Also, by  Theorem \ref{T3.4} (a) and Theorem \ref{T3.3}, (c)
$(U_{\Gamma_0},\| \cdot\| _{0,p(  \cdot)  ,\nabla})  $ is compactly embedded
in $(  L^{q(\cdot)}(\Omega)  ,\| \cdot\| _{0,q(  \cdot)})  $.
According to \cite[Theorem 4.6 a)]{CDM2}, $\Psi'$ satisfies condition $(  S)  _2 $.

Obviously $\mathcal{G}(  0)  =0$ and taking into account
Propositions \ref{P4.2} and \ref{P4.3}, it follows that $\mathcal{G}$ is
$\mathcal{C}^{1}$ and that the hypothesis (ii) of Theorem \ref{T2.1} is fulfilled.

Let us prove that hypothesis (iii) of Theorem \ref{T2.1} is fulfilled. For the
first term in \eqref{4.10}, we have \eqref{4.24} for all $u\in U_{\Gamma_0}$
with $\| u\| _{0,p(  \cdot)  ,\nabla}<1$.

Arguing as in \cite[p. 239]{DJM}, from (H3) we deduce that there exists
\begin{equation}
0<\mu<(  \varphi^{\ast}\Phi(  1) /2)  \lambda_{1,\varphi^{\ast}} \label{4.32}
\end{equation}
and $\underline{s}>0$ such that
\begin{equation}
G(x,s)<(  \mu/\varphi^{\ast})  | s|
^{\varphi^{\ast}},\quad \text{for }x\in\Omega,0<| s|
<\underline{s}. \label{4.20}
\end{equation}


Now, let us consider $| s| \in[\underline{s},\infty)$. The function
$| s| ^{q(  x)  -1}$ being
increasing as function of $| s| $, we have
\[
| s| \leq\frac{1}{\underline{s}^{q(  x)  -1}}| s| ^{q(  x)  }.
\]
Since the function $a$ in \eqref{4.4} is assumed to be bounded, it follows
from \eqref{4.29} that
\[
| G(x,s)| \leq c_{3}\cdot s^{q(  x)  },\quad \text{for }| s| \geq\underline{s}\,,
\]
where $c_{3}:=C_1/q^{-}+\| a\| _{\infty}/\underline {s}^{q^{-}-1}$.

Now, we denote $\underline{\Omega}=\{x\in\Omega: | u(x)|\geq\underline{s}\}$.
 Then, for every $u\in L^{q(  \cdot)  }
(\Omega)$, we have
\begin{equation}
{\int_{\underline{\Omega}}} G(  x,u(x))
\mathrm{d}x\leq c_{3}{\int_{\Omega}} |
u(x)| ^{q(  x)  }\mathrm{d}x. \label{4.30}
\end{equation}
But $U_{\Gamma_0}$ is continuously imbedded in $L^{q(  \cdot)
}(  \Omega)  $ (Lemma \ref{L1}), therefore there exists a positive
constant $\underline{c}$ such that
\[
\| u\| _{0,q(\cdot)}\leq\underline{c}\| u\|
_{0,p(\cdot),\nabla}\quad \text{for all }u\in U_{\Gamma_0}.
\]
Consequently, for all $u\in U_{\Gamma_0}$ with
$\| u\|_{0,p(\cdot),\nabla}<1/\underline{c}$ it follows that
$\| u\|_{0,q(  \cdot)  }<1$. Therefore, taking into account \eqref{4.30}
and Theorem \ref{T3.2} (d), we obtain
\begin{equation}
{\int_{\underline{\Omega}}} G(  x,u(x))
\mathrm{d}x\leq c_{3}\| u\| _{0,q(  \cdot)  }
^{q^{-}}, \label{4.33}
\end{equation}
for all $u\in U_{\Gamma_0}$ with $\| u\| _{0,p(\cdot),\nabla}<1/\underline{c}$.

