\documentclass[reqno]{amsart} \usepackage{hyperref} \AtBeginDocument{{\noindent\small \emph{Electronic Journal of Differential Equations}, Vol. 2017 (2017), No. 188, pp. 1--17.\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-2017/188\hfil Existence and multiplicity of solutions] {Existence and multiplicity of solutions for anisotropic elliptic problems with variable exponent and nonlinear Robin boundary conditions} \author[B. Ellahyani, A. El Hachimi \hfil EJDE-2017/188\hfilneg] {Brahim Ellahyani, Abderrahmane El Hachimi} \address{Brahim Ellahyani \newline Department of Mathematics, Faculty of Sciences, Mohammed V University, P.B. 10014, Rabat, Morocco} \email{ellahyani.brahim.1991@gmail.com} \address{Abderrahmane El Hachimi \newline Department of Mathematics, Faculty of Sciences, Mohammed V University, P.B. 10014, Rabat, Morocco} \email{aelhachi@yahoo.fr} \dedicatory{Communicated by Jesus Ildefonso Diaz} \thanks{Submitted May 24, 2016. Published July 24, 2017.} \subjclass[2010]{35J20, 35J25, 35J62} \keywords{Anisotropic elliptic problems; Robin boundary conditions \hfill\break\indent existence and multiplicity} \begin{abstract} This article presents sufficient conditions for the existence of solutions of the anisotropic quasilinear elliptic equation with variable exponent and nonlinear Robin boundary conditions, \begin{gather*} -\sum_{i=1}^{N}\frac{\partial }{\partial x_{i}} \Big(\big| \frac{\partial u}{\partial x_{i}}\big|^{p_{i}(x)-2} \frac{\partial u}{\partial x_{i}}\Big) +\sum_{i=1}^{N}| u|^{p_{i}(x)-2}u+\lambda| u|^{m(x)-2}u =\gamma g(x,u)\\ \text{in } \Omega,\\ \sum_{i=1}^{N}\big| \frac{\partial u}{\partial x_{i}}\big|^{p_{i}(x)-2} \frac{\partial u}{\partial x_{i}}\upsilon_{i} =\mu| u|^{q(x)-2}u \quad\text{on } \partial\Omega. \end{gather*} Under appropriate assumptions on the data, we prove some existence and multiplicity results. The methods are based on Mountain Pass and Fountain theorems. \end{abstract} \maketitle \numberwithin{equation}{section} \newtheorem{theorem}{Theorem}[section] \newtheorem{lemma}[theorem]{Lemma} \newtheorem{proposition}[theorem]{Proposition} \newtheorem{remark}[theorem]{Remark} \allowdisplaybreaks \section{Introduction} Many problems in physics and mechanics can be modeled with sufficient accuracy using classical Lebesgue and Sobolev spaces, $L^{p} (\Omega)$ and $W^{1,p}(\Omega)$, where $p$ is a fixed constant and $\Omega$ is an appropriate domain. But for the electrorheological fluids (Smart fluids), this is not adequate but rather, the exponent should be able to vary. This leads to study the problem in the frame-work of variable exponent Lebesgue and Sobolev spaces, $L^{p(\cdot)}(\Omega)$ and $W^{1,p(\cdot)}(\Omega)$, where $p(\cdot)$ is a real-valued function; see, e.g.\ \cite{f3,f4}. On the other hand, it has been experimentally shown that the above-mentioned fluids may have their viscosity undergoing a significant change; see, e.g.\ \cite{B1}. Consequently, the mathematical modelling of such fluids requires the introduction of the so-called anisotropic variable spaces.Indeed, there is by now a large number of papers and increasing interest about anisotropic problems. With no hope of being complete, let us mention some pioneering works on anisotropic Sobolev spaces \cite{k1,r2} and some more recent regularity results for minimizers of anisotropic functionals \cite{A2,C2,m1}. Therefore, in the recent years, the study of various mathematical problems modeled by quasilinear elliptic and parabolic equations with both anisotropic and variable exponent has received considerable attention. Let us mention many works in that direction by Antontsev and Shmarev; see, e.g.\ \cite{A3} and the references therein. Our paper is mainly devoted to the existence and multiplicity of solutions of quasilinear elliptic equations under nonlinear Robin boundary condition such as \begin{equation}\label{e1.1} \begin{gathered} \begin{aligned} &-\sum_{i=1}^{N}\frac{\partial }{\partial x_{i}} \Big(\big|\frac{\partial u}{\partial x_{i}}\big|^{p_{i}(x)-2} \frac{\partial u}{\partial x_{i}}\Big) +\sum_{i=1}^{N}| u|^{p_{i}(x)-2}u+\lambda| u|^{m(x)-2}u \\ &=\gamma g(x,u)\quad\text{in }\Omega,\end{aligned}\\ \sum_{i=1}^{N}\big|\frac{\partial u}{\partial x_{i}}\big|^{p_{i}(x)-2} \frac{\partial u}{\partial x_{i}}\upsilon_{i}=\mu| u|^{q(x)-2}u \quad\text{on } \partial\Omega, \end{gathered} \end{equation} where $\Omega\subset\mathbb{R}^{n}$ is a bounded domain with $n\geq 2$, with smooth boundary $\partial\Omega$ and $\upsilon_{i}$ are the components of the outer normal unit vector and for $i\in\{1,\dots,N\}$, $p_{i},m \in\mathcal{C}(\bar{\Omega})$, $q\in\mathcal{C}(\partial\Omega)$. The functions $p_{i}$ and $g$ are supposed to satisfy some conditions to be specified below, while $\lambda,\gamma$, and $\mu $ are real parameters, with $\gamma,\mu> 0$. We shall give conditions under which problem \eqref{e1.1} has infinitely many solutions. According to the behaviour of $g$ and to the kind of results we want to prove, variational methods turn out to be more appropriate.