\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 $1
0$, 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
$\| u\| _X 0$, $c_2 >0$;
\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 $rd_0$.
\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)1$,
$q^{-}>1$ and \eqref{3.4} holds. For $\varphi^{\ast}
1$, 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)] $\varphi^{\ast}
0$ 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}
\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
1$, and let $p^{\ast }$ be given by
\[
p^{\ast }:=\frac{Np}{N-p}\text{ if }p
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