On the other hand, from \eqref{4.20}, for $u\in U_{\Gamma_0}$, we deduce
\begin{equation}
{\int_{\Omega\backslash\underline{\Omega}}} G(
x,u(x))  \mathrm{d}x\leq\frac{\mu}{\varphi^{\ast}}{\int
_{\Omega}} | u(x)| ^{\varphi^{\ast}}
\mathrm{d}x=\frac{\mu}{\varphi^{\ast}}\| u\| _{L^{\varphi
^{\ast}}(  \Omega)  }^{\varphi^{\ast}}. \label{4.11}
\end{equation}

Since $\varphi^{\ast}<q^{-}$, then $U_{\Gamma_0}$ is compactly imbedded in
$L^{\varphi^{\ast}}(  \Omega)  $ (Remark \ref{R1}). Taking into
account \eqref{4.11}, \eqref{4.32},\ and the definition \eqref{4.9} of
$\lambda_{1,\varphi^{\ast}}$, for $u\in U_{\Gamma_0}$, we obtain
\begin{equation}
{\int_{\Omega\backslash\underline{\Omega}}} G(
x,u(x))  \mathrm{d}x\leq\frac{\mu}{\varphi^{\ast}\lambda_{1,\varphi^{\ast
}}}\| u\| _{0,p(\cdot),\nabla}^{\varphi^{\ast}}\leq\frac
{\Phi(  1)  }{2}\| u\| _{0,p(\cdot),\nabla
}^{\varphi^{\ast}}\,. \label{4.31}
\end{equation}
Then, from Lemma \ref{L2}, \eqref{4.31}, \eqref{4.33}, we obtain
\begin{align*}
H(u)&>\Phi(  1)  \| u\| _{0,p(\cdot),\nabla
}^{\varphi^{\ast}}-\frac{\Phi(  1)  }{2}\| u\|
_{0,p(\cdot),\nabla}^{\varphi^{\ast}}-c_{3}\| u\|
_{0,q(  \cdot)  }^{q^{-}} \\
&=\frac{\Phi(  1)  }{2}\| u\| _{0,p(\cdot),\nabla
}^{\varphi^{\ast}}-c_{3}\| u\| _{0,q(  \cdot)
}^{q^{-}}\,,
\end{align*}
for all $u\in U_{\Gamma_0}$ with $\| u\| _{0,p(\cdot
),\nabla}<\min(  1,1/\underline{c})  $.
Therefore, the hypothesis (iii) of Theorem \ref{T2.1} is fulfilled.

Now, we shall verify the hypothesis (iv) of Theorem \ref{T2.1}.
 Let $\theta$ and $s_0$ be as in (H2). We shall deduce that one has
\begin{equation}
G(x,s)\geq\gamma(x)| s| ^{\theta},\quad
\text{for almost all $x\in\Omega$  and $| s| \geq s_0$}, \label{4.28}
\end{equation}
where the function $\gamma$ will be specified below. Indeed, it follows from
\cite[p. 236]{DJM} that
\begin{equation}
G(x,s)\geq(  G(x,s_0)/s_0^{\theta})  s^{\theta},\quad
\text{for almost all $x\in\Omega$ and $s\geq s_0$}. \label{4.26}
\end{equation}
On the other hand, for almost all $x\in\Omega$ and $\tau\leq-s_0$, from
\eqref{4.5}, we have
$G(x,s)>0$ for almost all $x\in\Omega$  and $| s|\geq s_0$,
and
\[
\frac{\theta}{\tau}\geq\frac{g(x,\tau)}{G(x,\tau)}.
\]
By integrating from $s\leq-s_0$ to $-s_0$, it follows that
\[
\frac{s_0^{\theta}}{| s| ^{\theta}}\geq\frac{G(x,-s_0
)}{G(x,s)},
\]
which implies
\begin{equation}
G(x,s)\geq(  G(x,-s_0)/s_0^{\theta})  | s|
^{\theta},\quad\text{for almost all $x\in\Omega$  and $s\leq-s_0$}.
\label{4.27}
\end{equation}
Setting
\[
\gamma(x)=\begin{cases}
(  G(x,s_0)/s_0^{\theta}), & \text{if }s\geq s_0\\
(  G(x,-s_0)/s_0^{\theta}), &  \text{if }s\leq-s_0,
\end{cases}
\]
from \eqref{4.26} and \eqref{4.27}, we obtain \eqref{4.28}.