\\ When $\lim_{s\to 0} g(x,s)/| s|^{\sigma}=0$, $\sigma$ to be made precise later, Mountain Pass theorem provides the existence of at least a solution of \eqref{e1.1} and, on the other hand, when $g$ is an odd function, Fountain's theorem yields the existence of infinitely many solutions. A host of publications exist for this type of problems when the boundary condition is replaced by $\frac{\partial u }{\partial\nu}=0$ on $\partial\Omega$ and $\gamma=0$ ; see, e.g.\ \cite{J1} and the references therein, where the authors obtained existence results by means of standard variational tools. The associated problem with Dirichlet boundary conditions has also been treated by many authors; see, e.g.\ \cite{B1,f6}. Furthermore, existence of positive solutions for nonlinear Robin problem involving the $p(x)$-Laplacian have been studied by S. G. Deng; in \cite{D3}, by using the sub-super solutions and variational methods. We consider here the case where $\mu$ is positive and $g$ satisfies more hypotheses than in \cite{g1}, to use the Mountain Passe and Fountain theorems. It turns out that the condition $q^{-}> P^{+}_{+}$ plays an important role in the proofs of our main results. This article is divided into four sections. In the second section, we introduce some basic properties of the generalized Lebesgue-Sobolev space $W^{1,p(x)}(\Omega)$ and anisotropic Sobolev spaces $W^{1,\overrightarrow{p}(x)}(\Omega)$ , and state the existence and multiplicity results concerning the problem \eqref{e1.1}. The third section is devoted to the proofs of the main results and finally, in the fourth section we deal with a generalized equation related to our problem \eqref{e1.1}. \section{Preliminaries and main results} To study problem \eqref{e1.1}, we need to introduce the notions of Sobolev space $W^{1,p(x)}(\Omega)$ and anisotropic Sobolev spaces $W^{1,\overrightarrow{p}(x)}(\Omega)$, with variable exponent. For convenience, we only recall some basic facts which will be used later. Let $\Omega\subset\mathbb{R}^{N}$ be a measurable subset with $\operatorname{meas}(\Omega)> 0$. We write \begin{gather*} \mathcal{C}(\bar{\Omega})=\{u:u \text{ is a continuous function in } \bar{\Omega}\}, \\ \mathcal{C}_{+}(\bar{\Omega})=\{u\in\mathcal{C}(\bar{\Omega}): \operatorname{ess\,inf}_{\Omega}u\geq 1\}. \end{gather*} Suppose that $\Omega$ is a bounded domain of $\mathbb{R}^{N} $ with a smooth boundary $\partial\Omega$, and $p\in\mathcal{C}(\bar{\Omega},\mathbb{R})$ with $p(x)> 1$, for any $x\in\Omega$. Denote $p^{-}={\inf_{x\in\Omega}} p(x)$ and $p^{+}={\sup_{x\in\Omega}} p(x)$; then we have, $p^{-}> 1$ and $p^{+}<\infty$. Denote by $\mathcal{M}$ be either $\Omega$ or $\partial\Omega$. Define the variable exponent Lebesgue space $$ L^{p(x)}(\mathcal{M})=\big\{ u\mid u:\mathcal{M}\to\mathbb{R} \text{ is measurable and } \int_{\mathcal{M}} | u(x)|^{p(x)} \, \mathrm{d}x <\infty\big\} , $$ endowed with the Luxembourg norm $$ | u|_{p(x)}=| u|_{L^{p(x)}(\mathcal{M})} =\inf\Big\{ \tau> 0;{\int_{\mathcal{M}}} \big|\frac{u(x)}{\tau}\big|^{p(x)} \,\mathrm{d}x\leq 1 \Big\}. $$ \begin{proposition}[\cite{D2}] \label{prop1} Let $\rho(u)= \int_{\mathcal{M}}\big|\frac{u(x)}{\tau}\big|^{p(x)} \, \mathrm{d}x$. For $u,u_{k}\in L^{p(x)}(\mathcal{M})(k=1,2,\dots)$, we have: \begin{enumerate} \item $ | u|_{L^{p(x)}(\mathcal{M})}\leq 1 \Rightarrow | u|_{L^{p(x)}(\mathcal{M})}^{p^{+}}\leq\rho(u) \leq | u|_{L^{p(x)}(\mathcal{M})}^{p^{-}}$. \item $ | u|_{L^{p(x)}(\mathcal{M})}> 1 \Rightarrow | u|_{L^{p(x)}(\mathcal{M})}^{p^{-}}\leq\rho(u)\leq | u|_{L^{p(x)} (\mathcal{M})}^{p^{+}}$. \item $ | u_{k}|_{L^{p(x)}(\mathcal{M})}\to 0 \Leftrightarrow \rho(u_{k})\to 0$. \item $ | u_{k}|_{L^{p(x)}(\mathcal{M})}\to\infty\Leftrightarrow \rho(u_{k})\to\infty$. \end{enumerate} \end{proposition} We define the variable exponent Sobolev space $$ W^{1,p(x)}(\Omega)=\{ u\in L^{p(x)}(\Omega): |\nabla u|\in L^{p(x)}(\Omega) \}, $$ endowed with the norm $$ \|u\|=\inf\Big\{ \tau> 0;{\int_{\Omega}} {\Big( \big|\frac{\nabla u(x)}{\tau}\big|^{p(x)} +\big|\frac{u(x)}{\tau}\big|^{p(x)}\Big)} \, \mathrm{d}x\leq 1 \Big\}. $$ \begin{proposition}[See \cite{f3}] \label{prop2} Both $(L^{p(x)}(\mathcal{M})),|\cdot|_{p(x)})$ and $ (W^{1,p(x)}(\Omega),\|\cdot\|)$ are separable, reflexive and uniformly convex Banach spaces. \end{proposition} \begin{proposition}[See \cite{f3}] \label{prop3} The H\"older inequality holds, namely $$ \int_{\mathcal{M}}| uv| \, \mathrm{d}x\leq 2| u|_{p(x)}| v|_{q(x)};\quad \forall u\in L^{p(x)}(\mathcal{M}),\; \forall v\in L^{q(x)}(\mathcal{M}) , $$ where $\frac{1}{p(x)}+\frac{1}{q(x)}=1$. \end{proposition} \begin{proposition}[See \cite{D2}] \label{prop4} Let $\rho(u)=\int_{\Omega}( |\nabla u(x)|^{p(x)}+| u(x)|^{p(x)}) \, \mathrm{d}x$. For $u,u_{k}\in W^{1,p(x)}(\Omega) (k=1,2,\dots)$, we have \begin{enumerate} \item $ \|u\|\leq 1 \Rightarrow \|u\|^{p^{+}}\leq\rho(u)\leq\|u\|^{p^{-}}$. \item $ \|u\|> 1 \Rightarrow \|u\|^{p^{-}}\leq\rho(u)\leq\|u\|^{p^{+}}$. \item $\|u_{k}\|\to 0 \Leftrightarrow \rho(u_{k})\to 0$. \item $\|u_{k}\|\to\infty\Leftrightarrow \rho(u_{k})\to\infty$. \end{enumerate} \end{proposition} Let $\overrightarrow{p}(\cdot):\overline{\Omega}\to\mathbb{R}^{N}$ be a vectorial function, $\overrightarrow{p}(\cdot)=(p_{1}(\cdot),p_{2}(\cdot), \dots,p_{N}(\cdot))$ such that $2\leq p_{i}\leq N$ and put $$ p_{M}(x)=\max\{p_{1}(x),\dots,p_{N}(x)\}. $$ The anisotropic Sobolev space with variable exponent is defined by $$ W^{1,\overrightarrow{p}(x)}(\Omega)=\{u\in L^{p_{M}(x)}(\Omega): \frac{\partial u}{\partial x_{i}}\in L^{p_{i}(\cdot)}(\Omega), \forall i\in\{1,\dots,N\}\}, $$ endowed with the norm $$ \|u\|_{\overrightarrow{p}(\cdot)} =\sum_{i=1}^{N}{\big|\frac{\partial u}{\partial x_{i}}\big|_{p_{i}(\cdot)}} +\sum_{i=1}^{N}| u|_{p_{i}(\cdot)}\,. $$ For convenience, we denote: $$ P^{-}_{-}=\inf\{p^{-}_{1},p^{-}_{2},\dots,p^{-}_{N}\},\quad P^{+}_{+}=\sup\{p^{+}_{1},p^{+}_{2},\dots,p^{+}_{N}\} $$ and write $X=W^{1,\overrightarrow{p}(x)}(\Omega)$. We know that $X$ is reflexive if $P^{-}_{-}> 1$, (see e.g \cite{n1}). We define \begin{gather*} J(u)= \int_{\Omega}\Big( \sum^{N}_{i=1}\frac{1}{p_{i}(x)} \big|\frac{\partial u}{\partial x_{i}}\big|^{p_{i}(x)} +\sum^{N}_{i=1}\frac{1}{p_{i}(x)}| u|^{p_{i}(x)}\Big) \, \mathrm{d}x, \\ G(x,u)=\int_{0}^{u}g(x,s)\mathrm{d}s. \end{gather*} We have \[ (J' u,v)= \int_{\Omega} \Big( \sum^{N}_{i=1}\big|\frac{\partial u}{\partial x_{i}}\big|^{p_{i}(x)-2} \frac{ \partial u}{\partial x_{i}} \frac{\partial v}{\partial x_{i}}+ \sum^{N}_{i=1}| u|^{p_{i}(x)-2}uv\Big)\, \mathrm{d}x, \] for all $v\in X$. In all this paper $C$, $C_{i}(i=0,1,2,\dots)$ represents different positive real constants. We make the following assumptions on the functions $q$ and $g$. \begin{itemize} \item[(H0)]$q\in\mathcal{C}(\partial\Omega)$ satisfies : $1\leq q(x)\leq \frac{(N-1)P^{-}_{-}}{N-P^{-}_{-}}$ for all $x\in\partial\Omega$ and $q^{-}
0$ and a function $\alpha\in\mathcal{C}(\bar{\Omega})$ such that: $1<\alpha(x)<\frac{NP^{-}_{-}}{N-P^{-}_{-}},$ for all $x\in\bar{\Omega}$ and
$$| g(x,s)|\leq C(1+| s|^{\alpha(x)-1})\quad\text{for all } (x,s)\in\Omega\times\mathbb{R}.$$
\item[(H2)] There exists $M >0$ and $\theta_{\lambda}\geq m^{+}$ (resp $\theta_{\lambda}\leq m^{-}$ ) if $\lambda\geq 0$ (resp $\lambda< 0$ ). such that for all $s$ with $| s|\geq M$ and $ x\in\Omega$, we have
$$0<\theta_{\lambda}G(x,s)\leq sg(x,s) .$$
\item[(H3)]$g(x,s)=\circ(| s|^{P^{+}_{+}})$ as $s\to 0$ and uniformly for $x\in\Omega$,
\item[(H4)]$g(x,-s)=-g(x,s)$, $x\in\Omega, s\in\mathbb{R}$.
\end{itemize}
We say that $u\in X$ is a weak solution of \eqref{e1.1} if
\begin{align*}
&\int_{\Omega}\Big( \sum^{N}_{i=1}\big|\frac{\partial u}{\partial x_{i}}
\big|^{p_{i}(x)-2}\frac{\partial u}{\partial x_{i}}
\frac{\partial v}{\partial x_{i}}+
\sum^{N}_{i=1}| u|^{p_{i}(x)-2}uv\Big)\, \mathrm{d}x
+\lambda \int_{\Omega} | u|^{m(x)-2} uv\, \mathrm{d}x \\
&= \int_{\Omega}\gamma g(x,u)v\, \mathrm{d}x
+\mu \int_{\partial\Omega} | u|^{q(x)-2} uv\, \mathrm{d}x,
\end{align*}
for all $v\in X$.
The energy functional associated with problem \eqref{e1.1} is
\begin{equation} \label{e2.1}
\begin{aligned}
\Phi(u)&=\int_{\Omega}\Big( \sum^{N}_{i=1}\frac{1}{p_{i}(x)}\big|
\frac{\partial u}{\partial x_{i}}\big|
+\sum^{N}_{i=1}\frac{1}{p_{i}(x)}| u|^{p_{i}(x)}\Big)\, \mathrm{d}x\\
&\quad +\lambda\int_{\Omega}\frac{1}{m(x)}| u|^{m(x)}\, \mathrm{d}x
-\gamma\int_{\Omega}G(x,u)\, \mathrm{d}x
-\mu\int_{\partial\Omega}\frac{1}{q(x)}| u|^{q(x)}\, \mathrm{d}x.
\end{aligned}
\end{equation}
\begin{proposition}[See \cite{f5,f2}] \label{prop5}\quad
\begin{itemize}
\item[(1)] $L\equiv J':X\to X^{*}$ is a continuous, bounded and strictly
monotone operator;
\item[(2)] $L$ is a mapping of type $(S_{+})$, i.e. if
$u_{n}\rightharpoonup u$ in $X$, and
$\overline{\lim}_{n \to +\infty}(L(u_{n})-L(u),u_{n}-u)\leq 0$, then
$u_{n}\to u$ in $X$;
\item[(3)] $L:X\to X^{*}$ is a homeomorphism.
\end{itemize}
\end{proposition}
The following are embedding results on anisotropic generalized Sobolev spaces
and will be used later.
\begin{proposition}[See \cite{m1}] \label{prop6}
Suppose $\Omega\subset\mathbb{R}^{N}$ is a bounded domain with smooth
boundary. For any $q\in\mathcal{C}_{+}(\overline{\Omega})$ satisfying
$q(x)<\frac{NP^{-}_{-}}{N-P^{-}_{-}}$ for all $x\in\overline{\Omega}$, the embedding
$$
W^{1,\overrightarrow{p}(x)}(\Omega)\hookrightarrow L^{q(x)}(\Omega)
$$
is continuous and compact.
\end{proposition}
\begin{proposition}[See \cite{m1}] \label{prop7}
Assume that the boundary of $\Omega$ possesses the cone property and
$p_{i}\in\mathcal{C}(\overline{\Omega}), 2\leq p_{i}< N$ for all
$i\in\{1,2,\dots,N\}$.
If $q\in\mathcal{C}(\partial\Omega)$ satisfies the hypothesis $1< q(x)<\frac{(N-1)P^{-}_{-}}{N-P^{-}_{-}}$ for all $x\in\partial{\Omega}$, then the embedding
$$W^{1,\overrightarrow{p}(x)}(\Omega)\hookrightarrow L^{q(x)}(\partial\Omega)$$
is continuous and compact.
\end{proposition}
The main results of this article are as follows:
\begin{theorem} \label{thm1}
Suppose that {\rm (H0)--(H2), (H4)} hold with
$m^{+}<\frac{NP^{-}_{-}}{N-P^{-}_{-}}$ and $P^{+}_{+}<\min(\alpha^{-}, m^{-})$.
Then, for any $\lambda\in\mathbb{R}$ and $\mu,\gamma> 0$, problem \eqref{e1.1}
has at least a nontrivial weak solution.
\end{theorem}
\begin{theorem} \label{thm2}
Suppose that {\rm (H0)--(H2) , (H5)} hold with
$m^{+}<\frac{NP^{-}_{-}}{N-P^{-}_{-}}$ and $P^{+}_{+}<\min(\alpha^{-}, m^{-})$.
