\documentclass[reqno]{amsart}
\usepackage{hyperref}
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\AtBeginDocument{{\noindent\small
\emph{Electronic Journal of Differential Equations},
Vol. 2017 (2017), No. 223, pp. 1--12.\newline
ISSN: 1072-6691. URL: http://ejde.math.txstate.edu or http://ejde.math.unt.edu}
\thanks{\copyright 2017 Texas State University.}
\vspace{8mm}}

\begin{document}
\title[\hfilneg EJDE-2017/223\hfil Bifurcation analysis of elliptic equations]
{Bifurcation analysis of elliptic equations described by nonhomogeneous
differential operators}

\author[H. M\^aagli, R. Alsaedi, N. Zeddini \hfil EJDE-2017/223\hfilneg]
{Habib M\^aagli, Ramzi Alsaedi, Noureddine Zeddini}

\address{Habib M\^aagli (corresponding author)\newline
Department of Mathematics,
College of Sciences and Arts,
King Abdulaziz University,
 Rabigh Campus P.O. Box 344,
Rabigh 21911, Saudi Arabia}
\email{maaglihabib@gmail.com}

\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}

\address{Noureddine Zeddini \newline
Department of Mathematics,
Faculty of Sciences,
Taibah University,
Medina, Saudi Arabia}
\email{noureddinezeddini@yahoo.fr}

\dedicatory{Communicated by Vicentiu D. Radulescu}

\thanks{Submitted June 10, 2017. Published September 19, 2017.}
\subjclass[2010]{35J20, 35P30, 46E35}
\keywords{Variable exponent; nonhomogeneous differential operator;
\hfill\break\indent  Ekeland variational principle; energy estimates}

\begin{abstract}
 In this article, we are concerned with a class of nonlinear partial
 differential elliptic equations  with Dirichlet boundary data.
 The key feature of this paper consists in competition effects of two
 generalized differential operators, which extend the standard operators
 with variable exponent. This class of problems is motivated by phenomena
 arising in non-Newtonian fluids or image reconstruction, which deal with
 operators and nonlinearities with variable exponents. We establish an
 existence property in the framework of small perturbations of the reaction
 term with indefinite potential. The mathematical analysis developed in this
 paper is based on the theory of anisotropic function spaces.
 Our analysis combines variational arguments with energy estimates.
\end{abstract}

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

\section{Introduction}

Partial differential equations driven by nonhomogeneous differential operators
have been a very productive and rich research field in the last few decades because of
the multiple relevant applications in various fields.
We mainly refer to nonlinear stationary problems with associated energy that changes
pointwise its growth properties and ellipticity. Problems with this structure have
been comprehensively analyzed. We refer, e.g., to the seminal works of
Halsey \cite{halsey} and Zhikov \cite{zh1, zh2}, in close connection with the
qualitative and quantitative mathematical analysis of some classes of  anisotropic
materials and their applications to  fields like homogenization and nonlinear
elasticity.

In the  framework of materials with non-homogeneous structure, the standard abstract
analytic approach relying on the classical theory of $L^p$ and $W^{k,p}$
function spaces (Lebesgue and Sobolev) is not satisfactory. We refer to
electro-rheological fluids (also called ``smart fluids") as well as to
image processing, which should enable that the exponent $p$ is varying;
see Chen, Levine and Rao \cite{clr} and Ruzicka \cite{ruzi}.
For instance, we refer to the Winslow effect of some fluids (like
lithium polymetachrylate) in which the viscosity in a certain magnetic or
 electric range is inversely proportional to the field strength.
This corresponds to non-Newtonian electro-rheological fluids, which are
mathematically understood by means of nonlinear equations with one or more
variable exponents.

Such a study corresponds to the abstract setting of  Lebesgue and Sobolev
function spaces  $L^{p(x)}$ and $W^{1,p(x)}$. Here,  $p$ is a nonconstant
smooth real-valued function with given properties.
The abstract theory of function spaces with variable exponent was studied
by Diening, H\"asto, Harjulehto and Ruzicka
 \cite{die} while the recent book by R\u adulescu and Repov\v{s} \cite{radrep}
is devoted to the careful mathematical analysis of some models of nonlinear
problems with one or more variable exponents; see also  Harjulehto, H\"ast\"o,
Le and Nuortio \cite{har1} and R\u adulescu \cite{radnla}. We also refer to
Alsaedi {\it et al.} \cite{alsaedi, amz1}, Mingione {\it et al.}
\cite{mingi1, mingi2, mingi3}, Pucci {\it et al.} \cite{aupucci, puccizh},
Repov\v{s} {\it et al.} \cite{cencel, repovs} for related results.


Recently,  Kim and Kim \cite{kim} introduced an extended class of non-homogeneous
differential operators. The main feature of their work is in relationship with
the thorough mathematical understanding of nonlinear models with lack of
uniform convexity. More precisely, Kim and Kim \cite{kim} studied some classes
 of the boundary-value problems
\begin{equation} \label{pro1}
\begin{gathered}
-\operatorname{div} (\phi (x,|\nabla u|)\nabla u)=f(x,u) \quad\text{in } \Omega\\
u=0\quad\text{on } \partial\Omega,
\end{gathered}
\end{equation}
where $\Omega$ is a smooth bounded domain in $\mathbb{R}^N$.