For $v\in U_{\Gamma_0}$, we define
\[
\Omega_{\geq}:=\{x\in\Omega: | v(x)| \geq s_0\},\Omega_{<}
:=\Omega\backslash\Omega_{\geq}.
\]
From \eqref{4.28} it follows that
\begin{align*}
{\int_{\Omega}} G(x,v(x))\mathrm{d}x
&\geq{\int_{\Omega_{\geq}}} \gamma(x)| v(x)| ^{\theta
}\mathrm{d}x+{\int_{\Omega_{<}}} G(x,v(x))\mathrm{d}x
\\
&={\int_{\Omega}} \gamma(x)| v(x)|
^{\theta}\mathrm{d}x+{\int_{\Omega_{<}}} G(x,v(x))\mathrm{d}
x-{\textstyle\int_{\Omega_{<}}} \gamma(x)| v(x)|
^{\theta}\mathrm{d}x
\end{align*}
Since
\[
{\int_{\Omega_{<}}} \gamma(x)| v(x)|
^{\theta}\mathrm{d}x\leq\| \gamma\| _{\infty}
s_0^{\theta} \operatorname{vol}(\Omega),
\]
we have
\[
{\int_{\Omega}} G(x,v(x))\mathrm{d}x\geq{\int
_{\Omega}} \gamma(x)| v(x)| ^{\theta}
\mathrm{d}x+{\int_{\Omega_{<}}} G(x,v(x))\mathrm{d}x-k,
\]
where $k:=\| \gamma\| _{\infty} s_0^{\theta}
\operatorname{vol}(\Omega)$. On the other hand, it follows from \eqref{4.29} that
\[
{\int_{\Omega_{<}}} G(x,v(x))\mathrm{d}x\leq\|
a\| _{\infty}s_0+c_{4}\max(  s_0^{q^{+}},s_0^{q^{-}
})  \operatorname{vol}(\Omega),
\]
where $c_{4}=c_1/q^{-}$. Therefore
\[
{\int_{\Omega}} G(x,v(x))\mathrm{d}x\geq{\int
_{\Omega}} \gamma(x)| v(x)| ^{\theta} \mathrm{d}x-K,
\]
where $K:=k+\| a\| _{\infty}s_0+c_{4}\max(
s_0^{q^{+}},s_0^{q^{-}})  \operatorname{vol}(\Omega)$.
Consequently,
\[
H(v)\leq\Phi(  \| v\| _{0,p(  \cdot),\nabla})  -{\int_{\Omega}} \gamma(x)|
v(x)| ^{\theta}\mathrm{d}x+K,
\]
where $K$ is a positive constant and $\theta$ is given by (H)$_2 $. Taking
into account Lemma \ref{L2}, for
$\| v\|_{0,p(  \cdot)  ,\nabla}>1$ we have
\begin{equation}
H(v)\leq\Phi(1)\| v\| _{0,p(  \cdot)  ,\nabla
}^{\varphi^{\ast}}-{\int_{\Omega}} \gamma(x)|
v(x)| ^{\theta}\mathrm{d}x+K. \label{4.25}
\end{equation}
Now, the functional $\| \cdot\| _{\gamma}:U_{\Gamma_0
}\to\mathbb{R} $ defined by
\[
\| v\| _{\gamma}=\Big(  {\int_{\Omega
}} \gamma(x)| v(x)| ^{\theta}\mathrm{d}x\Big)  ^{1/\theta}
\]
is a norm on $U_{\Gamma_0}$. Let $X_1$ be a finite dimensional subspace of
$U_{\Gamma_0}$. Since the tow norms $\| \cdot\| _{0,p(
\cdot)  ,\nabla}$ and $\| \cdot\| _{\gamma}$
are equivalent on the finite dimensional subspace $X_1$, there is a constant
$\delta=\delta(X_1)>0$ such that
\[
\| v\| _{0,p(  \cdot)  ,\nabla}\leq\delta \| v\| _{\gamma}.
\]
Therefore,  from \eqref{4.25} it follows that
\[
H(v)\leq\Phi(1)\| v\| _{0,p(  \cdot)  ,\nabla
}^{\varphi^{\ast}}-\frac{1}{\delta^{\theta}}\| v\|
_{0,p(  \cdot)  ,\nabla}^{\theta}+K,
\]
if $v\in X_1$, $\| v\| _{0,p(  \cdot)  ,\nabla }>1$, that is the hypothesis
(iv) is fulfilled.

