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

\AtBeginDocument{{\noindent\small
\emph{Electronic Journal of Differential Equations},
Vol. 2016 (2016), No. 295, pp. 1--12.\newline
ISSN: 1072-6691. URL: http://ejde.math.txstate.edu or http://ejde.math.unt.edu}
\thanks{\copyright 2016 Texas State University.}
\vspace{8mm}}

\begin{document}
\title[\hfilneg EJDE-2016/295\hfil Perturbed subcritical Dirichlet problems]
{Perturbed subcritical Dirichlet problems with variable exponents}

\author[R. Alsaedi \hfil EJDE-2016/295\hfilneg]
{Ramzi Alsaedi}

\address{Ramzi Alsaedi \newline
Department of Mathematics, Faculty of Sciences,
King Abdulaziz University, P.O. Box 80203, Jeddah 21589, Saudi Arabia}
\email{ramzialsaedi@yahoo.co.uk}

\thanks{Submitted September 8, 2016. Published November 16, 2016.}
\subjclass[2010]{35J60, 58E05}
\keywords{Nonhomogeneous elliptic problem; variable exponent;
\hfill\break\indent Dirichlet boundary condition; mountain pass theorem}

\begin{abstract}
 We study a class of nonhomogeneous elliptic problems with Dirichlet
 boundary condition and involving the $p(x)$-Laplace operator and
 power-type nonlinear terms with variable exponent.
 The main results of this articles establish sufficient conditions for
 the existence of nontrivial weak solutions, in relationship with the
 values of certain real parameters.
 The proofs combine the Ekeland variational principle, the  mountain pass
 theorem and energy arguments.
\end{abstract}

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

\section{Introduction}

Let $\Omega\subset\mathbb{R}$ be a bounded domain with smooth boundary.
In a pioneering paper, Ambrosetti and Rabinowitz \cite{ambrab} consider
the  subcritical elliptic problem
\begin{equation}\label{ambro}
\begin{gathered}
 -\Delta u=|u|^{p-2}u,\quad x\in\Omega\\
 u=0,\quad x\in\partial\Omega,
\end{gathered}
\end{equation}
where $1<p<2N/(N-2)$ if $N\geq 3$ and $1<p<\infty$ if $N\in\{1,2\}$.
 Problem \eqref{ambro} illustrates in \cite{ambrab} the celebrated mountain
pass theorem, which yields the existence of a nontrivial solution of \eqref{ambro}.
This result has been a rich source of valuable extensions to several classes
of nonlinear elliptic equations described by various types of differential
operators and involving nonlinear terms fulfilling or not the
 Ambrosetti-Rabinowitz growth condition. We refer, for instance, to
Pucci and R\u{a}dulescu \cite{pucci}, where it is established that a related
existence property remains true, provided that the (linear) operator in
the left-hand side of \eqref{ambro} is perturbed in a suitable manner.
More precisely, \cite[Theorem 9]{pucci} (see also Theorem 10) asserts
that the perturbed problem
\begin{gather*}
 -\Delta u-\lambda u=|u|^{p-2}u,\quad x\in\Omega\\
 u=0, \quad x\in\partial\Omega
\end{gather*}
has a nontrivial solution for all $\lambda<\lambda_1$, where $\lambda_1$
is the principal eigenvalue of the Laplace operator in $H^1_0(\Omega)$.
A related problem is studied in \cite{mrpams} in the framework of
differential operators and nonlinear terms involving variable exponents.
The main result in \cite{mrpams} establishes the existence of nontrivial
solutions in the case of small perturbations, namely with respect
to the values of a suitable parameter.

The main purpose of this article is to show that the perturbation results
established in \cite[Theorem 9]{pucci} and \cite[Theorem 2.1]{mrpams}
can be extended to the case of variable exponents. However, the problem
studied in this article has a different structure with those studied
in \cite{pucci} and \cite{mrpams} because of the several nonlinearities with
nonhomogeneous behaviour.

We refer to Cencelj, Repov\v{s} and Virk \cite{cencel},
Repov\v{s} \cite{repovs} for related results and to the monographs by
Diening, H\"asto,  Harjulehto and Ruzicka \cite{die} and R\u{a}dulescu
and Repov\v{s} \cite{radrep} for the basic functional framework used
in the present paper.

\section{Function spaces with variable exponent}

In this section we recall some basic definitions and properties
concerning the Lebesgue and Sobolev spaces with variable exponent.
Consider the set
$$
C_+(\overline\Omega)=\{h\in C(\overline\Omega),\;h(x)>1
\text{ for all } x\in\overline\Omega\}.
$$
For all $h\in C_+(\overline\Omega)$ we define
$$
h^+=\sup_{x\in\Omega}h(x)\quad\text{and}\quad
h^-=\inf_{x\in\Omega}h(x).
$$
The real numbers $h^+$ and $h^-$ will play a crucial role in our arguments
and usually the gap between these quantities produces new results,
which are no longer valid for constant exponents.

For any $p\in C_+(\overline\Omega)$, we define the \emph{variable exponent
Lebesgue space}
$$
L^{p(x)}(\Omega)=\{u: u \text{ is  measurable and }
\int_\Omega|u(x)|^{p(x)}\,dx<\infty\}.
$$
This vector space is a Banach space if it is endowed with the
\emph{Luxemburg norm}, which is defined by
$$
|u|_{p(x)}=\inf\big\{\mu>0;\;\int_\Omega|
\frac{u(x)}{\mu}|^{p(x)}\,dx\leq 1\big\}.
$$
Then $L^{p(x)}(\Omega)$ is reflexive if and only if $1 < p^-\leq p^+<\infty$
 and continuous functions with compact support
are dense in $L^{p(x)}(\Omega)$ if $p^+<\infty$.