The reaction term $f:\Omega\times\mathbb{R}\to\mathbb{R}$ fulfills a Carath\'eodory-type
 hypothesis and the function $\phi(x,t)$ behaves as $|t|^{p(x)-2}$ with
$p:\overline\Omega\to (1,\infty)$ continuous. In the case where
$\phi(x,t)=|t|^{p(x)-2}$, then the operator involved in problem \eqref{pro1}
reduces to the $p(x)$-Laplace operator.

In many papers (see, e.g., \cite[Hypothesis (A4), p. 2629]{mrroyal}),
the functional $\Phi$ induced by the principal part of problem \eqref{pro1}
 is assumed to be uniformly convex. This means that there exists $k>0$
such that for each $(x,\xi,\psi)\in\Omega\times\mathbb{R}^N\times\mathbb{R}^N$,
$$
\Phi\Big(x,\frac{\xi+\psi}{2} \Big)
\leq\frac 12\, \Phi (x,\xi)+\frac 12\,\Phi (x,\psi)-k\,|\xi-\psi|^{p(x)}.
$$
However, since the function $\Psi (x,s)=s^p$ is not uniformly convex for
$s\in(0,\infty)$ for $1<p<2$, this condition is not applicable to all
$p$-Laplacian problems. A feature of the abstract setting developed in \cite{kim}
 is that the main results are obtained without any uniform convexity assumption.
 Related properties can be found in the recent paper of Baraket, Chebbi, Chorfi
and R\u adulescu \cite{baraket}.

We  study  some nonlinear phenomena driven by non-homogeneous differential operators.
Our main purpose in this paper is to establish some qualitative properties of
solutions in the framework of small perturbations.

\section{Terminology and preliminary results}\label{sec2}

We suppose that $\Omega\subset\mathbb{R}^N$ is a smooth bounded domain.
Define
$$
C_+(\overline\Omega)=\{p\in C(\overline\Omega): p>1\text{in }\overline\Omega\}.
$$
For $p\in C_+(\overline\Omega)$ we define
$$
p^+=\sup_{x\in\Omega}p(x);\quad  p^-=\inf_{x\in\Omega}p(x).
$$
We define the Banach space
$$
L^{p(x)}(\Omega)=\{u: u \text{ is  measurable and }
\int_\Omega|u|^{p(x)}\,dx<\infty\}
$$
 with the associated Luxemburg norm
$$
|u|_{p(x)}=\inf\big\{\mu>0: \int_\Omega|
\frac{u(x)}{\mu}|^{p(x)}\,dx\leq 1\big\}.
$$
According to \cite{radrep}, $L^{p(x)}(\Omega)$ is reflexive if and only if
 $1 < p^-\leq p^+<\infty$.

The usual continuous embedding property of Lebesgue function spaces extends
to variable exponent spaces. More precisely, if
$\Omega$ has finite measure and $p_1$, $p_2$ are two functions satisfying
$p_1\leq p_2$  in $\Omega$ then there exists a continuous embedding
$L^{p_2(x)}(\Omega)\hookrightarrow L^{p_1(x)}(\Omega)$.


Let $L^{p'(x)}(\Omega)$ denote the conjugate space
of $L^{p(x)}(\Omega)$, where $1/p(x)+1/p'(x)=1$. Then for all
$u\in L^{p(x)}(\Omega)$ and all $v\in L^{p'(x)}(\Omega)$
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 {\it modular} of  $L^{p(x)}(\Omega)$ has a crucial role in
arguments dealing with variable exponent Lebesgue spaces. This {\it modular}
is the map
 $\rho_{p(x)}:L^{p(x)}(\Omega)\to\mathbb{R}$ defined by
$$
\rho_{p(x)}(u)=\int_\Omega|u|^{p(x)}\,dx.
$$
If $u$, $(u_n)\in L^{p(x)}(\Omega)$ and $p^+<\infty$ then the following
properties are true:
\begin{gather}\label{L4}
\text{if $|u|_{p(x)}>1$ then } |u|_{p(x)}^{p^-}\leq\rho_{p(x)}(u)
\leq|u|_{p(x)}^{p^+}, \\
\label{L5}
\text{if $|u|_{p(x)}<1$, then } |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}
Let
$$
W^{1,p(x)}(\Omega)=\{u\in L^{p(x)}(\Omega): |\nabla u|\in L^{p(x)} (\Omega) \}.
$$
This Banach space is usually equipped with the norm
$$
\|u\|_{p(x)}=|u|_{p(x)}+|\nabla u|_{p(x)}
$$
or
$$
\|u\|_{p(x)}=\inf\Big\{\mu>0:\int_\Omega\Big(| \frac{\nabla
u}{\mu}|^{p(x)}+|
\frac{u}{\mu}|^{p(x)}\Big)\,dx\leq 1\Big\}\,.
$$

Zhikov \cite{zh2} showed that smooth functions are  not always dense in
$W^{1,p(x)}(\Omega)$. This property is in relationship with the
{\it Lavrentiev phenomenon}. Roughly speaking, this phenomenon
asserts that there are problems with variational structure such that
the infimum over the family of
smooth functions is  bigger than the infimum over the set of all functions
satisfying the same boundary conditions. We refer to
\cite[pp. 12-13]{radrep} for more details.