Then, for any $\lambda\in\mathbb{R}$ and $\mu,\gamma> 0$, problem \eqref{e1.1}
has infinite many pairs of weak solutions.
\end{theorem}
\section{Proofs of main results}
To prove Theorem \ref{thm1}, we shall use the Mountain Pass theorem \cite{W1}.
We first start with the following lemmas.
\begin{lemma} \label{lem1}
If {\rm (H0)--(H2)} hold, then for any $\lambda\in\mathbb{R}$, the
functional $\Phi$ satisfies the Palais Smale condition (PS).
\end{lemma}
\begin{proof}
Suppose that $(u_{n})\subset X$ is a Palais Smale sequence , ie,
$$
\sup|\Phi(u_{n})|\leq C, \Phi'(u_{n})\to 0 \text{, as } n\to\infty.
$$
We shall prove that $(u_{n})$ has a convergent subsequence.
Let us show that $(u_{n})$ is bounded in $X$. Denote by
$ \widetilde{m}:\equiv m^{+}$ if $\lambda> 0$ and $\widetilde{m}:\equiv m^{-}$
if $\lambda\leq 0$. Since $\Phi(u_{n})$ is bounded, then by using (H1),
we have for large $n$,
\begin{align*}
C+C\|u_{n}\|
&\geq \Phi(u_{n})-\theta_{\lambda}\Phi'(u_{n})\\
&\geq \Big(\frac{1}{P^{+}_{+}}-\frac{1}{\theta_{\lambda}}\Big)\sum^{N}_{i=1}
\int_{\Omega}\Big(
\big|\frac{\partial u}{\partial x_{i}}\big|^{p_{i}(x)}+| u_{n}|^{p_{i}(x)}\Big)
\, \mathrm{d}x\\
&\quad +\lambda\Big(\frac{1}{\widetilde{m}}-\frac{1}{\theta_{\lambda}}\Big)
\int_{\Omega}| u_{n}|^{m(x)}\, \mathrm{d}x
-\gamma \int_{\Omega}\left(G(x,u_{n})-\theta_{\lambda} g(x,u_{n})u_{n}\right)\, \mathrm{d}x\\
&\quad -\frac{1}{\theta_{\lambda}}\langle\Phi'(u_{n}),u_{n}\rangle
+\mu\int_{\partial\Omega}\Big(\frac{1}{\theta_{\lambda}}-\frac{1}{q(x)}
\Big)| u_{n}|^{q(x)}\, \mathrm{d}x\\
&\geq\Big(\frac{1}{P^{+}_{+}}-\frac{1}{\theta_{\lambda}}\Big)\sum^{N}_{i=1}
\big|\frac{\partial u}{\partial x_{i}}\big|^{P^{-}_{-}}_{p_{i}
(x)}-\frac{1}{\theta_{\lambda}}(\Phi'(u_{n}),u_{n})\\
&\quad +\mu\Big(\frac{1}{\theta_{\lambda}}-\frac{1}{q^{-}}\Big)
\int_{\partial\Omega}| u_{n}|^{q(x)}\, \mathrm{d}x.
\end{align*}
Now, according to \cite[page 6]{B2}, we have
\begin{align*}
\frac{\|u\|^{P^{-}_{-}}_{\overrightarrow{p}(\cdot)}}{2^{P^{-}_{-}-1}N^{P^{-}_{-}-1}}
& \leq\sum^{N}_{i=1}\Big(\big|\frac{\partial u}{\partial x_{i}}
\big|^{P^{-}_{-}}_{p_{i}(\cdot)}+| u|^{P^{-}_{-}}_{p_{i}(\cdot)}\Big)\\
&\leq\sum^{N}_{i=1}\int_{\Omega}
\Big(\big|\frac{\partial u}{\partial x_{i}}\big|^{p_{i}(x)}+| u|^{p_{i}(x)}\Big)\,
\mathrm{d}x.
\end{align*}
Then,
$$
C+C\|u_{n}\|\geq \frac{1}{2^{P^{-}_{-}-1}N^{P^{-}_{-}-1}}
\Big(\frac{1}{P^{+}_{+}}-\frac{1}{\theta_{\lambda}}\Big)
\|u_{n}\|^{P^{-}_{-}}_{\overrightarrow{p}(\cdot)}
-\frac{C_{1}}{\theta_{\lambda}}\|u_{n}\|_{\overrightarrow{p}(\cdot)}-C.
$$
Since $\mu> 0$, then by using condition (H2) and the inequality above,
we deduce that $u_{n}$ is bounded in $X$. The proof is complete.
\end{proof}
\begin{lemma} \label{lem2}
There exist $r_{1},C'> 0$ such that $\Phi(u)\geq C'$, for all $u\in X$
such that $\|u\|=r_{1}$.
\end{lemma}
\begin{proof}
Conditions (H0), (H1) and (H2) ensure that, for any $\epsilon >0$, we have
$$
| G(x,s)|\leq \epsilon| s|^{P^{+}_{+}}+C(\epsilon)| s|^{\alpha(x)},\quad
\text{for all } (x,s)\in\Omega\times\mathbb{R}.
$$
For $\|u\|$ small enough, we thus obtain
\begin{equation} \label{e3.1}
\begin{aligned}
\Phi(u)
&\geq\frac{1}{P^{+}_{+}}\sum^{N}_{i=1}\int_{\Omega}
\Big(\big|\frac{\partial u}{\partial x_{i}}\big|^{p_{i}(x)}+| u|^{p_{i}(x)}\Big)\,
\mathrm{d}x+\lambda\int_{\Omega}\frac{1}{m(x)}| u|^{m(x)}\, \mathrm{d}x\\
&\quad -\int_{\Omega}(\epsilon| u|^{P^{+}_{+}}+C(\epsilon)| u|^{\alpha(x)})
\, \mathrm{d}x
-\frac{\mu}{q^{-}}\int_{\partial\Omega}| u|^{q(x)}\, \mathrm{d}x \\
&\geq\frac{1}{P^{+}_{+}2^{P^{+}_{+}-1}N^{P^{+}_{+}-1}}
\|u\|^{P^{+}_{+}}_{\overrightarrow{p}(\cdot)}
-\frac{|\lambda|}{m^{-}}\int_{\Omega}| u|^{m(x)}\, \mathrm{d}x
-\int_{\Omega}\epsilon| u|^{P^{+}_{+}}\, \mathrm{d}x\\
&\quad -\int_{\Omega}C(\epsilon)| u|^{\alpha(x)}\, \mathrm{d}x
-\frac{\mu}{q^{-}}\int_{\partial\Omega}| u|^{q(x)}\, \mathrm{d}x.
\end{aligned}
\end{equation}
Since $P^{+}_{+}<\alpha^{-}\leq\alpha(x)<\frac{NP^{-}_{-}}{N-P^{-}_{-}}$,
for all $x\in\Omega$ and $q(x)< \frac{(N-1)P^{-}_{-}}{N-P^{-}_{-}}$, for all
$x\in\partial\Omega$; then, we have
$$
W^{1,\overrightarrow{p}(x)}(\Omega)\hookrightarrow L^{P^{+}_{+}}(\Omega)\quad
\text{and}\quad
W^{1,\overrightarrow{p}(x)}(\Omega)\hookrightarrow L^{q(x)}(\partial\Omega),
$$
with continuous and compact embeddings.
