\documentclass[reqno]{amsart}
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\AtBeginDocument{{\noindent\small
Two nonlinear days in Urbino 2017,\newline
\emph{Electronic Journal of Differential Equations},
Conference 25 (2018), pp. 27--37.\newline
ISSN: 1072-6691. URL: http://ejde.math.txstate.edu or http://ejde.math.unt.edu}
\thanks{\copyright 2018 Texas State University.}
\vspace{8mm}}


\begin{document} \setcounter{page}{27}
\title[\hfilneg EJDE-2018/conf/25\hfil Generalized biharmonic problems]
{Generalized biharmonic problems with variable exponent and Navier boundary condition}

\author[R. Alsaedi, V. D. R\u{a}dulescu \hfil EJDE-2018/conf/25\hfilneg]
{Ramzi Alsaedi, Vicen\c{t}iu D. R\u{a}dulescu}

\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{Vicen\c{t}iu D. R\u{a}dulescu \newline
Faculty of Applied Mathematics,
AGH University of Science and Technology,
al. Mickiewicza 30, 30-059 Krak\'ow, Poland. \newline
Department of Mathematics, University of Craiova,
Street A.I. Cuza No. 13, 200585 Craiova, Romania.\newline
Institute of Mathematics, Physics and Mechanics, 
Jadranska 19, 1000 Ljubljana, Slovenia}
\email{radulescu@inf.ucv.ro}

\thanks{Published Sepember 15, 2018.}
\subjclass[2010]{35J60, 35J20, 46E35}
\keywords{Generalized biharmonic operator; Navier boundary condition; 
\hfill\break\indent variable exponent}

\begin{abstract}
 We study a class of biharmonic problems with Navier boundary condition
 and involving a generalized differential operator and competing
 nonlinearities with variable exponent. The main result of this paper
 establishes a sufficient condition for the existence of nontrivial
 weak solutions, in relationship with the values of a positive parameter.
 The proofs combine variational methods with analytic arguments.
 The approach developed in this paper allows the treatment of several
 classes of nonhomogeneous biharmonic problems with variable growth arising
 in applied sciences, including the capillarity equation and the mean
 curvature problem.
\end{abstract}

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

\section{Introduction}

The interest in recent years to the mathematical analysis of
partial differential equations driven by nonhomogeneous differential
operators is motivated by their numerous applications to various fields.
We refer, e.g., to phenomena in the applied sciences that are characterized 
by the fact that the associated energy density changes its ellipticity
and growth properties according to the point. Such models have been studied
starting with the pioneering papers by Halsey \cite{halsey} and 
Zhikov \cite{zh1, zh2}, in close relationship with the qualitative mathematical 
analysis of strongly anisotropic materials
in the context of the homogenization and nonlinear elasticity.

In the framework of materials with non-homogeneities, the standard approach
based on the classical theory of $L^p$ and $W^{1,p}$ Lebesgue and Sobolev 
spaces is inadequate. We refer to electrorheological (smart) fluids or to 
phenomena in 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 an electrical field is 
inversely proportional to the strength of the field. The field induces 
string-like formations in the fluid,
which are parallel to the field. They can raise the viscosity by as much as five
orders of magnitude. This corresponds to electrorheological (non-Newtonian) 
fluids, which are mathematically described by means of nonlinear equations 
with variable exponent.
Such a study corresponds to the abstract setting of variable exponents Lebesgue 
and Sobolev spaces, $L^{p(x)}$ and $W^{1,p(x)}$, where $p$ is a real-valued function.
The theory of function spaces with variable exponent has been rigorously developed 
in the monograph of Diening, H\"asto,  Harjulehto and Ruzicka
\cite{die} while the recent book by R\u{a}dulescu and Repov\v{s} \cite{radrep} 
is devoted to the thorough variational and topological analysis of several 
classes of problems with one or more variable exponents; see also  the survey
 papers of Harjulehto, H\"ast\"o,  Le and Nuortio \cite{har1} and 
R\u{a}dulescu \cite{radnla}. We also refer to  Mingione  et al.\
 \cite{mingi1, mingi2, mingi3},
Cencelj, R\u{a}dulescu and Repov\v{s} \cite{cencel0}
Cencelj, Repov\v{s} and Virk \cite{cencel}, and Repov\v{s} \cite{repovs} 
for related results. The abstract setting of $p(x)$-biharmonic problems with 
singular weights has been recently considered by Kefi and R\u{a}dulescu \cite{kefi} 
in relationship with microelectromechanical phenomena, surface diffusion on solids, 
thin film theory, flow in Hele-Shaw cells and phase field models
of multiphasic systems. The present paper complements  some results contained 
in \cite{kefi} to more general operators. In such a way, we extend the approach 
developed in Chorfi and R\u{a}dulescu \cite{chorfi} to generalized biharmonic 
operators.