Let $W_0^{1,p(x)}(\Omega)$ denote the closure with respect to  $\|u\|_{p(x)}$
of the family of all $W^{1,p(x)}$-functions with compact support.
In the case where  smooth functions are dense, we can use as alternative
approach the closure of the function space
$C_0^\infty(\Omega)$ in $W^{1,p(x)}(\Omega)$. We also point out that
Poincar\'e's inequality enables to define, equivalently, the space
$W_0^{1,p(x)}(\Omega)$ as the closure of $C_0^\infty(\Omega)$ with respect to
$$
\|u\|_{p(x)}=|\nabla u|_{p(x)}.
$$

The vector space $(W^{1,p(x)}_0(\Omega),\|\cdot\|)$ is a reflexive and
separable  Banach space.
Moreover, if $\Omega$ has finite measure and $p_1$, $p_2$ are two functions
satisfying $p_1\leq p_2$  in $\Omega$ then there is a continuous embedding
$W^{1,p_2(x)}_0(\Omega)\hookrightarrow W^{1,p_1(x)}_0(\Omega)$.

Let
\begin{equation} \label{rho2}
\varrho_{p(x)}(u)=\int_\Omega |\nabla u(x)|^{p(x)}\,dx.
\end{equation}

Assume that $(u_n)$, $u\in W^{1,p(x)}_0(\Omega)$. 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
$$
p^*(x)=\begin{cases}
\frac{Np(x)}{N-p(x)}& \text{for } p(x)<N \\
+\infty& \text{for } p(x)\geq N.
\end{cases}
$$

We recall that if $p$ and $q$ belong to $ C_+(\overline\Omega)$
and $q(x)<p^\star(x)$ for every $x\in\overline{\Omega}$ then the
continuous embedding
$W_0^{1,p(x)}(\Omega)\hookrightarrow L^{q(x)}(\Omega)$
is compact.

If the function $p$ is constant, then variable exponent Lebesgue and Sobolev
spaces reduce to the standard Lebesgue and Sobolev spaces.

From \cite{radrep}, some
curious properties are valid in this framework, such as:

(i)
If $p$ is a smooth function, then the following coarea formula
$$
\int_\Omega |w(x)|^pdx=p\int_0^\infty t^{p-1}\,|\{x\in\Omega ;\ |w(x)|>t\}|\,dt
$$
is no longer valid for variable exponent spaces.

(ii) Suppose that $p$ is a nonconstant smooth (continuous) function in a ball
$B$. Then there exists $w\in L^{p(x)}(B)$ such that $w(x+h)\not\in L^{p(x)}(B)$
for all $h\in\mathbb{R}^N$, provided that the norm of $h$ is sufficiently small.

\section{Main result}\label{sec3}

Assume that $p_1, p_2\in C_+(\overline\Omega)$ and let
$\phi,\psi:\Omega\times [0,\infty)\to [0,\infty)$ be functions that
satisfy the following growth assumptions:
\begin{itemize}
\item[(H1)] the functions $\phi(\cdot,\xi)$, $ \psi (\cdot,\xi)$
are measurable in the domain $\Omega$ for every $\xi\geq 0$ and the
mappings $\phi(x,\cdot)$, $\psi(x,\cdot)$ are locally absolutely continuous
 in $[0,\infty)$ for almost all $x\in\Omega$;

\item[(H2)] there are $a_1\in L^{p_1'}(\Omega)$, $a_2\in L^{p_2'}(\Omega)$
 and $b>0$ such that
$$
|\phi (x,|v|)v|\leq a_1(x)+b|v|^{p_1(x)-1},\quad
|\psi (x,|v|)v|\leq a_2(x)+b|v|^{p_2(x)-1}
$$
for almost all $x\in\Omega$ and for every $v\in\mathbb{R}^N$;

\item[(H3)] there exists a positive real number $c$ such that
\begin{gather*}
\phi(x,\xi)\geq c\xi^{p_1(x)-2},\quad
\phi(x,\xi)+\xi\frac{\partial\phi}{\partial\xi}(x,\xi)\geq c\xi^{p_1(x)-2}, \\
\psi(x,\xi)\geq c\xi^{p_2(x)-2},\quad \psi(x,\xi)
+\xi\frac{\partial\psi}{\partial\xi}(x,\xi)\geq c\xi^{p_2(x)-2}
\end{gather*}
for almost all $x\in\Omega$ and for all $\xi>0$.

\end{itemize}
An interesting consequence of these assumptions is that $\phi$ and $\psi$
satisfy a Simon-type inequality. More precisely, if we denote
$$
\Omega_1:=\{x\in\Omega:\ 1<p(x)<2\}\quad\text{and}\quad
\Omega_2:=\{x\in\Omega;\ p(x)\geq 2\},
$$
then
\begin{equation} \label{simonvari}
\begin{aligned}
&\langle\phi (x,|u|)u-\phi (x,|v|)v,u-v\rangle\\
&\geq\begin{cases}
c(|u|+|v|)^{p(x)-2}|u-v|^2 &\text{if } x\in\Omega_1 \text{ and } (u,v)\neq (0,0)\\
4^{1-p^+}c|u-v|^{p(x)} &\text{if } x\in\Omega_2
\end{cases}
\end{aligned}
\end{equation}
is valid for all $u,v\in\mathbb{R}^N$, where $c$ is the constant given in hypothesis (H3).
This inequality is used in \cite{kim} to show that
$A':{W_0^{1,p(x)}(\Omega)}\to W^{-1,p'(x)}(\Omega)$ is both a nonlinear monotone operator and a
$(S_+)$ mapping.
We refer to Simon \cite{simon} for the initial version of inequality
\eqref{simonvari} in the framework of the $p$-Laplace operator.