Consequently, there exist two constants $C'_{1}> 0$ and $C'_{2}> 0$ such that
\begin{equation} \label{e3.2}
| u|_{L^{P^{+}_{+}}(\Omega)}\leq C'_{2}\|u\|,\quad
| u|_{L^{q(x)}(\Omega)}\leq C'_{1}\|u\|,\text{ for all } u\in X.
\end{equation}
By using \eqref{e3.2} for $\|u\|$ small enough, we obtain from \eqref{e3.1} that
\begin{align*}
\Phi(u)&\geq\frac{1}{P^{+}_{+}2^{P^{+}_{+}-1}N^{P^{+}_{+}-1}}\|u\|^{P^{+}_{+}}
-\frac{|\lambda|}{m^{-}}\max\{| u|^{m^{+}}_{L^{m(x)}(\Omega)},
| u|^{m^{-}}_{L^{m(x)}(\Omega)}\}\\
&\quad -\epsilon C'_{2}\|u\|^{P^{+}_{+}}-
C(\epsilon)C'_{3}\|u\|^{\alpha^{-}}-\frac{\mu}{q^{-}}C'_{1}\|u\|^{q^{-}}.
\end{align*}
Since $W^{1,\overrightarrow{p}}(\Omega)\hookrightarrow L^{m^{+}}(\Omega)$, we have
\begin{align*}
\Phi(u)
&\geq\frac{1}{P^{+}_{+}2^{P^{+}_{+}-1}N^{P^{+}_{+}-1}}\|u\|^{P^{+}_{+}}
-\frac{|\lambda| C}{m^{-}}\max\{\|u\|^{m^{+}},\|u\|^{m^{-}}\}\\
&\quad -\epsilon {C'_{2}}^{P^{+}_{+}}\|u\|^{P^{+}_{+}}-
C(\epsilon)C'_{3}\|u\|^{\alpha^{-}}-\frac{\mu}{q^{-}}C'_{1}\|u\|^{q^{-}}.
\end{align*}
Now, let $\epsilon> 0$ be small enough so that:
\[
0<\epsilon {C'_{2}}^{P^{+}_{+}}\leq\frac{1}{2P^{+}_{+}2^{P^{+}_{+}-1}N^{P^{+}_{+}-1}}
=: c_0.
\]
We have
\begin{align*}
\Phi(u)
&\geq c_{0}\|u\|^{P^{+}_{+}}-\frac{|\lambda| C}{m^{-}}\max\{\|u\|^{m^{+}},
\|u\|^{m^{-}}\}-C(\epsilon)\|u\|^{\alpha^{-}}
-\frac{\mu C'_{1}}{q^{-}}\|u\|^{q^{-}}\\
&\geq\|u\|^{P^{+}_{+}}\Big(c_{0}-\frac{|\lambda| C}{m^{-}}
\max\{\|u\|^{m^{+}-P^{+}_{+}},\|u\|^{m^{-}-P^{+}_{+}}\}\Big)\\
&\quad -\|u\|^{P^{+}_{+}}\Big(C(\epsilon)\|u\|^{\alpha^{-}-P^{+}_{+}}
+\frac{\mu C'_{1}}{q^{-}}\|u\|^{q^{-}-P^{+}_{+}}\Big).
\end{align*}
Since $P^{+}_{+}<\min{(\alpha^{-},m^{-},q^{-})}$, then there exist
$r_{1}> 0$ and $C'> 0$ such that
$$
\Phi(u)\geq C'> 0 ,\quad \text{for any } u\in X.
$$
Hence, the proof is complete.
\end{proof}
\begin{proof}[Proof of Theorem \ref{thm1}]
To apply the Mountain Pass theorem (\cite{W1}), we have to prove that
$\Phi(tu)\to-\infty$ as $t\to+\infty$, for some $u\in X$.
From (H2), it follows that
$$
G(x,s)\geq C| s|^{\theta_{\lambda}}, \quad \forall x\in\bar{\Omega},
\forall| s|\geq M.
$$
For $u\in X$ and $t> 1$, we have
\begin{align*}
\Phi(tu)
&\leq\frac{1}{P^{-}_{-}}\sum^{N}_{i=1}\int_{\Omega}
\Big(\big|\frac{\partial (tu)}{\partial x_{i}}\big|^{p_{i}(x)}
+| tu|^{p_{i}(x)}\Big)\, \mathrm{d}x
+\lambda\int_{\Omega}\frac{1}{m(x)}| tu|^{m(x)}\, \mathrm{d}x\\
&\quad -\int_{\Omega}G(x,tu)\, \mathrm{d}x-\frac{\mu}{q^{+}}
\int_{\partial\Omega}| tu|^{q(x)}\, \mathrm{d}x\\
&\leq\frac{t^{P^{+}_{+}}}{P^{-}_{-}}\sum^{N}_{i=1}\int_{\Omega}
\Big(\big|\frac{\partial (tu)}{\partial x_{i}}\big|^{p_{i}(x)}
+| u|^{p_{i}(x)}\Big)\, \mathrm{d}x
+\lambda t^{\widetilde{m}} \int_{\Omega}\frac{1}{m(x)}| u|^{m(x)}\, \mathrm{d}x\\
&\quad -Ct^{\theta_{\lambda}}\int_{\Omega}| u|^{\theta_{\lambda}}\, \mathrm{d}x
-\frac{\mu t^{q ^{-}}}{q^{+}}\int_{\partial\Omega}| u|^{q(x)}\, \mathrm{d}x,
\end{align*}
where again $\widetilde{m}=m^{+}$ if $\lambda> 0$ and $\widetilde{m}=m^{-}$
if $\lambda\leq 0$.
By (H0) and (H2), it follows that, for any $\lambda\in\mathbb{R}$,
$\Phi(tu)\to-\infty\text{ as }(t\to +\infty)$.
Since $\Phi(0)=0$, it follows that $\Phi$ satisfies the condition of the
Mountain Pass lemma, and so $\Phi$ admits at least one nontrivial critical
point $u_{0}\in X$; which is characterized by
$$
\tau=\inf_{h\in\Gamma}\sup_{t\in[0,1]}\Phi(h(t)),
$$
where
$$
\Gamma=\{h\in\mathcal{C}([0,1],X); h(0)=0 \text{ and } h(1)=e\}.
$$
\end{proof}
\subsection*{Proof of Theorem \ref{thm2}}
Let $X$ be a reflexive and separable Banach space. It is well know
(see, e.g.\ \cite{A2}) that there are $\{e_{j}\}^{\infty}_{j=1}\subset X$ and
$\{e^{*}_{j}\}^{\infty}_{j=1}\subset X^{*}$
(where $X^{*}$ is the topological dual of $X$) such that
$$
X=\overline{\operatorname{span}}\{e_{j}:1,2,\dots\},\quad
X^{*}=\overline{\operatorname{span}}\{e^{*}_{j}:1,2,\dots\} , $$
and
\begin{equation}
\langle e^{*}_{j},e_{i}\rangle
=\begin{cases}
1&\text{ if } i=j,\\
0&\text{ if } i\neq j.