Taking into account Theorem \ref{T2.1}, it follows that the functional $F$
possesses a sequence of critical positive values. By Proposition \ref{P4.2},
equation
\[
J_{\varphi}u=g(x,u)
\]
has a sequence of solutions in $U_{\Gamma_0}$ or, equivalently, the problem
\eqref{4.1}, \eqref{4.2} possesses a sequence of weak solutions in
$U_{\Gamma_0} $.
\end{proof}

Taking into account Remark \ref{RR}, if $p( x) =p=$const. and
$\varphi (t)=t^{r-1}$, $r>1$, from Theorem \ref{T4.1} it follows:

\begin{corollary} \label{coro4}
Let $\Omega $ be a domain in $\mathbb{R}^{N}$ $(N\geq 2)$,
$p\in ( 1,\infty ) $, and let $p^{\ast }$ be given by
\[
p^{\ast }:=\frac{Np}{N-p}\text{ if }p<N\quad \text{and}\quad
p^{\ast }:=\infty \text{ if }p\geq N,
\]
Let there be given a Carath\'{e}odory function
$g:\Omega \times \mathbb{R}\to \mathbb{R}$ satisfying the hypotheses:
\begin{itemize}
\item[(1)] there exists a function $q( \cdot ) \in \mathcal{
C}( \overline{\Omega }) $ that satisfies
\[
1\leq q( x) <p^{\ast }\quad \text{for each }x\in \overline{\Omega }
\]
such that
\[
| g( x,s) | \leq C_1| s|
^{q( x) /q'( x) }+a( x),\quad
\text{for almost all $x\in \Omega$ and all $s\in \mathbb{R}$},
\]
where $\frac{1}{q(x)}+\frac{1}{q'(x)}=1$,
$a$ is a bounded function, $a( x) \geq 0$ for almost all $x\in \Omega $, and
$C_1 $ is a constant, $C_1>0$;

\item[(2)] there exist $s_0>0$ and $\theta >r$ such that
\eqref{4.5} holds for almost every $x\in \Omega $ and all $s$ with
$| s| \geq s_0$, where $G$ is given by \eqref{4.6}.
\end{itemize}
Also assume that
\begin{itemize}
\item[(3)]
\[
\limsup_{s\to 0}\frac{g( x,s) }{|
s| ^{r-2}s}<\frac{\lambda _{1,r}}{2}
\]
uniformly with respect to almost all $x\in \Omega $, where $\lambda _{1,r}$
is given by \eqref{4.9}.

\item[(4)]  $r<q^{-}$.
\end{itemize}
Let $N_{g}:L^{q( \cdot ) }( \Omega ) \to
L^{q'( \cdot ) }( \Omega )$, with
$(N_{g}u) ( x) =g( x,u( x) )$  for almost all $x\in \Omega$,
denote the Nemytskij operator generated by $g$.
Under these assumptions, the problem
\begin{gather}
-\operatorname{div}\big( \| | \nabla u| \|
_{L^p( \Omega ) }^{r-p}| \nabla u|
^{p-2}\nabla u\big) =g(x,u)\quad \text{in }\Omega ,  \label{5.1}\\
u=0\quad \text{on }\Gamma _0\subset \partial \Omega ,  \label{5.2}
\end{gather}
has a weak non-trivial solution in the space $U_{\Gamma _0}$. Moreover, if
$g$ is odd in the second argument: $g( x,-s) =-g(x,s)$,
$s\in \mathbb{R}$, then  problem \eqref{5.1}, \eqref{5.2} has a
sequence of weak solutions.
\end{corollary}