The inclusion between Lebesgue spaces with variable exponent generalizes
the classical framework, namely if
$0 <|\Omega|<\infty$ and $p_1$, $p_2$ are variable exponents so that
$p_1\leq p_2$  in $\Omega$ then there exists the continuous embedding
$L^{p_2(x)}(\Omega)\hookrightarrow L^{p_1(x)}(\Omega)$.

Let $L^{p'(x)}(\Omega)$ be the conjugate space
of $L^{p(x)}(\Omega)$, where $1/p(x)+1/p'(x)=1$. For any
$u\in L^{p(x)}(\Omega)$ and $v\in L^{p'(x)}(\Omega)$  the following
H\"older-type inequality holds:
\begin{equation}\label{Hol}
\big|\int_\Omega uv\,dx\big|\leq\Big(\frac{1}{p^-}+
\frac{1}{p'^-}\Big)|u|_{p(x)}|v|_{p'(x)}\,.
\end{equation}


The \emph{modular} of  $L^{p(x)}(\Omega)$ is the mapping
 $\rho_{p(x)}:L^{p(x)}(\Omega)\to\mathbb{R}$ defined by
$$
\rho_{p(x)}(u)=\int_\Omega|u|^{p(x)}\,dx.
$$
If $(u_n)$, $u\in L^{p(x)}(\Omega)$ and $p^+<\infty$ then the following
relations are true:
\begin{gather}\label{L4}
|u|_{p(x)}>1\;\Rightarrow\;|u|_{p(x)}^{p^-}\leq\rho_{p(x)}(u)
\leq|u|_{p(x)}^{p^+}, \\
\label{L5}
|u|_{p(x)}<1\;\Rightarrow\;|u|_{p(x)}^{p^+}\leq
\rho_{p(x)}(u)\leq|u|_{p(x)}^{p^-}, \\
\label{L6}
|u_n-u|_{p(x)} \to 0\; \Leftrightarrow\;\rho_{p(x)}
(u_n-u)\to 0.
\end{gather}

We define the variable exponent  Sobolev space by
$$
W^{1,p(x)}(\Omega)=\{u\in L^{p(x)}(\Omega):|\nabla u|\in L^{p(x)} (\Omega) \}.
$$
On $W^{1,p(x)}(\Omega)$ we may consider one of the following
equivalent norms
$$
\|u\|_{p(x)}=|u|_{p(x)}+|\nabla u|_{p(x)}
$$
or
$$\|u\|=\inf\big\{\mu>0;\;\int_\Omega\Big(| \frac{\nabla
u(x)}{\mu}|^{p(x)}+|\frac{u(x)}{\mu}|^{p(x)}\Big)\,dx\leq 1\big\}\,.
$$
We  define $W_0^{1,p(x)}(\Omega)$ as the closure of the set of compactly
supported $W^{1,p(x)}$-functions with respect to the norm $\|u\|_{p(x)}$.
When smooth functions are dense, we can also use the closure of
$C_0^\infty(\Omega)$ in $W^{1,p(x)}(\Omega)$. Using the Poincar\'e inequality,
the space $W_0^{1,p(x)}(\Omega)$ can be  defined, in an equivalent manner,
as the closure of $C_0^\infty(\Omega)$ with respect to the norm
$$
\|u\|_{p(x)}=|\nabla u|_{p(x)}.
$$
The space $(W^{1,p(x)}_0(\Omega),\|\cdot\|)$ is a separable and
reflexive Banach space.
Moreover, if $0 <|\Omega|<\infty$ and $p_1$, $p_2$ are variable exponents
so that $p_1\leq p_2$  in $\Omega$ then there exists the continuous embedding
$W^{1,p_2(x)}_0(\Omega)\hookrightarrow W^{1,p_1(x)}_0(\Omega)$.
Set
\begin{equation}\label{rho2}
\varrho_{p(x)}(u)=\int_\Omega |\nabla u(x)|^{p(x)}\,dx.
\end{equation}
If $(u_n)$, $u\in W^{1,p(x)}_0(\Omega)$ then the following
properties are true:
\begin{gather}\label{M4}
\|u\|>1\;\Rightarrow\;\|u\|^{p^-}\leq \varrho_{p(x)}(u) \leq\|u\|^{p^+}\,, \\
\label{M5}
\|u\|<1\;\Rightarrow\;\|u\|^{p^+}\leq \varrho_{p(x)}(u) \leq\|u\|^{p^-}\,, \\
\label{M6}
\|u_n-u\|\to 0\;\Leftrightarrow\;\varrho_{p(x)} (u_n-u)\to 0\,.
\end{gather}
Set
$$
p^*(x)=\begin{cases}
\frac{Np(x)}{N-p(x)} &\text{if $p(x)<N$}\\
 +\infty &\text{if $p(x)\geq N$}.
\end{cases}
$$
We point out that if $p,q\in C_+(\overline\Omega)$
and $q(x)<p^\star(x)$ for all $x\in\overline\Omega$ then the
embedding
$W_0^{1,p(x)}(\Omega)\hookrightarrow L^{q(x)}(\Omega)$
is compact.