The study of  elliptic problems with variable exponent has been recently 
extended by Kim and Kim \cite{kim} to a new class of non-homogeneous 
differential operators. Their contribution is a step forward in the analysis 
of nonlinear problems with variable exponent since it enables the 
understanding of problems with possible lack of uniform convexity. 
More precisely, in \cite{kim} they studied problems of the type
\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 $\Omega\subset\mathbb R^N$ is a bounded domain with smooth boundary. 
The nonlinear term $f:\Omega\times\mathbb R\to\mathbb R$ satisfies the Carath\'eodory 
condition and the function $\phi(x,t)$ is of type $|t|^{p(x)-2}$ with 
$p:\overline\Omega\to (1,\infty)$ continuous.

 In the special case when $\phi(x,t)=|t|^{p(x)-2}$, the operator involved 
in problem \eqref{pro1} reduces to the $p(x)$-Laplacian, that is, 
$$
\Delta_{p(x)} u=\operatorname{div} (|\nabla u|^{p(x)-2}\nabla u).
$$ 
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, namely, there exists $k>0$ such that for 
all $x\in\Omega$ and all $\xi,\psi\in\mathbb R^N$,
$$
\Phi\Big(x,\frac{\xi+\psi}{2} \Big)
\leq\frac 12\, \Phi (x,\xi)+\frac 12\,\Phi (x,\psi)-k\,|\xi-\psi|^{p(x)}.
$$
However, since the function $\Psi (x,t)=t^p$ is not uniformly convex for 
$t>0$ and $1<p<2$, this condition is not applicable to all $p$-Laplacian problems. 
An important feature of the abstract setting developed in \cite{kim} 
is that the main results are obtained without any uniform convexity assumption.

In this article, we extend some results of \cite{kefi} in the framework 
of general biharmonic operators with variable exponent, as studied in \cite{kim}. 
We develop the study of biharmonic problems with Navier boundary condition 
for equations driven by the operator
$\Delta (\phi (x,|\Delta u|)\Delta u)$, 
where $\phi$ is as in \eqref{pro1}. Notice that if $\phi(x,t)=|t|^{p(x)-2}$, 
then we obtain the $p(x)$-biharmonic operator defined by 
$\Delta^2_{p(x)}u=\Delta(|\Delta u|^{p(x)-2}\Delta u)$. 



\section{Abstract framework and preliminary results}\label{sec2}

Throughout this article we assume that $\Omega\subset\mathbb R^N$ is a bounded 
domain with smooth boundary.
Set
$$
C_+(\overline\Omega)=\{h\in C(\overline\Omega),\;h(x)>1\text{ for all }
 x\in\overline\Omega\}.
$$
Assume that $p\in C_+(\overline\Omega)$ and let
$$
p^+=\sup_{x\in\Omega}p(x)\quad\text{and}\quad
p^-=\inf_{x\in\Omega}p(x).
$$
We define the Lebesgue space with variable exponent  by
$$
L^{p(x)}(\Omega)=\big\{u: u \text{ is measurable and }
\int_\Omega|u(x)|^{p(x)}\,dx<\infty\big\}.
$$
This function space is a Banach space if it is endowed with the norm
$$
|u|_{p(x)}=\inf\big\{\mu>0;\;\int_\Omega|
\frac{u(x)}{\mu}|^{p(x)}\,dx\leq 1\big\}.
$$
This norm is also called the Luxemburg norm.
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 standard inclusion between Lebesgue spaces generalizes to the framework 
of spaces with variable exponent, namely if
$0 <|\Omega|<\infty$ and $p_1$, $p_2$ are variable exponents such 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 $L^{p'(x)}(\Omega)$ 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 $v\in L^{p'(x)}(\Omega)$  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}