In particular, if $r=p$ and $q( x) =q=$const., we obtain a
result similar to \cite[Theorem 18, p. 370]{DJM}:

\begin{corollary} \label{coro5}
Let $\Omega $ be a domain in $\mathbb{R}^{N}$ $(N\geq 2)$,
let $p\in \mathbb{R}$ be such that $p>1$, and let $p^{\ast }$ be given by
\[
p^{\ast }:=\frac{Np}{N-p}\text{ if }p<N,\quad\text{and}\quad
p^{\ast }:=\infty \text{ if }p\geq N,
\]
Let there be given a Carath\'{e}odory function $g:\Omega \times \mathbb{R}
\to \mathbb{R}$ satisfying the hypotheses:
\begin{itemize}
\item[(1)] there exists  $q\in ( 1,p^{\ast }) $
such that
\[
| g( x,s) | \leq C_1| s|
^{q-1}+a( x),\quad \text{for almost all $x\in \Omega$  and all
$s\in \mathbb{R}$},
\]
where $\frac{1}{q}+\frac{1}{q'}=1$, $a$ is a bounded function,
$a(x) \geq 0$ for almost all $x\in \Omega $, and $C_1$ is a constant,
$C_1>0$;

\item[(2)] there exist $s_0>0$ and $\theta >p$ such that
\eqref{4.5} holds for almost every $x\in \Omega $ and all $s$ with $
| s| \geq s_0$, where $G$ is given by \eqref{4.6}.
\end{itemize}
Also assume that
\begin{itemize}
\item[(3)]
\[
\limsup_{s\to 0}\frac{g( x,s) }{|
s| ^{p-2}s}<\frac{\lambda _{1,p}}{2}
\]
uniformly with respect to almost all $x\in \Omega $, where $\lambda _{1,p}$
is given by \eqref{4.9}.

\item[(4)] $p<q$.
\end{itemize}
Let $N_{g}:L^{q}( \Omega ) \to L^{q'}( \Omega
)$, with $( N_{g}u) ( x) =g( x,u(x) )$ for almost all
$x\in \Omega$, denote the Nemytskij operator generated by $g$.
Under these assumptions, the problem
\begin{gather}
-\operatorname{div}( | \nabla u| ^{p-2}\nabla u)
=g(x,u)\quad \text{ in }\Omega ,  \label{5.3} \\
u=0\quad \text{on }\Gamma _0\subset \partial \Omega ,  \label{5.4}
\end{gather}
has a weak non-trivial solution in the space $U_{\Gamma _0}$.
 Moreover, if $g$ is odd in the second argument, then
 problem \eqref{5.3}, \eqref{5.4} has a
sequence of weak solutions.
\end{corollary}

Now, let us consider the gauge function $\varphi :\mathbb{R}_{+}\to
\mathbb{R} _{+}$, $\varphi (t)=t^{r-1}$ln$( 1+t) $, $r>1$. From
\eqref{1.10} we have
\[
\Phi (t)=\frac{t^{r}}{r}\ln ( 1+t) -\frac{1}{r}
\int_0^{t}\frac{\tau ^{r}}{1+\tau }\mathrm{d}\tau ,t>0.
\]
According to \cite[p. 54]{CGMS}, $\varphi ^{\ast }=r+1$. We shall apply
Theorem \ref{T4.1} with $\varphi ^{\ast }=r+1$. From definition of $\varphi
^{\ast }$ it follows that
\[
\varphi ^{\ast }\Phi ( 1) \geq \varphi ( 1) =\ln 2\,.
\]
From Theorem \ref{T4.1} we have the following result.

\begin{theorem} \label{thm9}
Let $\Omega $ be a domain in $\mathbb{R}^{N}$ $(N\geq 2)$,
let $p\in \mathcal{C}( \overline{\Omega }) $ be a function such
that $p^{-}>1$, and let $p^{\ast }(\cdot )$ be given by \eqref{3.3}.
Let us consider the function
\begin{equation}
\varphi :\mathbb{R}_{+}\to \mathbb{R}_{+},\quad
\varphi (t)=t^{r-1} \ln ( 1+t) ,r>1.  \label{5.5}
\end{equation}
Let there be given a Carath\'{e}odory function $g:\Omega \times \mathbb{R}
\to \mathbb{R}$ satisfying the hypotheses:
\begin{itemize}
\item[(1)] there exists a function
$q( \cdot ) \in \mathcal{C}( \overline{\Omega }) $ that satisfies
\eqref{3.4} such that
\[
| g( x,s) | \leq C_1| s|
^{q( x) /q'( x) }+a( x),\quad
\text{for almost all $x\in \Omega$  and all $s\in \mathbb{R}$,}
\]
where $\frac{1}{q(x)}+\frac{1}{q'(x)}=1$, a is a bounded function,
$a( x) \geq 0$ for almost all $x\in \Omega $, and $C_1$ is a
constant, $C_1>0$;