For a constant function $p$, the variable exponent Lebesgue and Sobolev
spaces coincide with the standard Lebesgue and Sobolev spaces.
Cf. \cite{radrep}, the function spaces with variable exponent have some
striking properties, such as:
\begin{itemize}
\item[(i)]
If $1<p^-\leq p^+<\infty$ and $p:\overline\Omega\to [1,\infty)$ is smooth,
 then the formula
$$
\int_\Omega |u(x)|^pdx=p\int_0^\infty t^{p-1} |\{x\in\Omega ; |u(x)|>t\}|\,dt
$$
has  no variable exponent analogue.

\item[(ii)] Variable exponent Lebesgue spaces do not have the
\emph{mean continuity property}: if $p$ is continuous and nonconstant
in an open ball $B$, then there exists a function $u\in L^{p(x)}(B)$
such that $u(x+h)\not\in L^{p(x)}(B)$ for all $h\in\mathbb{R}^N$
with arbitrary small norm.

(iii) An argument in the development of the theory of function spaces
with variable exponent is the fact that these spaces are never translation
invariant.  The use
of convolution is also limited, for instance  the Young inequality
$$
| f*g|_{p(x)}\leq C\, | f|_{p(x)}\, \| g\|_{L^1}
$$
holds if and only if $p$ is constant.
\end{itemize}
We refer to R\u{a}dulescu \cite{radnla} and the monographs by Diening,
H\"asto, Harjulehto and  Ruzicka \cite{die} and R\u{a}dulescu and
Repov\v{s} \cite{radrep} for additional properties of function spaces with
variable exponent and for a thorough variational analysis of these problems.

\section{Main results}

Let $\Delta_{p(x)}$ denote the \emph{$p(x)$-Laplace operator}, namely
$$
\Delta_{p(x)} u:=\operatorname{div} (|\nabla u|^{p(x)-2}\nabla u).
$$

In this article we study of the  perturbed nonhomogeneous Dirichlet problem
\begin{equation}\label{prob}
\begin{gathered}
 -\Delta_{p(x)} u=\lambda |u|^{p(x)-2}u+|u|^{q(x)-2}u,\quad \text{in } \Omega\\
 u=0,\quad \text{on } \partial\Omega.
\end{gathered}
\end{equation}
We suppose that $p,q\in C_+(\overline\Omega)$ satisfy
\begin{equation} \label{pq}
 p^+<q^-\quad\text{and}\quad
\max\{p(x), q(x)\}<p^*(x)\ \text{for all}\ x\in\overline\Omega.
\end{equation}

The energy functional associated with \eqref{prob} is
$\mathcal{E}:W^{1,p(x)}_0(\Omega)\to\mathbb{R}$ defined by
$$
\mathcal{E}(u)=\int_\Omega\frac{1}{p(x)}(|\nabla u|^{p(x)}-\lambda |u|^{p(x)})dx
 -\int_\Omega\frac{|u|^{q(x)}}{q(x)}dx,
$$
for all $ u\in W^{1,p(x)}_0(\Omega)$.

 Hypothesis \eqref{pq} implies that $\mathcal{E}$ is well-defined and
$\mathcal{E}\in C^1(W^{1,p(x)}_0(\Omega), \mathbb{R})$. We also observe that
$$
\langle \mathcal{E}'(u),v\rangle=\int_\Omega
\big( |\nabla u|^{p(x)-2}\nabla u\cdot\nabla v-\lambda|u|^{p(x)-2} u v \big)dx
-\int_\Omega |u|^{q(x)-2}uv\,dx,
$$
for all $v\in W^{1,p(x)}_0(\Omega)$.

We say that $u$ is a \emph{weak solution} of  \eqref{prob} if $u\in W^{1,p(x)}_0(\Omega)\setminus\{0\}$
and
$$
\int_\Omega \big( |\nabla u|^{p(x)-2}\nabla u\cdot\nabla v-\lambda|u|^{p(x)-2} u v \big)dx
-\int_\Omega |u|^{q(x)-2}uv\,dx=0,
$$
for all $v\in W^{1,p(x)}_0(\Omega)$.

Thus, weak solutions of  \eqref{prob} correspond to the critical points of
the energy functional $\mathcal{E}$.
Let
$$
\lambda^*=\inf\Big\{ \int_\Omega |\nabla u|^{p(x)}dx;\ u\in W^{1,p(x)}_0(\Omega),\ |u|_{p(x)}=1\Big\}.
$$
Our first result in this paper establishes the following existence property.

\begin{theorem}\label{t1}
Assume that hypothesis \eqref{pq} is satisfied and $\lambda<\lambda^*$.
Then problem \eqref{prob} has at least one solution.
\end{theorem}

We recall that the functional $\mathcal{E}:W^{1,p(x)}_0(\Omega)\to\mathbb{R}$ satisfies the
Palais-Smale condition if any sequence $(u_n)\subset W^{1,p(x)}_0(\Omega)$ such that
\begin{equation}\label{ps}
\mathcal{E}(u_n)=
O(1) \text{ and } \|\mathcal{E}'(u_n)\|_{W^{-1,p'(x)}}=o(1)\quad \text{as } n\to\infty ,
\end{equation}
is relatively compact.