An important role in analytic arguments on Lebesgue spaces with variable exponent 
is played by the \emph{modular} of  $L^{p(x)}(\Omega)$, which
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 $(u_n)$, $u\in L^{p(x)}(\Omega)$ and $p^+<\infty$ then the following
properties hold:
\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
$$
\|u\|_{p(x)}=\inf\big\{\mu>0;\;\int_\Omega\Big(| \frac{\nabla
u(x)}{\mu}|^{p(x)}+|
\frac{u(x)}{\mu}|^{p(x)}\Big)\,dx\leq 1\big\}\,.
$$
Zhikov \cite{zh2} showed that smooth functions are in general not dense in
$W^{1,p(x)}(\Omega)$. This property is in relationship with the 
\emph{Lavrentiev phenomenon}, which
asserts that there exist variational problems for which the infimum over the
smooth functions is strictly greater than the infimum over all functions that
satisfy 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 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 vector 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 
such that $p_1\leq p_2$  in $\Omega$ then there exists a 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 hold:
\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 any positive integer $k$,  let 
$$
W^{k,p(x)}(\Omega)=\{u\in L^{p(x)}(\Omega):
 D^{\alpha}u\in L^{p(x)}(\Omega), |\alpha|\leq k\},
$$ 
where $\alpha=(\alpha_1,\alpha_2,\dots ,\alpha_N) $ is a multi-index, 
$|\alpha|=\sum_{i=1}^{N}\alpha_i$ and
$$
D^{\alpha}u=\frac{\partial^{|\alpha|}u}{\partial^{\alpha_1}x_1
\ldots\partial^{\alpha_N}x_N}.
$$
Then $W^{k,p(x)}(\Omega)$ is a separable and reflexive Banach space equipped
 with the norm 
$$
\|u\|_{k,p(x)}=\sum_{|\alpha|\leq k}|D^{\alpha}u|_{p(x)}.
$$
The space $W_{0}^{k,p(x)}(\Omega)$ is the closure of $C_{0}^{\infty}(\Omega)$ 
in $W^{k,p(x)}(\Omega)$.

Next, we recall some properties of the space 
$$
\mathcal{X} := W^{1,p(x)}_{0}(\Omega)\cap W^{2,p(x)}(\Omega).
$$
For any $u\in \mathcal{X}$ we have $\|u\| = \|u\|_{1,p(x)}+\|u\|_{2,p(x)}$, 
thus  
$$
\|u\| =|u|_{p(x)}+|\nabla u|_{p(x)}+\sum_{|\alpha|=2}|D^{\alpha}u|_{p(x)}.
$$
In Zang and Fu \cite{ZF}, the equivalence of the norms was proved, 
and they even established that the norm
$|\Delta u|_{p(x)}$ is equivalent to the norm $\|u\| $ 
(see \cite[Theorem 4.4]{ZF}).
 Note that $(\mathcal{X}, \|\cdot\|)$ is  a separable and reflexive Banach space.

We recall that the critical Sobolev exponent is defined as 
 $$
p^*(x)=\begin{cases}
\frac{Np(x)}{N-2p(x)},&\text{if }\ p(x)<\frac{N}{2},\\
+\infty, &\text{if } p(x)\geq \frac{N}{2}. 
\end{cases}
$$
Assume that $q \in C^+(\overline{\Omega})$ and $q(x) < p^*(x)$ for any $x\in\Omega$. 
Then, by \cite[Theorem 3.2]{AA}, the function space $\mathcal{X}$ is
continuously and compactly embedded in $L^{q(x)}(\Omega)$.

For a constant function $p$, the variable exponent Lebesgue and Sobolev
spaces coincide with the standard Lebesgue and Sobolev spaces. 
As pointed out in \cite{radrep}, the function spaces with variable exponent
 have some striking properties, such as:

(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.

(ii) Variable exponent Lebesgue spaces do \emph{not} have the \emph{mean continuity
property}. More precisely, 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) The function spaces with variable exponent
 are \emph{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.

\section{Main result}\label{sec3}

In this article we assume that $\Omega\subset\mathbb R^N$ is a bounded domain with 
smooth boundary.