\end{cases}
\end{equation}
For convenience, we write $X_{j}=\operatorname{span}\{e_{j}\}$,
$Y_{k}=\oplus^{k}_{j=1}X_{j}$ and $Z_{k}=\oplus^{\infty}_{j=k} X_{j}$.
Denote
\[
p^{*}(x)=\begin{cases}
Np(x)/(N-p(x)) &\text{if } p(x) 0$ such that
$$
| f(x,s)|\leq C(1+| s|^{\beta(x)-1}),\quad\forall
(x,s)\in\partial\Omega\times\mathbb{R};
$$
where $\beta(x)\in\mathcal{C}(\partial\Omega)$ with
$ 1<\beta^{-}\leq\beta^{+}< P^{-}_{-}$ and
$\beta(x)<\frac{(N-1)P^{-}_{-}}{N-P^{-}_{-}}$ for all $x\in\partial\Omega$.
\item[(ii)] There exist $R> 0$, such that for all $| s|\geq R$ and
$x\in\partial\Omega$
$$
0<\sigma F(x,s)\leq f(x,s)s.
$$
\end{itemize}
\item[(H10)] There exist $\delta_{2}> 0$, $C_{4}> 0$ and
$q_{4}(x)\in\mathcal{C}(\partial\Omega)$ such that
$$
F(x,s)\geq C_{4}| s|^{q_{4}(x)}, \quad
\forall x\in\partial\Omega,\; \forall| s|\leq\delta_{2},
$$
where $1< q_{4}<\frac{(N-1)P^{-}_{-}}{N-P^{-}_{-}}$ and
$ q^{+}_{4}< P^{-}_{-} $ for all $ x\in\partial\Omega$.
\item[(H11)] For $i=1,2$, $g_{i}(x,-s)=-g_{i}(x,s)$ for all
$(x,s)\in\Omega\times\mathbb{R}$, and $f(x,-s)=-f(x,s)$ for all
$(x,s)\in\partial\Omega\times\mathbb{R}$.
\end{itemize}
We denote
\begin{gather*}
g(x,s)=\lambda g_{1}(x,s)+\gamma g_{2}(x,s),\quad
G_{i}(x,s)=\int_{0}^{s}g_{i}(x,t)\, \mathrm{d}t \quad (i=1,2) ,\\
G(x,s)=\int_{0}^{s}g(x,t)\, \mathrm{d}t, \quad
F(x,s)=\int_{0}^{s}f(x,t)\, \mathrm{d}t;
\end{gather*}
and the associated functional
\[
\Phi(u)=\sum^{N}_{i=1}\int_{\Omega}\frac{1}{p_{i}(x)}
\Big(\big|\frac{\partial u}{\partial x_{i}}\big|^{p_{i}(x)}+| u|^{p_{i}(x)}\Big)\,
\mathrm{d}x
-\int_{\Omega}G(x,u)\, \mathrm{d}x
-\mu\int_{\partial\Omega}F(x,u)\, \mathrm{d}x.
\]
\begin{proposition} \label{prop8}
If {\rm (H5), (H6)} and {\rm (H9)} hold, then for every
$\lambda,\gamma, \mu\geq 0$ the functional $\Phi$ satisfies the Palais
Small condition (PS).
\end{proposition}
\begin{proof}
We use the following inequalities:
For $x\in\Omega, s\in\mathbb{R}$
\begin{gather*}
\sigma G_{2}(x,s)\leq g_{2}(x,s)s+C_{3}, \\
\begin{aligned}
\sigma G(x,s)-g(x,s)s
&\leq\left[\sigma G_{1}(x,s)-s g_{1}(x,s)\right]
+\left[\sigma G_{2}(x,s)-s g_{2}(x,s)\right]\\
&\leq (C_{1}+C_{2}| s|^{q_{1}(x)})+C_{3}.
\end{aligned}
\end{gather*}
Suppose that $(u_{n})\subset X$ is a (PS) sequence ; i.e,
$$
\sup|\Phi(u_{n})|\leq C, \Phi'(u_{n})\to 0,\text{ as } n\to\infty.
$$
Let us show that $(u_{n})$ is bounded in $X$. Since $\Phi(u_{n})$ is bounded,
then by using hypothesis (H6) and (H9), we have for $n$ large enough
\begin{equation} \label{e4.2}
\begin{aligned}
C+C\|u_{n}\|
&\geq\sigma\Phi(u_{n})-\Phi' (u_{n}) \\
&\geq\Big(\frac{1}{P^{+}_{+}}-\frac{1}{\sigma}\Big)
\sum^{N}_{i=1}\int_{\Omega}\Big(
\big|\frac{\partial u}{\partial x_{i}}\big|^{p_{i}(x)}+| u_{n}|^{p_{i}(x)}\Big)\,
\mathrm{d}x\\
&\quad -\int_{\Omega}\left(\sigma G(x,u_{n})-g(x,u_{n})u_{n}\right)\, \mathrm{d}x \\
&\quad -\mu\int_{\partial\Omega}\left(\sigma f(x,u_{n})-f(x,u_{n})u_{n}\right)\, \mathrm{d}x\\
&\geq\frac{1}{2^{P^{-}_{-}-1}N^{P^{-}_{-}-1}}
\Big(\frac{1}{p^{+}_{+}}-\frac{1}{\sigma}\Big)
\|u_{n}\|^{P^{-}_{-}}_{\overrightarrow{p}(\cdot)}
-C'\int_{\Omega}| u_{n}|^{q_{1}}\, \mathrm{d}x\\
&\quad -C'-\int_{\partial\Omega}\left(\sigma f(x,u_{n})-f(x,u_{n})u_{n}\right)\,
\mathrm{d}x.
\end{aligned}
\end{equation}
Applying (H9) for $\|u_{n}\|$ large enough, we then get
$$
C+C\|u_{n}\|\geq\frac{1}{2^{P^{-}_{-}-1}N^{P^{-}_{-}-1}}
\Big(\frac{1}{p^{+}_{+}}-\frac{1}{\sigma}\Big)
\|u_{n}\|^{P^{-}_{-}}-C'\|u_{n}\|^{q^{+}_{1}}-C'_{3}.
$$
Now, as $W^{1,\overrightarrow{p}(x)}(\Omega)\hookrightarrow L^{q^{+}_{1}}(\Omega)$
is a continuous and compact embedding,
from the inequality above, we deduce that $u_{n}$ is bounded in $X$.
The proof is complete.
\end{proof}
\begin{remark} \label{rmk1} \rm
It follows from (H6) that
$$
G_{2}(x,s)\geq C_{5}| s|^{1/\sigma}-C_{6}, \quad
\forall x\in\Omega,\; \forall s\in\mathbb{R}.
$$
\end{remark}
The main results of this section are as follows:
\begin{proposition}[\cite{f5}] \label{prop9}
Assume that $\psi:X\to\mathbb{R}$ is weakly-strongly continuous and that
$\psi(0)=0$. Let $\nu> 0$ be given. Set
$$
\beta_{k}=\beta_{k}(\nu)=\sup_{u\in Z_{k},\|u\|\leq\nu}|\psi(u)|.
$$
Then $\beta_{k}\to 0$ as $k\to\infty$.