An important role in the proof of Theorem \ref{t1} is played by the
mountain pass theorem of Ambrosetti and Rabinowitz \cite{ambrab}.

\begin{theorem}\label{th1}
	Assume that $X$ is a Banach space, $\varphi\in C^1(X,\mathbb{R})$
satisfies the Palais-Smale condition, $u_0,\, u_1\in X$, $\|u_1-u_0\|>\rho>0$
	$$
\max\{\varphi(u_0),\varphi(u_1)\}<\inf[\varphi(u):\|u-u_0\|=\rho]=m_{\rho}
$$
	and
$$c=\inf_{\gamma\in\Gamma}\max_{0\leq t\leq 1}\varphi(\gamma(t))\quad
\text{with}\quad \Gamma=\{\gamma\in C([0,1],X):\gamma(0)=u_0,\gamma(1)=u_1\}.
$$
	Then $c\geq m_{\rho}$ and $c$ is a critical value of $\varphi$.
\end{theorem}

As pointed out by Brezis and Browder \cite{brebro}, the mountain pass theorem
 ``extends ideas already present in Poincar\'e and Birkhoff".
More generally, this result	 is in fact true in Banach-Finsler manifolds.

Assumption \eqref{pq} guarantees that the energy functional associated with
 \eqref{prob} has a mountain pass geometry. We study in what follows a related
perturbed problem, provided that Theorem \ref{th1} cannot be applied.
Consider the problem
\begin{equation}\label{prob1}
\begin{gathered}
 -\Delta_{p(x)} u=\lambda |u|^{p(x)-2}u+\mu |u|^{q(x)-2}u, \quad \text{in } \Omega\\
 u=0, \quad \text{on } \partial\Omega.
\end{gathered}
\end{equation}

We say that $u$ is a \emph{weak solution} of problem \eqref{prob1} if
$u\in W^{1,p(x)}_0(\Omega)\setminus\{0\}$ and
$$
\int_\Omega \big( |\nabla u|^{p(x)-2}\nabla u\cdot\nabla v-\lambda|u|^{p(x)-2} u v \big)dx
-\mu\int_\Omega |u|^{q(x)-2}uv\,dx=0,
$$
for all $v\in W^{1,p(x)}_0(\Omega)$.

We assume that $p,q\in C_+(\overline\Omega)$ satisfy
\begin{equation} \label{pq1}
q^-<p^-\ \text{and}\
\max\{p(x), q(x)\}<p^*(x)\ \text{for all}\ x\in\overline\Omega.
\end{equation}
Under this assumption, the functional $\mathcal{E}$ satisfies the  mountain condition
 of the mountain pass theorem near the origin for small values of the parameter
 $\mu$. However, the existence of a valley for large values of $\mu$
is not guaranteed, hence we do not have a mountain pass geometry.
The second existence result of this paper extends
\cite[Theorem 2.1]{mrpams} and is stated in the following theorem.

\begin{theorem}\label{t2}
Assume that hypothesis \eqref{pq1} is fulfilled and $\lambda<\lambda^*$.
Then there exists $\mu^*>0$ such that for all $\mu<\mu^*$ problem \eqref{prob1}
has at least one solution.
\end{theorem}

The key ingredient in the proof of Theorem \ref{t2} is the Ekeland variational
principle, which asserts the existence of almost critical points of $\mathcal{E}$.
The subcritical framework of the problem yields a nontrivial critical point
of $\mathcal{E}$, hence a weak solution of problem \eqref{prob1}.
We point out that the Ekeland variational principle can be viewed as the
nonlinear version of the Bishop-Phelps theorem \cite{bis1}.

The arguments developed in this paper show that a similar result
 holds if the $p(x)$-Laplace operator is replaced with other nonhomogeneous
differential operators with variable exponent, for instance the
\emph{generalized mean curvature operator} defined by
$$
\operatorname{div} \Big((1+|\nabla u|^2)^{[p(x)-2]/2}\nabla u \Big).
$$

\section{Proof of Theorem \ref{t1}}
The proof strongly relies on the mountain pass theorem in relationship
with some ideas developed in
 \cite{mora} and \cite{pucci}.

We start with the verification of the geometric hypotheses of the mountain pass
theorem. We observe that $\mathcal{E}(0)=0$ and we show the existence of a mountain
near the origin, namely there exist positive numbers $r$ and $\eta$ such that
$\mathcal{E}(u)\geq\eta$ for all $u\in W^{1,p(x)}_0(\Omega)$ with $\|u\|=r$. We first observe that
the definition of $\lambda^*$ combined with the fact that $\lambda<\lambda^*$
imply that there exists $\delta>0$ such that
$$
\int_\Omega\frac{1}{p(x)}(|\nabla u|^{p(x)}-\lambda |u|^{p(x)})dx
\geq \delta |\nabla u|_{p(x)},\quad\text{for all}\ u\in W^{1,p(x)}_0(\Omega).
$$
But
$$
\int_\Omega\frac{|u|^{q(x)}}{q(x)}dx\leq\frac{1}{q^-}|u|_{q(x)},
\quad\text{for all } u\in W^{1,p(x)}_0(\Omega).
$$
Combining these inequalities, we deduce that
\begin{equation} \label{enough}
\mathcal{E}(u)\geq \delta\,|\nabla u|_{p(x)}-\frac{1}{q^-}\,|u|_{q(x)}\,.
\end{equation}

Fix $r\in(0,1)$ and $u\in W^{1,p(x)}_0(\Omega)$ with $\|u\|=r$.
Then relations \eqref{M5}, \eqref{enough} and the Sobolev embedding
$W^{1,p(x)}_0(\Omega)\hookrightarrow L^{q(x)}(\Omega)$ yield
$$
\mathcal{E}(u)\geq\delta\,\|u\|^{p^+}-\frac{C}{q^-}\,\|u\|^{q^-}.
$$
Choosing eventually $r\in(0,1)$ smaller if necessary, we conclude that
there exists $\eta>0$ such that $\mathcal{E}(u)\geq\eta$ for all $u\in W^{1,p(x)}_0(\Omega)$ with
$\|u\|=r$.