Let $p\in C_+(\overline\Omega)$ and consider the function 
$\phi:\Omega\times [0,\infty)\to [0,\infty)$ satisfying the following hypotheses:
\begin{itemize}

\item[(H1)] the mapping $\phi(\cdot,\xi)$ is measurable on $\Omega$ for all 
$\xi\geq 0$ and $\phi(x,\cdot)$ is locally absolutely continuous on $[0,\infty)$ 
for almost all $x\in\Omega$;

\item[(H2)] there exists  $b>0$ such that
$$
|\phi (x,|v|)v|\leq b|v|^{p(x)-1}
$$
for almost all $x\in\Omega$ and for all $v\in\mathbb R^N$;

\item[(H3)] there exists $c>0$ such that
$$
\phi(x,\xi)\geq c\xi^{p(x)-2},\quad \phi(x,\xi)
+\xi\frac{\partial\phi}{\partial\xi}(x,\xi)\geq c\xi^{p(x)-2}
$$
for almost all $x\in\Omega$ and for all $\xi>0$.
\end{itemize}

An interesting consequence of theses assumptions is that $\phi$ satisfies 
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 the following estimate holds for all $u,v\in\mathbb R^N$
\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 $x\in\Omega_1$ and } (u,v)\not=(0,0)\\
4^{1-p^+}c|u-v|^{p(x)}&\text{if } x\in\Omega_2,
\end{cases}
\end{aligned}
\end{equation}
where $c$ is the positive constant from hypothesis (H3).

Let $A:W_0^{1,p(x)}(\Omega)\to\mathbb R$ defined by
 $$
A(u)=\int_\Omega\int_0^{|\nabla u(x)|}s\phi(x,s)\,ds\,dx.
$$
  Inequality \eqref{simonvari} was used in \cite{kim} to show that 
$A'\colon{W_0^{1,p(x)}(\Omega)}\to W^{-1,p'(x)}(\Omega)$ is both a monotone operator and a 
mapping of type $(S_+)$.
We refer to Simon \cite{simon} for the initial version of inequality 
\eqref{simonvari} in the framework of the $p$-Laplace operator.

We study the following  biharmonic problem with variable growth, competing 
reaction terms, and Navier boundary condition
\begin{equation} \label{problem}
\begin{gathered}
\Delta (\phi(x,|\Delta u|)\Delta u)+\phi(x,| u|) u
 =|u|^{q(x)-2}u-  |u|^{r(x)-2}u \quad \text{in}\ \Omega, \\
u=\Delta u=0,    \quad \text{on }\partial\Omega,
\end{gathered}
\end{equation}
where $q$, $r$ are continuous functions.

 If $\phi (x,\xi)=\xi^{p(x)-2}$ then we obtain the standard 
$p(x)$-Laplace biharmonic operator, that is, 
$\Delta^2_{p(x)}u:=\Delta (|\Delta u|^{p(x)-2}\Delta u)$.

Our abstract setting includes the case $\phi (x,\xi)=(1+\xi^2)^{(p(x)-2)/2}$, 
which corresponds to the generalized biharmonic mean curvature operator
$$
\Delta [(1+|\Delta u|^2)^{(p(x)-2)/2}\Delta u ].
$$
The biharmonic capillarity equation corresponds to
$$
\phi(x,\xi)=\Big(1+\frac{\xi^{p(x)}}{\sqrt{1+\xi^{2p(x)}}}\Big)\xi^{p(x)-2},\quad
 x\in\Omega,\; \xi>0,
$$ 
hence the corresponding capillary phenomenon is described by the differential 
operator
$$
\Delta \Big[\Big( 1+\frac{|\Delta u|^{p(x)}}{\sqrt{1+|\Delta u|^{2p(x)}}} 
\Big)|\Delta u|^{p(x)-2}\Delta u\Big].
$$

We say that $u$ is a solution of problem \eqref{problem} if 
$u\in \mathcal{X}\setminus\{0\}$ with $\Delta u=0$ on $\partial\Omega$ and
$$
\int_\Omega \left[\phi(x,|\Delta u|)\Delta u\Delta v+\phi(x,|u|)uv\right]dx
=\int_\Omega |u|^{q(x)-2}uvdx-\int_\Omega |u|^{r(x)-2}uv,
$$
for all $v\in \mathcal{X}$.
The main result of this paper is the following.

\begin{theorem} \label{thm1.1}
Assume that hypotheses{\rm  (H1)--(H3)} are fulfilled and that
\begin{equation} \label{ipoteza}
1<q(x)<r(x)<p(x)<p^*(x)\quad\text{for all}\ x\in\overline\Omega.
\end{equation}
Then problem \eqref{problem} has at least one nontrivial weak solution
 with negative energy.
\end{theorem}