Consider the  problem
\begin{equation} \label{problem}
\begin{gathered}
\begin{aligned}
&-\operatorname{div} (\phi(x,|\nabla u|)\nabla u)
 -\operatorname{div} (\psi(x,|\nabla u|)\nabla u) \\
&=\lambda a(x)|u|^{r(x)-2}u- b(x)|u|^{s(x)-2}u, \quad x\in \Omega
\end{aligned}\\
u=0, \quad x\in \partial\Omega.
\end{gathered}
\end{equation}
This problem extends in a general setting results that are valid for
standard operators with variable exponent, such as
the $p(x)$-Laplace operator, the mean curvature equation with variable exponent, or
the nonhomogeneous capillarity equation.

We assume that $\lambda$ is a positive parameter and
$r,s\in C_+(\overline\Omega)$. We study problem \eqref{problem} under
the following hypotheses:
\begin{itemize}
\item[(H4)] $a\in L^{q_1(x)}(\Omega)$ and there exists
 $\omega\Subset\Omega$, $|\omega|>0$ such that $a>0$ in $\omega$;
 $b\in L^{q_2(x)}(\Omega)$, $b>0$ almost everywhere in $\Omega$;

\item[(H5)] we have $\max\{ r(x),s(x)\}<\max\{p_1(x),p_2(x)\}
\leq N<\min\{q_1(x),q_2(x)\}$ for all $x\in\overline\Omega$;

\item[(H6)] we have $\inf_{x\in\omega}r(x)<\inf_{x\in\omega}
 (p_1\wedge p_2\wedge s)(x)$.

\end{itemize}
When $p_1, p_2$ are the exponents introduced in (H2) and (H3), we set
$$
p(x):=\max\{ p_1(x),p_2(x)\}\quad\text{for all } x\in\overline\Omega.
$$
Throughout this paper, we say that  $u$ is a (weak) solution of
problem \eqref{problem} if $u\in W_0^{1,p(x)}(\Omega)\setminus\{0\}$ and
$$
\int_\Omega [\phi(x,|\nabla u|)+\psi (x,|\nabla u|)]\nabla u\cdot\nabla v\,dx
=\lambda\int_\Omega a(x)|u|^{r(x)-2}uvdx-\int_\Omega b(x)|u|^{s(x)-2}uv,
$$
for all functions $v\in W_0^{1,p(x)}(\Omega)$.

Our main result of the present paper establishes that problem \eqref{problem}
 has solutions in the case of small perturbation of the reaction term in the
 right-hand side of \eqref{problem}.

\begin{theorem}\label{t1}
Assume that hypotheses {\rm (H1)--(H6)} are fulfilled.
Then there exists a positive real number $\Lambda$ such that \eqref{problem}
has at least one solution for all $\lambda\in(0,\Lambda)$.
\end{theorem}


\section{Proof of Theorem \ref{t1}}

For $x\in\overline\Omega$ we set
$$
\alpha_1(x)=\frac{q_1(x)r(x)}{q_1(x)-r(x)}\quad\text{and}\quad
\alpha_2(x)=\frac{q_2(x)s(x)}{q_2(x)-s(x)}\,.
$$
By hypothesis (H5), $\alpha_1(x)$ and $\alpha_2(x)$ are positive numbers.
Assumption (H5) also yields that
\begin{gather} \label{crr1}
\max\{\alpha_1(x),\alpha_2(x)\}<p^*(x)\quad \text{for } x\in\overline\Omega, \\
\label{crr2}
\max\{ q_1'(x)\alpha_1(x),q_2'(x)\alpha_2(x)\}<p^*(x).
\end{gather}
It follows that  ${W_0^{1,p(x)}(\Omega)}$ is compactly embedded into the spaces
$L^{\alpha_j(x)}(\Omega)$ and $L^{q_j'(x)\alpha_j(x)}(\Omega)$, $j=1,2$.

For functions $\phi$ and $\psi$ satisfying  (H1)-(H3), we define
\begin{equation} \label{A0def}
A_0(x,t):=\int_0^t[\phi (x,s)+\psi(x,s)]s ds.
\end{equation}
Consider the associated functional $A:{W_0^{1,p(x)}(\Omega)}\to\mathbb{R}$ defined by
\begin{equation} \label{Adef}
A(u):=\int_\Omega A_0(x,|\nabla u|)dx,
\end{equation}
where $A_0$ is introduced in \eqref{A0def}.

By \cite[Lemma 3.2]{kim} and since hypotheses (H1) and (H2) are fulfilled,
we obtain that $A\in C^1({W_0^{1,p(x)}(\Omega)},\mathbb{R})$ and its G\^ateaux directional derivative
is given by
\begin{equation} \label{aderiv}
A'(u)(v)=\int_\Omega [\phi (x,|\nabla u|)+\psi(x,|\nabla v|)]
\nabla u\cdot\nabla vdx\quad\text{ for all $u,v\in{W_0^{1,p(x)}(\Omega)}$}.
\end{equation}
Moreover, since conditions (H1)--(H3) are satisfied,
\cite[Lemma 3.4]{kim} implies that the nonlinear mapping
$A:{W_0^{1,p(x)}(\Omega)}\to W^{-1,p'(x)}(\Omega)$ is a strictly monotone operator.
Moreover, this is a $(S_+)$ mapping; namely, if
$$
u_n\rightharpoonup u \ \text{in ${W_0^{1,p(x)}(\Omega)}$ as $n\to\infty$ and }
 \limsup_{n\to\infty}\langle A'(u_n)-A'(u),u_n-u\rangle\leq 0,
$$
then
$$
u_n\to u \text{ in ${W_0^{1,p(x)}(\Omega)}$ as $n\to\infty$.}
$$
It is straightforward that the nonlinear mapping $A$ is weakly lower
semicontinuous, see \cite{kim} for details and proofs.