Next, we argue the existence of a valley over the chain of mountains.
For this purpose, we fix $s>1$ and $w\in W^{1,p(x)}_0(\Omega)\setminus\{0\}$. It follows that
\begin{equation} \label{abcc}
\begin{aligned}
 \mathcal{E} (sw)
& =\int_\Omega \frac{s^{p(x)}}{p(x)}\left( |\nabla w|^{p(x)}
 -\lambda |w|^{q(x)}\right)dx-\int_\Omega\frac{s^{q(x)}}{q(x)}\,|w|^{q(x)}dx\\
&\leq A \frac{s^{p^+}}{p^-}-B \frac{s^{q^-}}{q^+},
\end{aligned}
\end{equation}
where
$$
A=\int_\Omega \left( |\nabla w|^{p(x)}-\lambda |w|^{q(x)}\right)dx\quad\text{and}\quad
B=\int_\Omega |w|^{q(x)}dx.
$$
Using hypothesis \eqref{pq}, relation \eqref{abcc} yields $\mathcal{E} (sw)<0$ for $s$
large enough.


To apply Theorem \ref{th1} to our problem \eqref{prob} it remains to check
that the energy functional $\mathcal{E}$  satisfies the Palais-Smale compactness condition.
Let $(u_n)\subset W^{1,p(x)}_0(\Omega)$ be an arbitrary Palais-Smale sequence for $\mathcal{E}$, namely
\begin{equation}\label{ps1}
\mathcal{E} (u_n)=O(1)\quad\text{as } n\to\infty
\end{equation}
and
\begin{equation} \label{ps2} 
\|\mathcal{E}' (u_n)\|_{W^{-1,p'(x)}(\Omega)}=o(1)\quad\text{as } n\to\infty\,.
\end{equation}
We claim that
\begin{equation} \label{claim}
\text{the sequence $(u_n)$ is bounded in } W^{1,p(x)}_0(\Omega) .
\end{equation}
Relations \eqref{ps1} and \eqref{ps2} yield
\begin{equation} \label{ps3}
\int_\Omega\frac{1}{p(x)}\Big( |\nabla u_n|^{p(x)}-\lambda |u_n|^{p(x)}\Big)dx
-\int_\Omega\frac{1}{q(x)} |u_n|^{q(x)}dx=O(1)\quad\text{as } n\to\infty
\end{equation}
and for all $v\in W^{1,p(x)}_0(\Omega)$,
\begin{equation} \label{ps4}
\begin{aligned}
&\int_\Omega\left( |\nabla u_n|^{p(x)-2}\nabla u_n\cdot\nabla v
 -\lambda |u_n|^{p(x)-2}u_nv\right)dx-\int_\Omega |u_n|^{q(x)-2}u_nv\,dx \\
&= o(1)\|v\|\quad\text{as } n\to\infty\,.
\end{aligned}
\end{equation}
Choosing $v=u_n$ in \eqref{ps4} we deduce that
\begin{equation} \label{ps5}
\int_\Omega\Big( |\nabla u_n|^{p(x)}-\lambda |u_n|^{p(x)}\Big)dx
 -\int_\Omega |u_n|^{q(x)}dx=o(1)\|u_n\|\quad\text{as } n\to\infty\,.
\end{equation}
On the other hand, relation \eqref{ps3} implies
\begin{align*}
& O(1)+\frac{1}{p^+}\int_\Omega\big( |\nabla u_n|^{p(x)}-\lambda |u_n|^{p(x)}\big)dx \\
&\leq \int_\Omega\frac{1}{q(x)}\, |u_n|^{q(x)}dx \\
&\leq O(1)+\frac{1}{p^-}\int_\Omega\big( |\nabla u_n|^{p(x)}
 -\lambda |u_n|^{p(x)}\big)dx.
\end{align*}
Using now relation \eqref{ps5} we deduce that
\begin{align*}
& O(1)+o(1)\|u_n\|+\frac{1}{p^+}\int_\Omega |u_n|^{q(x)}dx \\
&\leq \int_\Omega\frac{1}{q(x)}\, |u_n|^{q(x)}dx \\
&\leq  O(1)+o(1)\|u_n\|+\frac{1}{p^-}\int_\Omega |u_n|^{q(x)}dx.
\end{align*}
It follows that
\begin{equation} \label{ps6}
\int_\Omega |u_n|^{q(x)}dx=O(1)+o(1)\|u_n\|\quad\text{as } n\to\infty\,.
\end{equation}
Returning to \eqref{ps3} and using relation \eqref{ps6} we deduce that
\begin{equation} \label{ps7}
\int_\Omega\frac{1}{p(x)}\big( |\nabla u_n|^{p(x)}-\lambda |u_n|^{p(x)}\big)dx
=O(1)+o(1)\|u_n\|\,.
\end{equation}
Taking into account the definition of $\lambda^*$ and the fact that
$\lambda<\lambda^*$, relation \eqref{ps7} implies that $(u_n)$ is bounded in
$W^{1,p(x)}_0(\Omega)$, hence Claim \eqref{claim} is argued. So, up to a subsequence
\begin{gather} \label{cow}
u_n\rightharpoonup u\quad\text{in } W^{1,p(x)}_0(\Omega), \\
\label{col}
u_n\to u\quad\text{in}\ L^{p(x)}(\Omega).
\end{gather}

We prove in what follows that
\begin{equation} \label{cra0}(u_n)
\text{ contains a strongly convergent subsequence in $W^{1,p(x)}_0(\Omega)$}.
\end{equation}
We first observe that relation \eqref{ps4} yields
\begin{equation} \label{cra1}
\int_\Omega |\nabla u_n|^{p(x)-2}\nabla u_n\cdot\nabla v\,dx
=\int_\Omega \varrho(x,u_n)v\,dx
+ o(1)\|v\|\quad\text{as}\ n\to\infty\,,
\end{equation}
for all $v\in W^{1,p(x)}_0(\Omega)$ where $\varrho (x,w)=\lambda |w|^{p(x)-2}w+|w|^{q(x)-2}w$.