In the present paper, problem \eqref{problem} is studied for the 
\emph{subcritical case}, namely under the basic hypothesis \eqref{ipoteza},
which is crucial for compactness arguments. We consider that a very 
interesting research direction is to study the same problem in the 
\emph{almost critical} setting, hence under the following assumption:
there exists $x_0\in\Omega$ such that
\begin{equation} \label{critical}
\begin{gathered}
 q(x)<r(x)<p(x)<p^*(x)\quad \text{for all } x\in\Omega\setminus\{x_0\} \text{ and}\\
 q(x_0)=r(x_0)=p(x_0)=p^*(x_0).
\end{gathered}
\end{equation}
Of course, this hypothesis is not possible if the functions $p$, $q$ and $r$ 
are \emph{constant}.
We conjecture that the result stated in Theorem \ref{thm1.1} remains true 
under assumption \eqref{critical}.

\section{Proof of Theorem \ref{thm1.1}} 
Denote 
$$
\Phi(x,t):=\int_0^ts\phi(x,s)ds\quad\text{for all}\ x\in\Omega.
$$
The energy functional associated to problem \eqref{problem} is 
$\mathcal{E}:\mathcal{X}\to\mathbb R$ defined by
$$
\mathcal{E} (u)=\int_\Omega\Phi(x,|\Delta u|)dx+\int_\Omega\Phi(x,| u|)dx
-\int_\Omega\frac{|u|^{q(x)}}{q(x)}dx+\int_\Omega\frac{|u|^{r(x)}}{r(x)}dx.
$$
By hypothesis \eqref{ipoteza},
the function space $\mathcal{X}$ is continuously embedded into $L^{q(x)}(\Omega)$, 
$L^{r(x)}(\Omega)$, and $L^{p(x)}(\Omega)$.
We deduce that $\mathcal{E}$ is well-defined.

On the other hand, with the same arguments as in \cite[Proposition 3.3]{kefi},
the energy functional $\mathcal{E}$ is sequentially lower semicontinuous and 
of class $C^1$. Moreover, the mapping $\mathcal{E}':\mathcal{X}\to\mathcal{X}^*$ is a strictly
 monotone, bounded homeomorphism and is of type $(S_+)$; that is, if
$$
u_n\rightharpoonup u \text{ in $\mathcal{X}$ and } 
\limsup_{n\to\infty}\mathcal{E}'(u_n)(u_n-u)\leq 0,
$$
then $u_n\to u$ in $\mathcal{X}$.

\begin{proof}
We split the proof of Theorem \ref{thm1.1} into several steps.
\smallskip

\noindent\textbf{Step 1.}
The energy functional $\mathcal{E}$ is coercive.
Using (H3), we have for all $u\in\mathcal{X}$
$$
\mathcal{E}(u)\geq c\int_\Omega\frac{|\Delta u|^{p(x)}}{p(x)}dx
+c\int_\Omega\frac{|u|^{p(x)}}{p(x)}dx
-\int_\Omega\frac{|u|^{q(x)}}{q(x)}dx+\int_\Omega\frac{|u|^{r(x)}}{r(x)}dx.
$$
Therefore,
\begin{align*}
\mathcal{E}(u)
&\geq \frac{c}{p^+}\int_\Omega |\Delta u|^{p(x)}dx
 +\frac{c}{p^+}\int_\Omega | u|^{p(x)}dx
 -\frac{1}{q^-}\int_\Omega |u|^{q(x)}dx
 +\frac{1}{r^+}\int_\Omega |u|^{r(x)}dx\\
&\geq\frac{c}{p^+}\int_\Omega |\Delta u|^{p(x)}dx
 +\frac{c}{p^+}\int_\Omega | u|^{p(x)}dx
 -\frac{1}{q^-}\int_\Omega |u|^{q(x)}dx.
\end{align*}
It follows that for all $u\in\mathcal{X}$ with $\|u\|>1$ we have
$$
\mathcal{E}(u)\geq\frac{c}{p^+}\,\|u\|^{p^-}-\frac{1}{q^-}\, |u|_{q(x)}^{q^+}dx.
$$
We conclude the proof of Step 1 by using hypothesis \eqref{ipoteza}, 
more precisely the fact that $q^-<p^+$.