Define the functionals $B, \mathcal{E}:{W_0^{1,p(x)}(\Omega)}\to\mathbb{R}$ by
\begin{gather*}
B(u)=\lambda\int_\Omega\frac{a(x)}{r(x)}\,|u|^{r(x)}dx
-\int_\Omega \frac{b(x)}{s(x)}\, |u|^{s(x)}dx ,\\
\mathcal{E} (u)=A(u)-B(u).
\end{gather*}

We argue in what follows that $B$ is well-defined in ${W_0^{1,p(x)}(\Omega)}$.
Indeed, for all $u\in{W_0^{1,p(x)}(\Omega)}$ we have
\begin{equation} \label{camp1}
\big|\int_\Omega\frac{a(x)}{r(x)}\big|u|^{r(x)}dx|
\leq\frac{1}{r^-}\, |a|_{q_1(x)}|\,|u|^{r(x)}\,|_{\alpha_1'(x)}
\leq\frac{1}{r^-}\, |a|_{q_1(x)}|u|^{k_1}_{r(x)\alpha_1'(x)}.
\end{equation}
Here, $k_1$ is a positive real number not depending on $u$.
Similarly, there exists $k_2>0$ such that for all $u\in{W_0^{1,p(x)}(\Omega)}$
\begin{equation} \label{camp2}
\big|\int_\Omega\frac{b(x)}{s(x)}\big|u|^{s(x)}dx|
\leq \frac{1}{s^-}\, |b|_{q_2(x)}|u|^{k_2}_{s(x)\alpha_2'(x)}.
\end{equation}
Relations \eqref{camp1} and \eqref{camp2} and the continuous
embeddings of ${W_0^{1,p(x)}(\Omega)}$  into the spaces $L^{r(x)\alpha_1'(x)}(\Omega)$ and
$L^{s(x)\alpha_2'(x)}(\Omega)$ imply that $B$ is well-defined.
Moreover, by standard computation we deduce that $B$ is of class $C^1$
and for all $u,v\in{W_0^{1,p(x)}(\Omega)}$
$$
B'(u)(v)=\lambda\int_\Omega a(x)|u|^{r(x)-2}uvdx-\int_\Omega b(x)|u|^{s(x)-2}uv\,dx.
$$
Returning to \eqref{aderiv}, we conclude that $\mathcal{E}$ is of class $C^1$ on ${W_0^{1,p(x)}(\Omega)}$ and
\begin{align*}
\mathcal{E}'(u)(v)
&=\int_\Omega [\phi (x,|\nabla u|)+\psi(x,|\nabla v|)]\nabla u\cdot\nabla vdx\\
&\quad -\lambda\int_\Omega a(x)|u|^{r(x)-2}uvdx+\int_\Omega b(x)|u|^{s(x)-2}uv\,dx.
\end{align*}
These arguments also show that $u\in{W_0^{1,p(x)}(\Omega)}$ is a nontrivial critical point of
the energy functional $\mathcal{E}$ if and only if $u$ is a (weak) solution of
problem \eqref{problem}.
\smallskip

\noindent\textbf{Step 1.}
 For all $\rho>0$ sufficiently small, we can find $\lambda^*, \eta>0$
such that $\mathcal{E}(u)\geq\eta$, provided that $\|u\|=\rho$ and
$\lambda\in (0,\lambda^*)$.

Using  (H3) and \eqref{L5} we observe that for all $u\in{W_0^{1,p(x)}(\Omega)}$ with $\|u\|<1$
$$
\int_\Omega\int_0^{|\nabla u|}\phi(x,s)s\,ds\,dx
\geq c\int_\Omega\int_0^{|\nabla u|}s^{p_1(x)-1}\,ds\,dx
\geq\frac{c}{p^+_1}\int_\Omega |\nabla u|^{p_1(x)}dx
$$
and
$$
\int_\Omega\int_0^{|\nabla u|}\psi(x,s)s\,ds\,dx
\geq \frac{c}{p^+_2}\int_\Omega |\nabla u|^{p_2(x)}dx.
$$
Thus, for all $u\in{W_0^{1,p(x)}(\Omega)}$ with $\|u\|<1$,
\begin{equation} \label{Eunu}
\begin{aligned}
A(u)&\geq \frac{c}{p^+}\int_\Omega \Big(|\nabla u|^{p_1(x)}+|\nabla u|^{p_2(x)}\big)dx\\
&\geq \frac{c}{p^+}\int_\Omega |\nabla u|^{p(x)}
\geq \frac{c}{p^+}\,\|u\|^{p^+}.
\end{aligned}
\end{equation}
Next, by  H\"older's inequality,
$$
\int_\Omega\frac{a(x)}{r(x)}\,|u|^{r(x)}dx
\leq\frac{1}{r^-}\int_\Omega a(x)|u|^{r(x)}dx\leq\frac{1}{r^-}|a|_{r(x)}
 |u|^{r^-}_{r(x)q_1'(x)}.
$$
Assumption (H5) implies that $r(x)q_1'(x)<p^*(x)$, hence ${W_0^{1,p(x)}(\Omega)}$ is
 continuously embedded in $L^{r(x)q_1'(x)}(\Omega)$.
Relation \eqref{M5} implies the existence of some $C_1>0$ such that for all
$u\in{W_0^{1,p(x)}(\Omega)}$ with sufficiently small norm we have
\begin{equation} \label{Edoi}
\int_\Omega\frac{a(x)}{r(x)}\,|u|^{r(x)}dx
\leq \frac{C_1}{r^-}\,|a|_{r(x)}\, \|u\|^{r^-}.
\end{equation}
Combining relations \eqref{Eunu} and \eqref{Edoi} we obtain that for every
 $u\in{W_0^{1,p(x)}(\Omega)}$ with sufficiently small norm we have
\begin{align*}
\mathcal{E}(u)
&\geq \frac{c}{p^+}\,\|u\|^{p^+}-\frac{\lambda C_1}{r^-} |a|_{r(x)}
 \|u\|^{r^-}\\
&=C_2\,\|u\|^{p^+}-\lambda C_3\, \|u\|^{r^-}\\
&=\|u\|^{r^-}\big( C_2\|u\|^{p^+-r^-}-\lambda C_3\big).
\end{align*}
Then step 1 follows by using hypothesis (H5).
\smallskip