We assume in what follows that $p^+<N$ (the same arguments can be developed
in the contrary case). Using \eqref{claim}, relation \eqref{cra1} implies that
 our claim \eqref{cra0} follows as soon as we show that $\varrho (x,u_n)$
is relatively compact in $W^{-1,p'(x)}(\Omega)$. Hence, by Sobolev embeddings,
it is enough to show that
\begin{equation} \label{cra2}
\text{$\varrho (x,u_n)$ is relatively compact in $L^{Np(x)/[Np(x)-N+p(x)]}(\Omega)$},
\end{equation}
 which is the dual space of $L^{Np(x)/[N-p(x)]}(\Omega)$.

Using \eqref{col} in combination with the Egorov theorem, we deduce that for
fixed $\eta>0$ there is a subset $A$ of $\Omega$ of measure less than $\eta$
and such that
$$
u_n\to u\quad\text{uniformly on}\ \Omega\setminus A.
$$
So, our claim \eqref{cra2} follows as soon as we show that for for all $\varepsilon>0$
and $n$ large enough
\begin{equation} \label{cra3}
\int_A|\varrho (x,u_n)-\varrho (x,u)|^{Np(x)/[Np(x)-N+p(x)]}dx\leq\varepsilon\,.
\end{equation}
Taking into account the subcritical growth of $\varrho$ we have for some $C>0$
$$
\int_A|\varrho (x,u)|^{Np(x)/[Np(x)-N+p(x)]}dx
\leq C \int_A(1+|u|)^{Np(x)/[N-p(x)]}dx,
$$
which can be made sufficiently small if we choose $\eta>0$ small enough.
 We conclude that
\begin{align*}
&\int_A|\varrho (x,u_n)-\varrho (x,u)|^{Np(x)/[Np(x)-N+p(x)]}dx \\
&\leq \delta \int_A|u_n-u|^{Np(x)/[N-p(x)]}dx+C_\delta |A|<\varepsilon,
\end{align*}
by choosing $\eta$ small enough and by using Sobolev embeddings and
\eqref{claim}. Our claim \eqref{cra0} is proved, which implies that $\mathcal{E}$
satisfies the Palais-Smale condition. Applying Theorem \ref{th1} we deduce
that \eqref{prob} has a nontrivial solution for all $\lambda<\lambda^*$.

The ideas developed in the proof of Theorem \ref{t1} allow to extend this
 result for more general perturbations.
Indeed, let us consider the   nonhomogeneous Dirichlet problem:
\begin{equation}\label{probbis}
\begin{gathered}
 -\Delta_{p(x)} u+a(x) |u|^{p(x)-2}u=|u|^{q(x)-2}u,\quad \text{in } \Omega\\
 u=0,\quad \text{on } \partial\Omega,
\end{gathered}
\end{equation}
where $a\in L^\infty(\Omega)$ and there exists $\alpha>0$ such that
\begin{equation} \label{pq11}
\int_\Omega\left(|\nabla u|^{p(x)}+a(x)|u|^{p(x)}\right)
\geq\alpha\rho_{p(x)}(\nabla u)\quad\text{for all}\ u\in W^{1,p(x)}_0(\Omega).
\end{equation}
The following property is the counterpart of \cite[Theorem 10]{pucci}
in the framework of operators and nonlinearities involving variable exponents.

\begin{theorem}\label{t1bis}
Suppose that hypotheses \eqref{pq} and \eqref{pq11} are satisfied.
Then  \eqref{probbis} has at least one nontrivial solution.
\end{theorem}

\section{Proof of Theorem \ref{t2}}

The weak solutions of \eqref{prob1} correspond to critical points of the
 associated energy functional
 $\mathcal{J}:W^{1,p(x)}_0(\Omega)\to\mathbb{R}$ defined by
$$
\mathcal{J}(u)=\int_\Omega\frac{1}{p(x)}(|\nabla u|^{p(x)}-\lambda |u|^{p(x)})dx
-\mu\int_\Omega\frac{|u|^{q(x)}}{q(x)}dx,
$$
for all $ u\in W^{1,p(x)}_0(\Omega)$.