The next step shows that the energy $\mathcal{E}$ does not satisfy one of 
the geometric hypotheses of the mountain pass theorem. More precisely, 
we show that there exists a ``valley" for $\mathcal{E}$ close to the origin, 
so not far away from the origin, as it is required by the Ambrosetti-Rabinowitz 
theorem.
\smallskip

\noindent\textbf{ Step 2.} 
There exists $v\in\mathcal{X}$ such that $\mathcal{E}(tv)<0$ for all small enough $t>0$.
Since $q^-<r^-$, let $\varepsilon>0$ be such that $q^-+\varepsilon<r^-$. By continuity, 
there exists $\omega\Subset \Omega$ such that
$$
|q(x)-q^-|\leq\varepsilon\quad\text{for all}\ x\in\omega.
$$
Let $v\in C^\infty_c(\Omega)$ such that $\operatorname{supp}(v)\subset\omega$, 
$0\leq v\leq 1$, and $v\equiv 1$ in a subset of $\operatorname{supp}(v)$.

Hypothesis (H2) yields that for all $u\in\mathcal{X}$,
\begin{gather*}
\Phi(x,|\Delta u|)\leq\big|\int_0^{|\Delta u|}b|s|^{p(x)-1}ds\big|
\leq b\frac{|\Delta u|^{p(x)}}{p(x)}, \\
\Phi(x,| u|)\leq\left|\int_0^{| u|}b|s|^{p(x)-1}ds\right|
\leq b\,\frac{| u|^{p(x)}}{p(x)}.
\end{gather*}
It follows that
$$
\int_\Omega [\Phi(x,|\Delta u|)+\Phi(x,| u|)]dx
\leq b\int_\Omega \frac{|\Delta u|^{p(x)}+|u|^{p(x)}}{p(x)}dx.
$$
 It follows that for all $t\in (0,1)$ we have
\begin{align*}
\int_\Omega \left[\Phi(x,|\Delta (tv)|)+\Phi(x,|tv|)\right]dx
&\leq b\int_\Omega t^{p(x)}\,\frac{|\Delta v|^{p(x)}+|v|^{p(x)}}{p(x)}dx\\
&\leq b\,\frac{t^{p^-}}{p^-}\int_\Omega \Big(|\Delta v|^{p(x)}+|v|^{p(x)}|\Big)dx\\
&=C_1t^{p^-}.
\end{align*}
Next, we have
\begin{gather*}
\int_\Omega\frac{|tv|^{q(x)}}{q(x)}dx
 \geq\frac{t^{q^-+\varepsilon}}{q^-}\int_\omega |v|^{q(x)}dx=C_2\,t^{q^-+\varepsilon}, \\
\int_\Omega\frac{|tv|^{r(x)}}{r(x)}dx\leq\frac{t^{r^-}}{r^-}
 \int_\Omega |v|^{r(x)}dx=C_3\,t^{r^-}.
\end{gather*}
We conclude that
\begin{equation} \label{newdeal}
\mathcal{E}(tv)\leq C_1t^{p^-}-C_2t^{q^-+\varepsilon}+C_3t^{r^-},
\end{equation}
where $C_1$, $C_2$ and $C_3$ are positive constants.

Since $q^-+\varepsilon<r^-<p^-$, relation \eqref{newdeal} implies that $\mathcal{E}(tv)<0$, 
provided that $t>0$ is small enough.
Since $\mathcal{E}$ is coercive and weakly lower semi-continuous,  it admits a global
minimizer $u_0$, which is a critical point of $\mathcal{E}$.
By step 2, we have $u_0\not =0$.

To show that $u_0$ is a solution of problem \eqref{problem}, it remains to 
show that $\Delta u_0=0$ on $\partial\Omega$.
\smallskip

\noindent\textbf{Step 3.}
 We have $\Delta u_0=0$ on $\partial\Omega$.