\noindent\textbf{Step 2.}
 There exist $w\in {W_0^{1,p(x)}(\Omega)}$ and $t_0>0$ such that $\mathcal{E} (tw)<0$ for all $t\in (0,t_0)$.

Let $\omega$ be the subdomain of $\Omega$ defined in hypothesis (H4)
and let $p^-_{1,\omega}$, $p^-_{2,\omega}$, $s^-_\omega$, and
$r^-_{\omega}$ denote the infima of $p_1$, $p_2$, $s$, and $r$ in $\omega$. Set
$$
\delta:=\min\{p^-_{1,\omega}, p^-_{2,\omega}, s^-_\omega\}.
$$
By hypothesis (H6), there exists $\varepsilon_0>0$ such that
\begin{equation} \label{finalhyp}
1<r^-_\omega+\varepsilon_0<\delta\,.
\end{equation}

We fix $\omega_1\subset\subset \omega$ such that
$$
r^-_\omega-\varepsilon_0\leq r(x)\leq r^-_\omega+\varepsilon_0.
$$
We also fix $w\in C^\infty_0(\Omega)$ such that
$$
\operatorname{supp} (w)\subset\omega_1\quad\text{and}\quad
0\leq w\leq 1\quad  \text{in } \omega_1.
$$

Let $t\in (0,1)$. We have
$$\begin{array}{ll}
A(tw)&=\int_\Omega A_0(x,t|\nabla w|)dx=\int_\Omega\int_0^{t|\nabla w|}[\phi +\psi ]s\,ds\,dx\\
&\int_\omega\int_0^{t|\nabla w|}[\phi +\psi ]s\,ds\,dx.\end{array}
$$
Using hypothesis (H2) we obtain
\begin{align*}
A(tw)&\leq \int_\omega\int_0^{t|\nabla w|}\big(|a_1(x)|s+bs^{p_1(x)} \big)\,ds\,dx \\
&\quad + \int_\omega\int_0^{t|\nabla w|}\big(|a_2(x)|s+bs^{p_2(x)} \big)\,ds\,dx\\
&\leq\int_\omega \big(|a_1(x)|t|\nabla w|+bt^{p_1(x)}|\nabla w|^{p_1(x)} \big)dx \\
&\quad + \int_\omega \big(|a_2(x)|t|\nabla w|+bt^{p_2(x)}|\nabla w|^{p_2(x)} \big)dx\\
&\leq bt^\delta\Big(\int_\omega |\nabla w|^{p_1(x)}
+\int_\omega |\nabla w|^{p_2(x)} \Big)+C_6t.
\end{align*}
On the other hand, we have
\begin{align*}
B(tw)
&=\lambda\int_\Omega\frac{a(x)}{r(x)}\, t^{r(x)}w^{r(x)}dx
 -\int_\Omega\frac{b(x)}{s(x)}\,t^{s(x)}w^{s(x)}dx\\
&=\lambda\int_\omega\frac{a(x)}{r(x)}\, t^{r(x)}w^{r(x)}dx
 -\int_\omega\frac{b(x)}{s(x)}\,t^{s(x)}w^{s(x)}dx\\
&\geq \lambda\,\frac{t^{r^-_\omega+\varepsilon_0}}{r^+}\int_\omega a(x)w^{r(x)}dx
 -\frac{t^{s^-_\omega}}{s^-}\int_\omega b(x)w^{s(x)}dx\\
&=\lambda C_{7}t^{r^-_\omega+\varepsilon_0}-\frac{t^{\delta}}{s^-}\int_\omega b(x)w^{s(x)}dx.
\end{align*}
We conclude that
\begin{align*}
\mathcal{E}(tw)&=A(tw)-B(tw)\\
&\leq t^\delta\Big[b\int_\omega\big(|\nabla w|^{p_1(x)}
 +|\nabla w|^{p_2(x)} \big)dx+
\frac{1}{s^-}\int_\omega b(x)w^{s(x)}dx\Big] \\
&\quad +C_6t-\lambda C_{7}t^{r^-_\omega+\varepsilon_0}.
\end{align*}
Recalling the choice of $\varepsilon_0$ and the definition of $\delta$
(see relation \eqref{finalhyp}), we deduce that $\mathcal{E}(tw)<0$ for all
 $t>0$ sufficiently small.