\item[(2)] there exist $s_0>0$ and $\theta >r+1$ such that
\[
0<\theta G(x,s)\leq sg(x,s),
\]
for almost every $x\in \Omega $ and all $s$ with $| s|
\geq s_0$, where
\[
G( x,s) :={\textstyle\int_0^{s}}g( x,\tau )
\mathrm{d}\tau .
\]
\end{itemize}
Also assume that
\begin{itemize}
\item[(3)]
\[
\limsup_{s\to 0}\frac{g( x,s) }{|
s| ^{r-1}s}<\frac{\ln 2}{2}\lambda _{1,r+1}
\]
uniformly with respect to almost all $x\in \Omega $, where $\lambda _{1,r+1}$
is given by \eqref{4.11}.

\item[(4)] $r+1<q^{-}$.
\end{itemize}
Let $N_{g}:L^{q( \cdot) }( \Omega) \to
L^{q'( \cdot) }( \Omega)$, with $(N_{g}u) ( x) =g( x,u( x) )$  for
almost all $x\in\Omega$,
denote the Nemytskij operator generated by $g$.
Under these assumptions,  problem \eqref{4.1}, \eqref{4.2},
where $\varphi $ is given by \eqref{5.5}, has a weak non-trivial
solution in the space $U_{\Gamma _0}$ (endowed with the norm
\eqref{3.2}. Moreover, if $g$ is odd in the second argument,
then problem \eqref{4.1}, \eqref{4.2} has a sequence of weak solutions.
\end{theorem}


\subsection*{Acknowledgements}
The author thanks the anonymous referee for his/her valuable suggestions
which contributed to improving the final form of this article.


\begin{thebibliography}{99}

\bibitem{AR} A. Ambrosetti, P. H. Rabinowitz;
\emph{Dual variational methods in critical point theory and applications},
J. Funct. Anal., 14 (1973), 349-381.

\bibitem{As} E. Asplund;
\emph{Positivity of duality mappings}, Bull. Amer. Math. Soc., 73 (1967), 200-203.

\bibitem{Br} F. E. Browder;
\emph{Probl\`{e}mes non-lin\'{e}aires}, Presses de
l'Universit\'{e} de Montr\'{e}al, 1964.

\bibitem{CDM1} P. G. Ciarlet, G. Dinca, P. Matei;
\emph{Fr\'{e}chet differentiability of the norm in a Sobolev space with a
variable exponent},
Analysis and Applications, 11, 4 (2013), 1350012 (31 pages),
DOI: 10.1142/S0219530513500127.

\bibitem{CDM2} P. G. Ciarlet, G. Dinca, P. Matei;
\emph{Operator equations and duality mappings in Sobolev space with
 variable exponents}, Chin. Ann. Math. B, 34, 5 (2013), 639-666.

\bibitem{CGMS} P. Cl\'{e}ment, M. Garc\'{\i}a-Huidobro, R. Man\'{a}sevich,
 K. Schmitt;
\emph{Mountain pass type solutions for quasilinear elliptic equations},
Calc. Var. Partial Differ. Equ., 11, 1,(2000), 33-62.

\bibitem{CR} J. Cringanu;
\emph{Existence results for operator equations involving duality mapping via
the mountain pass theorem}, Math. Reports, 11, 61 (2008), 11-20.

\bibitem{DHHR} L. Diening, P. Harjulehto, P. H\"{a}st\"{o},  M.
R\^{u}\v{z}i\v{c}ka;
\emph{Lebesgue and Sobolev spaces with variable exponents},
Lecture Notes in Mathematics, 2017, Springer Verlag, Berlin, 2011.