Since $u_0$ verifies \eqref{problem} in the weak sense, we deduce that $u_0$ 
satisfies, for all $v\in\mathcal{X}$,
\begin{equation} \label{15bis}
\int_\Omega \phi(x,|\Delta u_0|)\Delta u_0\Delta vdx
=\int_\Omega A(x)vdx,
\end{equation}
where
$$
A(x):=|u_0|^{q(x)-2}u_0-|u_0|^{r(x)-2}u_0-\phi(x,|u_0|)u_0.
$$
Let $z\in\mathcal{X}$ be the unique solution of the linear problem
\begin{equation} \label{16bis}
\begin{gathered}
\Delta z=A(x) \quad\text{in } \Omega\\
z=0\quad\text{on }\partial\Omega.
\end{gathered}
\end{equation}
It follows that for all $v\in\mathcal{X}$,
$$
\int_\Omega \phi(x,|\Delta u_0|)\Delta u_0\Delta vdx=\int_\Omega (\Delta z)vdx.
$$
By Green's formula we deduce that for all $v\in C^\infty_c(\Omega)\subset\mathcal{X}$
\begin{equation} \label{17bis}
\int_\Omega \phi(x,|\Delta u_0|)\Delta u_0\Delta vdx
=\int_\Omega z\Delta vdx.
\end{equation}

For all $w\in C^\infty_c(\Omega)$, let $v\in C^\infty_c(\Omega$ be the 
unique solution of the problem
\begin{gather*}
\Delta v=w\quad\text{in } \Omega\\
v=0\quad\text{on } \partial\Omega.
\end{gather*}
Returning to \eqref{17bis}, we deduce that for all $w\in C^\infty_c(\Omega)$
$$
\int_\Omega\left(\phi(x,|\Delta u_0|)\Delta u_0-z \right)wdx=0.
$$
Applying \cite[Lemma VIII.1]{brezis} we conclude that
\begin{equation} \label{mitica}
\phi(x,|\Delta u_0|)\Delta u_0-z=0\quad\text{in}\ \Omega.
\end{equation}
But $z=0$ on $\partial\Omega$. Using hypothesis (H3), relation \eqref{mitica} 
implies that $\Delta u_0=0$ on $\partial\Omega$. 
The proof of Theorem \ref{thm1.1} is now complete.
\end{proof}

A very interesting open problem concerns the same analysis if the left-hand side  
of problem \eqref{problem}
is replaced either by the differential operator
\begin{equation} \label{ene1} 
\Delta (\phi_1(x,|\Delta u|)\Delta u)+V(x) \Delta (\phi_2(x,|\Delta u|)
\Delta u)
\end{equation}
or by
\begin{equation} \label{ene2} 
\Delta (\phi_1(x,|\Delta u|)\Delta u)+V(x) \Delta (\phi_2(x,|\Delta u|)\Delta u)
\log (e+|x|),
\end{equation}
where $V$ is a nonnegative potential and $\phi_1$, $\phi_2$ satisfy hypotheses 
(H1)--(H3) corresponding to the variable exponents $p_1(x)$, $p_2(x)$ with 
 $p_1(x)\leq p_2(x)$ in $\Omega$.
Considering two different materials with power hardening exponents 
$p_1(x)$ and $p_2(x)$, respectively, the coefficient
$V(x)$ dictates the geometry of a composite of the
two materials. When $V(x)>0$ then
$p_2(x)$-material is present, otherwise the $p_1(x)$-material is the only
one making the composite.
  Composite materials with locally different hardening exponents
$p_1(x)$ and $p_2(x)$
can be described using the energies associated to the differential operators 
defined in \eqref{ene1} and \eqref{ene2}.

Problems of this type were also motivated
by applications to elasticity, homogenization, modelling of strongly anisotropic 
materials, Lavrentiev phenomenon, etc.
 In the case of constant exponents, we refer to the pioneering papers by 
Marcellini \cite{marce1, marce2} and Mingione  et al.\ \cite{mingi1, mingi2, mingi3}. 
Double phase problems with variable growth have been recently considered 
by Cencelj, R\u{a}dulescu and Repov\v{s} \cite{cencel0}, R\u{a}dulescu and 
Zhang \cite{radz}, and Shi, R\u{a}dulescu, Repov\v{s} and Zhang \cite{shi}.



\subsection*{Acknowledgements} 
V. D. R\u{a}dulescu was supported by the Slovenian Research Agency Grants
P1-0292, J1-8131, J1-7025, N1-0064, and N1-0083. 
The same author acknowledges the support through a grant of the Romanian 
Ministry of Research and Innovation, CNCS--UEFISCDI, project number 
PN-III-P4-ID-PCE-2016-0130, within PNCDI III.

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\end{document}