\begin{proof}[Proof of Theorem \ref{t1} concluded]
 Combining steps 1 and 2, we deduce that there exist $\lambda^*>0$  and
$\rho>0$  such that for every $\lambda\in (0,\lambda^*)$
$$
\inf_{\|u\|=\rho}\mathcal{E} (u)>0\quad\text{and}\quad \inf_{\|u\|\leq\rho}\mathcal{E} (u)<0.
$$
Fix $\varepsilon>0$ so that
\begin{equation} \label{varepsi}
\varepsilon<\inf_{\|u\| =\rho}\mathcal{E} (u)-\inf_{\|u\|\leq\rho}\mathcal{E} (u).
\end{equation}
Consider the energy functional $\mathcal{E}$ restricted to the complete metric space
 $\overline{B(0,\rho)}\subset{W_0^{1,p(x)}(\Omega)}$. Applying Ekeland's variational principle,
we find $u_\varepsilon\in {W_0^{1,p(x)}(\Omega)}$ with $\|u_\varepsilon\|\leq\rho$ such that
\begin{gather} \label{eki1}
\inf_{\|u\|\leq\rho}\mathcal{E} (u)\leq\mathcal{E}(u_\varepsilon)\leq \inf_{\|u\|\leq\rho}\mathcal{E} (u)+\varepsilon, \\
\label{eki2}
 \mathcal{E}(u)-\mathcal{E}(u_\varepsilon)+\varepsilon\,\|u-u_\varepsilon\|\geq 0\quad\text{for all } u\neq u_\varepsilon.
\end{gather}
The choice of $\varepsilon$ given in \eqref{varepsi} implies that $\|u_\varepsilon\|<\rho$,
hence $u_\varepsilon$ is an interior point of $B(0,\rho)$. Next, a standard
argument based on relation \eqref{eki2} implies that $\|\mathcal{E}'(u_\varepsilon)\|\leq\varepsilon$.

In conclusion, we obtain a bounded sequence $(u_n)\subset{W_0^{1,p(x)}(\Omega)}$ satisfying
$$
\mathcal{E}(u_n)\to \inf_{\|u\|\leq\rho}\mathcal{E} (u)\quad\text{and}\quad
\|\mathcal{E}'(u_n)\|\to 0\quad\text{as}\ n\to\infty.
$$
Thus, passing if necessary to a subsequence, we can assume that $(u_n)$
is weakly convergent to $u\in{W_0^{1,p(x)}(\Omega)}$.

We claim that the sequence $(u_n)\subset {W_0^{1,p(x)}(\Omega)}$ is strongly convergent.
The key argument for this purpose is that the nonlinear mapping
 $A':{W_0^{1,p(x)}(\Omega)}\to W^{-1,p'(x)}(\Omega)$ is an operator of type $(S_+)$.
For this purpose, we observe that the H\"older inequality yields
\begin{equation} \label{conve1}
\begin{aligned}
&\big| \int_\Omega a(x)|u_n|^{r(x)-2}u_n(u_n-u)dx\big| \\
&\leq |a|_{q_1(x)}\big| |u_n|^{r(x)-2}u_n(u_n-u)\big|_{q_1'(x)}\\
&\leq |a|_{q_1(x)}\big| |u_n|^{r(x)-2}u_n\big|_{r(x)/[r(x)-1]}
 |u_n-u|_{\alpha_1(x)}.
\end{aligned}
\end{equation}
Recall that $\alpha_1(x)<p^*(x)$. Thus, up to a subsequence,
the convergence of $(u_n)$ to $u$ is strong  in $L^{\alpha_1(x)}(\Omega)$.
Returning to inequality \eqref{conve1}, we obtain
\begin{equation} \label{conve2}
\int_\Omega a(x)|u_n|^{r(x)-2}u_n(u_n-u)dx\to 0\quad\text{as}\ n\to\infty.\end{equation}
A similar argument shows that
\begin{equation} \label{conve3}
\int_\Omega b(x)|u_n|^{s(x)-2}u_n(u_n-u)dx\to 0\quad\text{as } n\to\infty.
\end{equation}
Relations \eqref{conve2} and \eqref{conve3} combined with the fact that
 $\|\mathcal{E}'(u_n)\|\to 0$ as $n\to\infty$ imply that
\begin{equation} \label{conve4}
\mathcal{E}'(u_n)(u_n-u)-\mathcal{E}'(u)(u_n-u)\to 0\quad\text{as } n\to\infty.
\end{equation}
But
\begin{equation} \label{conve5}
\begin{aligned}
&\mathcal{E}'(u_n)(u_n-u)-\mathcal{E}'(u)(u_n-u) \\\
&=\int_\Omega(\phi (x,|\nabla u_n|)+\psi (x,|\nabla u_n|))\nabla u_n\nabla (u_n-u)dx\\
&\quad -\int_\Omega(\phi (x,|\nabla u|)+\psi (x,|\nabla u|))\nabla u\nabla (u_n-u)dx\\
&\quad -\lambda\int_\Omega a(x)(|u_n|^{r(x)-2}u_n-|u|^{r(x)-2}u)(u_n-u)dx\\
&\quad + \int_\Omega b(x)(|u_n|^{s(x)-2}u_n-|u|^{s(x)-2}u)(u_n-u)dx.
\end{aligned}\end{equation}
Combining relations \eqref{conve2}--\eqref{conve5} we deduce that
\begin{equation} \label{spre}
\begin{aligned}
&\langle A'(u_n)-A'(u),u_n-u\rangle \\
&=\int_\Omega\left(\phi (x,|\nabla u_n|)\nabla u_n+\psi (x,|\nabla u_n|)\nabla u_n
\right)\nabla (u_n-u)dx \\
&\quad -\int_\Omega\left(\phi (x,|\nabla u|)\nabla u+\psi (x,|\nabla u|)\nabla u\right)
 \nabla (u_n-u)dx \to 0\quad\text{as } n\to\infty.
\end{aligned}
\end{equation}
Recall that the operator $A'$ is a $(S_+)$-type mapping and
$u_n\rightharpoonup u$. Thus, using relation \eqref{spre},
 we deduce the strong convergence of $(u_n)$ to $u$. It follows that
$\mathcal{E}(u_n)\to \mathcal{E}(u)= \inf_{\|w\|\leq\rho}\mathcal{E} (w)<0$, hence $u$
is a nontrivial critical point of $\mathcal{E}$.
\end{proof}