We first establish a preliminary result, which asserts that the energy
functional $\mathcal{J}$ satisfies the geometric condition around the origin,
provided that $\lambda<\lambda^*$ and $\mu$ is small enough.

\begin{lemma}\label{lema1}
Let $\lambda<\lambda^*$. Then there exist positive numbers $\mu^*$, $r$
and $a$ such that for all $\mu<\mu^*$ we have $\mathcal{J}(u)\geq a$, provided that
 $u\in W^{1,p(x)}_0(\Omega)$ satisfies $\|u\|=r$.
\end{lemma}

\begin{proof}
 As in the proof of Theorem \ref{t1}, combining the definition of $\lambda^*$
 with the fact that $\lambda<\lambda^*$, we deduce that there is a positive
 number $\delta$ such that
\begin{equation} \label{amiens1}
\mathcal{J}(u)\geq \delta |\nabla u|_{p(x)}-\frac{\mu}{q^-}\,|u|_{q(x)}\quad
 \text{for all $u\in W^{1,p(x)}_0(\Omega)$}.
\end{equation}
Let $a_1>0$ denote the best constant corresponding to the continuous embedding
of $W^{1,p(x)}_0(\Omega)$ into $L^{q(x)}(\Omega)$.
Fix arbitrarily $r\in (0,1)$. Using \eqref{L5} we obtain
\begin{equation} \label{amiens2}
\int_\Omega |u|^{q(x)}dx\leq |u|^{q^-}_{q(x)}\leq a_1^{q^-}\|u\|^{q^-}
\quad\text{for all $u\in W^{1,p(x)}_0(\Omega)$ }, \|u\|=\frac{r}{a_1}.
\end{equation}

Next, using \eqref{M5} we obtain
\begin{equation} \label{amiens3}
|\nabla u|_{p(x)}\geq \|u\|^{p^+}\quad\text{for all $u\in W^{1,p(x)}_0(\Omega)$},\quad
 \|u\|=\frac{r}{a_1}.
\end{equation}

We notice that in \eqref{amiens2} and \eqref{amiens3} we can assume that
$a_1$ is large enough in order to have $r/a_1<1$.

Relations \eqref{amiens1}, \eqref{amiens2} and \eqref{amiens3} imply
that for all  $u\in W^{1,p(x)}_0(\Omega)$ with $\|u\|=\frac{r}{a_1}$ we have
\begin{align*}
\mathcal{J}(u)
&\geq\delta\|u\|^{p^+}-\frac{\mu}{q^-}\,a_1^{q^-}\|u\|^{q^-}\\
& =\delta r^{p^+}-\frac{\mu}{q^-}\, a_1^{q^-}r^{q^-}\\
&=r^{q^-}\Big( \delta r^{p^+-q^-}-\frac{\mu}{q^-}\,a_1^{q^-}\Big).
\end{align*}
This relation shows that the lemma follows after choosing
$$
\mu^*=\frac{\delta q^-r^{p^+-q^-}}{2a_1^{q^-}}\quad\text{and}\quad
a=\frac{\delta r^{p^+}}{2}\,.
$$
\end{proof}

The second auxiliary result shows that $\mathcal{J}$ has a valley around the origin,
 hence the hypotheses of the mountain pass theorem are not fulfilled.

\begin{lemma}\label{lema2}
Let $\lambda<\lambda^*$. Then there exists a smooth nonnegative function
$\phi\in W^{1,p(x)}_0(\Omega)$ such that $\mathcal{J}(t\phi)<0$ for all $t>0$ sufficiently small.
\end{lemma}

\begin{proof}
 The basic idea is to use our assumption \eqref{pq1}, more exactly $q^-<p^-$. Set
$$
\eta:=\frac{p^--q^-}{2}\,.
$$
Let $\omega$ be a subdomain of $\Omega$ such that
$$
q(x)\leq q^-+\eta<p^-\quad\text{for all $x\in\omega$}.
$$
Let $\varphi\in W^{1,p(x)}_0(\Omega)$ a smooth function with compact support such that
$0\leq\phi\leq 1$, $\varphi|_\omega =1$ and $\bar\omega\subset{\rm supp}\,(\phi)$.
It follows that for all $t\in (0,1)$ we have
\begin{align*}
\mathcal{J}(t\phi)
& =\int_\Omega\frac{t^{p(x)}}{p(x)}\big( |\nabla\phi|^{p(x)}
 -\lambda\phi^{p(x)}\big)dx
 -\mu\int_\Omega \frac{t^{q(x)}}{q(x)}\,\phi^{q(x)}dx\\
&\leq \frac{t^{p^-}}{p^-}\int_\Omega\big( |\nabla\phi|^{p(x)}-\lambda\phi^{p(x)}\big)dx
 -\frac{\mu}{q^+}\int_\Omega t^{q(x)}\phi^{q(x)}dx\\
&\leq \frac{t^{p^-}}{p^-}\int_\Omega\big( |\nabla\phi|^{p(x)}-\lambda\phi^{p(x)}\big)dx
 -\frac{\mu}{q^+}\int_{\omega} t^{q(x)}\phi^{q(x)}dx\\
&\leq A\frac{t^{p^-}}{p^-}-B\frac{\mu t^{q^-+\eta}}{q^+},
\end{align*}
where
$$
A:=\int_\Omega\left( |\nabla\phi|^{p(x)}-\lambda\phi^{p(x)}\right)dx>0\quad\text{and}\quad
 B:=\int_{\omega} \phi^{q(x)}dx>0.
$$
We deduce that $\mathcal{J}(t\phi)<0$, provided that $t\in(0,1)$ is small enough.
\end{proof}

Fix $\mu^*>0$ as established in Lemma \ref{lema1} and let $\mu<\mu^*$.
Using Lemmata \ref{lema1} and \ref{lema2} we deduce that there exists $r>0$
such that
$$
\inf_{u\in\overline{B_r(0)}}\mathcal{J} (u)<0<\inf_{u\in\partial B_r(0)}\mathcal{J} (u),
$$
where $B_r(0):=\{u\in W^{1,p(x)}_0(\Omega); \|u\|<r\}$.