\end{proposition}
\begin{theorem} \label{thm3}
Assume that {\rm (H5), (H6)} and {\rm (H9)} hold.
\begin{itemize}
\item[(1)] If in addition, {\rm (H10)} holds, then for every $\gamma,\mu> 0$,
there exists $r_{0}(\gamma)> 0$ such that when $0\leq\lambda,\mu\leq r_{0}(\gamma)$,
problem \eqref{e4.1} has a nontrivial solution $u_{1}$ such that $\Phi(u_{1})> 0$.
\item[(2)] If in addition, {\rm (H7)} and {\rm (H10)} hold, then for every
$\gamma,\mu> 0$, there exists $r_{0}(\gamma)> 0$ such that when
$0\leq\lambda,\mu\leq r_{0}(\gamma)$, problem \eqref{e4.1} has two nontrivial
solutions $u_{1},v_{1}$ such that $\Phi(u_{1})> 0$ and $\Phi(v_{1})< 0$.
\item[(3)] If in addition, {\rm (H7), (H10)} and {\rm (H11)} hold, then for
every $\lambda,\gamma,\mu> 0$, problem \eqref{e4.1} has a sequence of
solutions $\{\pm u_{k}\}$ such that $\Phi(\pm u_{k})\to +\infty$ as
$k\to +\infty$.
\end{itemize}
\end{theorem}
\begin{proof}
(1) We denote
$$
\psi_{1}(u)=\lambda\int_{\Omega}G_{1}(x,u(x))\, \mathrm{d}x,\quad
\psi_{2}(u)=\gamma\int_{\Omega}G_{2}(x,u(x))\, \mathrm{d}x.
$$
When the assumptions in (1) hold, then for sufficiently small $\|u\|$,
we get
$$
G_{2}(x,u)\leq\epsilon| u|^{P^{+}_{+}}+C(\epsilon)| u|^{q_{2}}, \quad
\forall(x,s)\in\Omega\times\mathbb{R},
$$
Then
$$
\psi_{2}(u)\leq\gamma\epsilon\int_{\Omega}| u|^{P^{+}_{+}}\, \mathrm{d}x
+\gamma C(\epsilon)\int_{\Omega}| u|^{q_{2}(x)}\, \mathrm{d}x.
$$
Since $1< q_{2}<\frac{NP^{-}_{-}}{N-P^{-}_{-}}$, for all $x\in\Omega$,
then we have
$$
W^{1,\overrightarrow{p}(x)}(\Omega)\hookrightarrow L^{P^{+}_{+}}(\Omega),\quad
\text{and}\quad
W^{1,\overrightarrow{p}(x)}(\Omega)\hookrightarrow L^{q_{2}(x)}(\Omega),
$$
with continuous and compact embeddings. This implies the existence of
$C_{1}, C_{2}> 0$ such that
$$
\psi_{2}(u)\leq\gamma\epsilon C_{1}\|u\|^{P^{+}_{+}}
+\gamma C(\epsilon)C_{2}\|u\|^{q^{-}_{2}}.
$$
Choose $\epsilon> 0$ small enough so that
$0<\gamma\epsilon C_{2}<\frac{1}{2^{P^{+}_{+}-1}N^{P^{+}_{+}-1}}$. Then, we have
\begin{align*}
&\sum^{N}_{i=1}\int_{\Omega}\frac{1}{p_{i}(x)}
\Big(\big|\frac{\partial u}{\partial x_{i}}\big|^{p_{i}(x)}+| u|^{p_{i}(x)}\Big)
\, \mathrm{d}x-\psi_{2}(u) \\
&\geq\frac{1}{2^{P^{+}_{+}}N^{P^{+}_{+}-1}}
\|u\|^{P^{+}_{+}}-\gamma C(\epsilon)C_{2}\|u\|^{q^{-}_{2}}.
\end{align*}
Since $q^{-}_{2}> P^{+}_{+}$, there exist $r_{1}> 0$ and $\alpha> 0$ such that
$$
\sum^{N}_{i=1}\int_{\Omega}\frac{1}{p_{i}(x)}
\Big(\big|\frac{\partial u}{\partial x_{i}}\big|^{p_{i}(x)}+| u|^{p_{i}(x)}\Big)
\, \mathrm{d}x-\psi_{2}(u)\geq\alpha> 0,\quad\text{for }\|u\|=r_{1}.
$$
We can find $r_{0}(\gamma)> 0$ such that when $\mu,\lambda\leq r_{0}(\gamma)$,
we obtain
$$
\psi_{1}(u)\leq\frac{\alpha}{2},\quad
\forall u\in S_{r_{1}}=\{u\in X;\|u\|=r_{1}\}.
$$
Therefore, $\lambda,\mu\leq r_{0}(\gamma)$. So, we obtain
$$
\Phi(u)\geq\frac{\alpha}{2}> 0,\quad\forall u\in S_{r_{1}}.
$$
Let $u\in X$ and $t> 1$, we have
\begin{equation} \label{e4.3}
\begin{aligned}
\Phi(tu)&=\sum^{N}_{i=1}\int_{\Omega}\frac{t^{p_{i}(x)}}{p_{i}(x)}
\Big(\big|\frac{\partial u}{\partial x_{i}}\big|^{p_{i}(x)}+| u|^{p_{i}(x)}\Big)\,
\mathrm{d}x-\lambda\int_{\Omega}G_{1}(x,tu)\, \mathrm{d}x\\
&\quad -\gamma\int_{\Omega}G_{2}(x,tu)\, \mathrm{d}x-\mu\int_{\partial\Omega}F(x,tu)\, \mathrm{d}x.
\end{aligned}
\end{equation}
From Remark \ref{rmk1}, we obtain
\begin{equation} \label{3.4}
\begin{aligned}
\Phi(tu)&\leq t^{P^{+}_{+}}\sum^{N}_{i=1}\int_{\Omega}\frac{1}{p_{i}(x)}
\Big(\big|\frac{\partial u}{\partial x_{i}}\big|^{p_{i}(x)}+| u|^{p_{i}(x)}\Big)\,
\mathrm{d}x-\lambda\int_{\Omega}G_{1}(x,tu)\, \mathrm{d}x\\
&\quad -\gamma C_{5}t^{1/\sigma}\int_{\Omega}| u|^{1/\sigma}\, \mathrm{d}x
-\mu\int_{\partial\Omega}F(x,tu)\, \mathrm{d}x.
\end{aligned}
\end{equation}
Now, since
$$
G_{1}(x,tu)= o((t| u|)^{q^{+}_{1}}),\quad
F(x,tu)= o((t| u|)^{\beta^{+}})\quad\text{when } t\to +\infty,
$$
(because $ P^{+}_{+}\leq q^{+}_{1}$ and
$\beta^{+}< P^{+}_{+}<\frac{1}{\sigma}$), we obtain
$$
\Phi(tu)\to -\infty,\quad\text{when } t\to +\infty.
$$
Hence, It follows that there exist $u_{0}\in X$ such that $\|u_{0}\|> r_{1}$
and $\Phi(u_{0})< 0$. Therefore, By the Mountain Pass theorem, problem \eqref{e4.1}
has a nontrivial solution $u_{1}$ such that $\Phi(u_{1})> 0$.