\bibitem{Di}J. Diestel;
\emph{Geometry of Banach Spaces-Selected Topics}, Lect.
Notes Math., 485, Springer-Verlag, 1975.

\bibitem{D} G. Dinca;
\emph{Some existence results for operator equations
involving duality mappings on Sobolev spaces with variable exponent}, Diff.
Int. Equat., 22, 9-10 (2009), 1019-1032.

\bibitem{DJ} G. Dinca, P. Jebelean;
\emph{Some existence results for a class of nonlinear equations involving
a duality mapping}, Nonlinear Analysis. Theory, Methods \& Applications,
46 (2001), 347-363.

\bibitem{DJM} G. Dinca, P. Jebelean, J. Mawhin;
\emph{A result of Ambrosetti-Rabinowitz type for \textit{p}-Laplacian},
in Qualitative Problems
for Differential Equations and Control Theory (C. Corduneanu, Ed.), World
Scientific, Singapore, 1995, 231-242.

\bibitem{DJM1} G. Dinca, P. Jebelean,  J. Mawhin;
\emph{Variational and topological methods for Dirichlet problems with $p$-Laplacian},
Portugaliae Mathematica, 58, 3 (2001), 339-377.

\bibitem{DM1} G. Dinca, P. Matei;
\emph{Multiple solutions for operator equations involving duality mappings
on Orlicz-Sobolev spaces}, Diff. Int. Equat., 21, 9-10 (2008), 891-916.

\bibitem{DM2} G. Dinca P. Matei;
\emph{Multiple solutions for operator equations involving duality mappings on
Orlicz-Sobolev spaces via the mountain pass theorem},
Rev. Roumaine Math. Pures Appl., 53, 5-6 (2008), 419-437.

\bibitem{DM3} G. Dinca, P. Matei;
\emph{Variational and topological methods for operator equations involving
duality mappings on Orlicz-Sobolev spaces}, E. Journal of Diff. Equat.,
93 (2007), 1-47.

\bibitem{DM4} G. Dinca, P. Matei;
\emph{Geometry of Sobolev spaces with variable exponent and a generalization
of the $p$-Laplacian}, Analysis and Applications, 7, 4 (2009), 373-390.

\bibitem{FZ} X. L. Fan, D. Zhao;
\emph{On the spaces $L^{p(x)}(\Omega)$ and $W^{m,p(x)}(\Omega)$},
J. Math. Anal. Appl., 263 (2001), 424-446.

\bibitem{KoR} O. Kov\'{a}\v{c}ik, J. R\'{a}kosn\'{\i}k;
\emph{On spaces $L^{p(x)}$ and $W^{k,p(x)}$},
Czechoslovak Math. J., 41, 4 (1991), 592-618.

\bibitem{KR} M. A. Krasnosel'skij, Ja. B. Rutitskij;
\emph{Convex functions and Orlicz spaces},
Groningen, Noordhoff, 1961.

\bibitem{LE}J. Lang, D. Edmunds;
\emph{Eigenvalues, embeddings and generalised trigonometric functions},
 Lecture Notes in Mathematics, 2016, Springer-Verlag, Berlin, 2011.

\bibitem{MW} J. Mawhin, M. Willem;
\emph{Critical point theory and Hamiltonian systems},
Springer Verlag, New York, 1989.

\bibitem{R} P. H. Rabinowitz;
\emph{Minimax methods in critical point theory with
applications to differential equations}, Conference Board of the Math. Sci.
Regional Conference Series in Math, no. 65, Amer. Math. Soc., Providence, RI, 1986.

\bibitem{Ru} M. R\^{u}\v{z}i\v{c}ka;
\emph{Electrorheological fluids: modelling
and mathematical theory}, Lecture Notes in Mathematics, 1748,
Springer Verlag, Berlin, 2000.

\bibitem{Ze} E. Zeidler;
\emph{Nonlinear functional analysis and its applications
II/B: Nonlinear monotone operators}, Springer, New York, 1990.

\end{thebibliography}

\end{document}