Fix $\varepsilon>0$ such that
$$
\varepsilon<\inf_{u\in\partial B_r(0)}\mathcal{J} (u)-\inf_{u\in\overline{B_r(0)}}\mathcal{J} (u).
$$
Applying Ekeland's variational principle \cite{eke} we find
$u_\varepsilon\in \overline{B_r(0)}$ such that
\begin{gather} \label{ivar1}
\mathcal{J} (u_\varepsilon)<\inf_{u\in\overline{B_r(0)}}\mathcal{J} (u)+\varepsilon, \\
\label{ivar2}
\mathcal{J} (u_\varepsilon)<\mathcal{J} (u)+\varepsilon \|u-u_\varepsilon\|\quad\text{for all $u\in W^{1,p(x)}_0(\Omega)\setminus\{u_\varepsilon\}$}.
\end{gather}
We claim that $u_\varepsilon\in B_r(0)$. Indeed, relations \eqref{ivar1} and
\eqref{ivar2} yield
$$
J(u_\varepsilon)<\inf_{u\in\overline{B_r(0)}}\mathcal{J} (u)+\varepsilon<\inf_{u\in\partial B_r(0)}\mathcal{J} (u),
$$
hence $u_\varepsilon\not\in\partial B_r(0)$.
A standard argument (see, e.g., \cite[p. 2934]{mrpams}) shows
that $\|\mathcal{J}'(u_\varepsilon)\|\leq\varepsilon$.

Let $c:=\inf_{u\in\overline{B_r(0)}}\mathcal{J} (u)$. It follows that $u_\varepsilon$
is an \emph{almost critical point} of $\mathcal{J}$ at the level $c$, that is,
$$
\lim_{\varepsilon\to 0}\mathcal{J} (u_{\varepsilon})=c\quad\text{and}\quad
\lim_{\varepsilon\to 0}\|\mathcal{J}' (u_{\varepsilon})\|=0.
$$
From now on, using the same argument as in \cite[p. 59]{gazrad}
(see also \cite[Theorem 3.1]{fan} and \cite[p. 2935]{mrpams}) we deduce
that there exists a subsequence $(v_n)$ of $(u_\varepsilon)$ that converges to
a nontrivial critical point $u$ of $\mathcal{J}$, hence $u$ is a weak solution of
problem \eqref{prob1}.


\section*{Perspectives and open problems}

Now we raise some open problems in relationship with the study 
developed in this paper.
\smallskip

\noindent\textbf{Open problem 1.} 
Problem \eqref{prob} has been studied in the  subcritical case, namely under 
the basic hypothesis \eqref{pq}, namely $\max\{p(x),q(x)\}<p^*(x)$ for all 
$x\in\overline\Omega$, which is crucial for the verification of the 
Palais-Smale compactness condition. We consider that a very interesting research 
direction is to study the same problem in the almost critical setting,
hence under the following weaker assumption: there exists $x_0\in\Omega$ such that
\begin{equation} \label{critical}
 q(x)<p^*(x)\quad\text{for all $x\in\overline\Omega\setminus\{x_0\}$ and
$q(x_0)=p^*(x_0)$}.
\end{equation}
Of course, this hypothesis is not possible if the functions $p$ and $q$ are
 constant.
We conjecture that the result stated in Theorem \ref{t1} remains true
under assumption \eqref{critical}.
\smallskip

\noindent\textbf{Open problem 2.}
\cite[Theorem 11]{pucci} studies with values of the parameter $\lambda$
 which are larger than the principal eigenvalue of the Laplace operator. 
In such a case it is established an existence property by using the 
 dual variational method. This is based on the introduction of a new 
unknown and uses in an essential manner the linearity of the Laplace operator. 
We consider that a very interesting open problem is the qualitative analysis 
of problem \eqref{prob} provided that $\lambda\geq\lambda^*$.
\smallskip

\noindent\textbf{Open problem 3.}
 We suggest to adapt the limited developments techniques introduced in the 
proof of Theorem 9 in \cite[pp. 30-32]{pucci} in order to obtain an 
alternative proof of Theorem \ref{t1} in this paper. 
In our framework the major difficulties arise due to the presence of  variable
 exponents, which is not the case in \cite{pucci}.
\smallskip

\noindent\textbf{Open problem 4.}
Inspired by the study developed in \cite[Chapter 3.3]{radrep}, we suggest the 
study of problems \eqref{prob} and \eqref{prob1} if the $p(x)$-Laplace operator
is replaced with a differential operator with several variable exponents,
for instance
$$
\operatorname{div}\Big((|\nabla u|^{p_1(x)-2}+|\nabla u|^{p_2(x)-2})\nabla u \Big).
$$

\noindent\textbf{Open problem 5.}
  We conjecture that the existence properties stated in Theorems \ref{t1} 
and \ref{t2} remain true if the $p(x)$-Laplace operator is replaced with 
a general class of Leray-Lions type operators, as defined in 
\cite[pp. 27-28]{radrep}. We refer to the pioneering paper by Leray and
 Lions \cite{leray} for basic properties of these operators and some 
relevant applications.
\smallskip

\noindent\textbf{Open problem 6.} 
Theorem \ref{t2} establishes an existence property for low perturbations, 
namely provided that the parameter $\mu$ is small enough. We consider that 
a complete bifurcation analysis for this problem, in relationship with 
the values of the variable exponents $p$ and $q$, is a very interesting 
open problem. We do not have partial results on this problem even for 
particular variable exponents or if $\Omega$ is a ball.

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\end{document}