(2) Under the assumpti9ns in (2) hold, (1), we know that there exist
$r_{0}(\gamma)> 0$ such that when $0\leq\lambda,\mu\leq r_{0}(\gamma)$,
problem has a nontrivial solution $u_{1}$ such that $\Phi(u_{1})> 0$.
For $t\in (0,1)$ small enough, and $v_{0}\in\mathcal{C}^{\infty}_{0}(\Omega)$
such that $0\leq v_{0}(x)\leq\min\{\delta_{1},\delta_{2}\}$, we have
\begin{equation} \label{e4.5}
\begin{aligned}
&\Phi(tv_{0}) \\
&=\sum^{N}_{i=1}\int_{\Omega}\frac{1}{p_{i}(x)}
\Big(\big|\frac{\partial (tv_{0})}{\partial x_{i}}\big|^{p_{i}(x)}
+| tv_{0}|^{p_{i}(x)}\Big)\, \mathrm{d}x
-\lambda\int_{\Omega}G_{1}(x,tv_{0})\, \mathrm{d}x\\
&\quad -\gamma\int_{\Omega}G_{2}(x,tv_{0})\, \mathrm{d}x
-\mu\int_{\partial\Omega}F(x,tv_{0})\, \mathrm{d}x\\
&\leq t^{P^{-}_{-}}\sum^{N}_{i=1}\int_{\Omega}\frac{1}{p_{i}(x)}
\Big(\big|\frac{\partial v_{0}}{\partial x_{i}}\big|^{p_{i}(x)}
+| v_{0}|^{p_{i}(x)}\Big)\, \mathrm{d}x-\lambda C_{3}\int_{\Omega}
| tv_{0}|^{q_{3}(x)}\, \mathrm{d}x\\
&\quad -\gamma\int_{\Omega}G_{2}(x,tv_{0})\, \mathrm{d}x
-\mu C_{4}\int_{\partial\Omega}| tv_{0}|^{q_{4}(x)}\, \mathrm{d}x.
\end{aligned}
\end{equation}
For $t\in(0,1)$ small enough, we obtain
$$
G_{2}(x,tv_{0})=o(| tv_{0}|^{P^{+}_{+}}), \quad \text{as } t\to\infty
$$
So, we have
\begin{equation} \label{e4.6}
\begin{aligned}
&\Phi(tv_{0}) \\
&\leq t^{P^{-}_{-}}\sum^{N}_{i=1}\int_{\Omega}\frac{1}{p_{i}(x)}
\Big(\big|\frac{\partial tv_{0}}{\partial x_{i}}\big|^{p_{i}(x)}
+| tv_{0}|^{p_{i}(x)}\Big)\, \mathrm{d}x
-\lambda C_{3}t^{q^{+}_{3}}\int_{\Omega}| v_{0}|^{q_{3}(x)}\, \mathrm{d}x\\
&\quad-\gamma Mt^{P^{+}_{+}}\int_{\Omega}v_{0}\, \mathrm{d}x
-\mu C_{4}t^{q^{+}_{4}}\int_{\partial\Omega}| v_{0}|^{q_{4}(x)}\, \mathrm{d}x.
\end{aligned}
\end{equation}
Since $\min(q^{+}_{3}, q^{+}_{4})< P^{-}_{-}$, by factoring the right side of
\eqref{e4.6} by $t^{q^{+}_{3}}$ if $ q^{+}_{4}> q^{+}_{3}$, and by
$t^{q^{+}_{4}}$ if $ q^{+}_{3}> q^{+}_{4}$, we obtain
$$
\lim_{t\to 0}\Phi(tv_{0})< 0.
$$
Then there exist $w\in X$ such that $\|w\|\leq r_{1}$, and $\Phi(w)< 0$.
(3) $\Phi$ is an even functional. We denote
$$
\psi(u)=\lambda\int_{\Omega}G_{1}(x,u)\, \mathrm{d}x
+\gamma\int_{\Omega}G_{2}(x,u)\, \mathrm{d}x
+\mu\int_{\partial\Omega}F(x,u)\, \mathrm{d}x
$$
As $\beta_{k}(\nu)$ is defined in Proposition \ref{prop6}, for each positive integer,
there exist a positive integer $k_{0}$ such that $\beta_{k}(n)\leq 1$
for all $k\geq k_{0}(n)$. We can assume $k_{0}(n)< k_{0}(n+1)$ for each $n$.
We define $\{\nu_{k}:k=1,2,\dots\}$ by
\begin{equation}
\nu_{k}= \begin{cases}
n &\text{if } k_{0}\leq k< k_{0}(n+1),\\
1 &\text{if } 1\leq k< k_{0}.
\end{cases}
\end{equation}
We see that $\nu_{k}\to\infty$ when $k\to\infty$, then for
$u\in Z_{k}\ with\ \|u\|=\nu_{k}$, we obtain
\begin{align*}
\Phi(u)&=\sum^{N}_{i=1}\int_{\Omega}\frac{1}{p_{i}(x)}
\Big(\big|\frac{\partial u}{\partial x_{i}}\big|^{p_{i}(x)}+| u|^{p_{i}(x)}\Big)
\, \mathrm{d}x -\psi(u)\\
&\geq\frac{1}{P^{+}_{+}2^{P^{-}_{-}-1}N^{P^{-}_{-}-1}}(\nu_{k})^{P^{-}_{-}}-1.
\end{align*}
Consequently,
$$
\inf_{u\in Z_{k},\|u\|=\nu_{k}}\Phi(u)\to\infty \quad \text{as } k\to\infty.
$$
So the hypotheses \eqref{D2} of Fountain theorem are satisfied.
Indeed, by (H6), (H9) and Remark \ref{rmk1}, for $\|u\|\geq 1$ we obtain
$$
\Phi(u)\leq\frac{C}{P^{-}_{-}}\|u\|^{P^{+}_{+}}
+C_{1}\lambda | u|_{q_{1}(x)}^{q^{+}_{1}}
-C_{5}\gamma| u|^{1/\sigma}_{\frac{1}{\sigma}}
+C_{6}\mu| u|^{\beta^{+}}_{\beta(x)}+C_{7}.
$$
As the space $Y_{k}$ has finite dimension i.e all norms are equivalents,
we then have
$$
\Phi(u)\leq\frac{C}{P^{-}_{-}}\|u\|^{P^{+}_{+}}
+C'_{1}\lambda \| u\|^{q^{+}_{1}}-C'_{5}\gamma\| u\|^{1/\sigma}
+C'\mu\| u\|^{\beta^{+}}+C_{7}.
$$
Since $\min(q^{+}_{1},q^{+}_{2})< P^{+}_{+}< \frac{1}{\sigma}$, we obtain
$\Phi(u)\to -\infty$ as $\|u\|\to +\infty$, $u\in Y_{k}$.
Finally, the proof of (3) is complete.
\end{proof}
\subsection*{Acknowledgements}
The authors would like to thank the anonymous referees for their helpful remarks.
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\end{document}