\subsection*{Perspectives}
 Problem \eqref{problem} has been studied in the {\it subcritical case},
see relations \eqref{crr1} and \eqref{crr2}. In our setting, these
assumptions are crucial  to establish that the bounded sequence of almost
 critical points of the energy functional $\mathcal{E}$ is, in fact,
strongly convergent (passing eventually to a subsequence) in ${W_0^{1,p(x)}(\Omega)}$.
We suggest to the reader the approach of a similar problem in the
{\it almost critical} framework, namely subject to the following hypothesis:
there are $x_0, x_1\in\Omega$ such that
\begin{equation} \label{crr3}
\max\{\alpha_1(x_0),\alpha_2(x_0)\}=p^*(x_0);\quad
\max\{\alpha_1(x),\alpha_2(x)\}<p^*(x)\quad \text{for all }
 x\in\overline\Omega\setminus\{x_0\}
\end{equation}
and
\begin{equation} \label{crr4}
\begin{gathered}
\max\{ q_1'(x_1)\alpha_1(x_1),q_2'(x_1)\alpha_2(x_1)\}=p^*(x_1); \\
\max\{ q_1'(x)\alpha_1(x),q_2'(x)\alpha_2(x)\}<p^*(x)\quad
 \text{for all } x\in\overline\Omega\setminus\{x_1\}
\end{gathered}
\end{equation}
In our opinion, the result established in Theorem \ref{t1} remains valid
if both hypotheses \eqref{crr3} and \eqref{crr4}
are fulfilled.


Motivated by the results developed by Chen, Levine and Rao \cite{clr}
in connection with models from image restoration, we consider that
a rich field of investigation concerns the study of energy functionals of the type
$$
{W_0^{1,p(x)}(\Omega)}\ni u\mapsto A(u)+\int_\Omega |u(x)-I(x)|^2dx,
$$
where $A$ is defined in \eqref{Adef}, $I$ is a given input corresponding
to shades of gray in the domain $\Omega$, and $1\leq p_1(x),p_2(x)\leq 2$.
Cf. \cite{clr}, the variable exponents $p_1$ and $p_2$ are close to 1
in regions where it is assumed to be edges, and
close to 2 in the contrary case. In order to have
information on the relative location
of edges, this can be performed either by smoothing the input data or by looking
for the region where the gradient is large.
We refer to \cite[pp. 5-6]{radrep} for related results, including the
staircase effect.

Another important step is to extend the approach
corresponds to {\it double phase} problems, as introduced and developed by
 Mingione {\it et al.} \cite{mingi1, mingi2,mingi3}.
In this framework the associated energy is either
$$
w\mapsto \int_\Omega [|\nabla w|^{p_1(x)}+V|\nabla w|^{p_2(x)}]dx
$$
or
$$
w\mapsto \int_\Omega [|\nabla w|^{p_1(x)}+V|\nabla w|^{p_2(x)}\log(e+|x|)]dx,
$$
where $p_1(x)\leq p_2(x)$, $p_1\neq p_2$, and $V(x)\geq 0$.
Considering two materials having corresponding hardening exponents
$p_1$ and $p_2$, the potential $V(x)$ characterizes the geometry of a
composite of these  materials. More precisely, if $V>0$ then the associated
$p_2(x)$-material is present in the composite.  In the contrary case,
the $p_1(x)$-material is the only that contributes to the structure of the composite.

Problems with this structure extend the pioneering contributions of
Paolo Marcellini \cite{marce1, marce2} concerning variational functionals as
$u\mapsto\int_\Omega F(x,\nabla u)dx$, where
 $F:\Omega\times{\mathbb R}^N\to {\mathbb R}$ fulfills asymmetrical growth
properties of the type
$$
|\eta|^p \lesssim F(x,\eta)\lesssim |\eta|^q,\quad
\text{for all $(x,\eta)\in\Omega\times{\mathbb R}^N$,}
$$
provided that $1<p<q$.

 We anticipate that the methods introduced in the present paper also work
in a more general framework corresponding to Orlicz-Sobolev-Musielak
function spaces (we refer to \cite[Chaper 4]{radrep} for a rigorous
treatment of several models  of stationary problems in Orlicz-Sobolev-Musielak
spaces).

\subsection*{Acknowledgments}
This work was funded by the Deanship of Scientific Research(DSR),
King Abdulaziz University, Jeddah, under grant No. 662-182-D1437.
The authors, therefore acknowledge with thanks DSR technical and financial support.


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\end{document}