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\date{}

\title{\sc {Anisotropic Neumann problems in Sobolev spaces with variable exponent}}
\author{\sc Maria-Magdalena Boureanu$\,^a$
and Vicen\c{t}iu D. R\u{a}dulescu$\,^{a,b}$\\
\small $^a\,$Department of Mathematics, University of Craiova,
200585 Craiova,
Romania\\
\small $^b\,$Institute of Mathematics ``Simion Stoilow" of the
Romanian Academy,
 014700 Bucharest, Romania\\
\small E-mail addresses: {\tt  mmboureanu@yahoo.com,}
  {\tt vicentiu.radulescu@math.cnrs.fr}}


\begin{document}
\baselineskip16pt \maketitle

\begin{center}{\it This paper is dedicated to Professor V.
Lakshmikantham on the occasion of his retirement as Editor of
Nonlinear Analysis: Theory, Methods and Applications}
\end{center}


\noindent{\small{\sc Abstract}. We prove a trace theorem that allows the treatment of  Neumann problems with nonlinearities on the boundary in anisotropic spaces with variable exponent. Then we proceed to the study of such a problem that involves general operators of the $\overset{\rightarrow} p(\cdot)$ - Laplace type. We deduce the existence of solutions and we bring into attention the situation when the solution is unique. \\
\small{\bf 2010 Mathematics
Subject Classification:}  35J25, 35J62, 35D30, 46E35, 35J20. \\
\small{\bf Key words:} anisotropic Sobolev space, variable exponent, Neumann elliptic problem,
weak solutions, existence, uniqueness, trace theorem.}


\section{Introduction}

In the last decades, one of the topics from the field of partial differential equations
 that continuously gained interest is the one concerning the Sobolev space with variable exponents, $W^{1,p(\cdot)}$ (where $p$ is a function depending on $x$), see for example the monograph \cite{dhhr} and the references therein. Naturally, problems involving the $p(\cdot)$ - Laplace operator
 \begin{equation}\label{elapl1}
 \Delta_{p(x)}u={\rm div}(|\nabla u|^{p(x)-2}\nabla u)
 \end{equation}
 were intensively studied. At the same time, due to the development of the theory regarding the anisotropic Sobolev space, $W^{1,\overset{\rightarrow} p}$ (where
$\overset{\rightarrow} p$ is a constant vector $\overset{\rightarrow} p=(p_1,\dots,p_N)$), there are also many authors discussing problems involving the $\overset{\rightarrow} p$ - Laplace operator,
\begin{equation}\label{elapl2}
 \Delta_{\overset{\rightarrow} p}(u)=\di\sum_{i=1}^N\di\partial_{x_i}\left(\left|\di\partial_{x_i}u
\right|^{p_i-2}\di\partial_{x_i}u \right),
 \end{equation}
see for example \cite{alvha, fraga} and the references therein. Furthermore, a new theory captures the attention when it is introduced the anisotropic space with variable exponent, $W^{1,\overset{\rightarrow} p(\cdot)}$ (where
$\overset{\rightarrow}
p(\cdot)=\left(p_1(\cdot), \dots,p_N(\cdot)\right)$ is a vector with variable components),  see \cite{patmv00, patmv, fananis}. Consequently, a new operator takes its place in the mathematical literature, namely
\begin{equation}\label{elapl3}
\Delta_{\overset{\rightarrow}
p(x)}(u)=\di\sum_{i=1}^N\di\partial_{x_i}\left(\left|\di\partial_{x_i}u
\right|^{p_i(x)-2}\di\partial_{x_i}u \right),
\end{equation}
see also \cite{ BPR, mmb}.
This operator will be referred to as the anisotropic variable
exponent $\overset{\rightarrow}
p(\cdot)$ - Laplace operator and it is closely related to \eqref{elapl1} and \eqref{elapl2}. To be more specific, when choosing
$p_1(\cdot)= \dots =p_N(\cdot)=p(\cdot)$ we obtain an operator with similar properties to the
variable exponent $p(\cdot)$ - Laplace operator \eqref{elapl1}, while when choosing $p_1, \dots, p_N$ to be constant functions we arrive at the anisotropic $\overset{\rightarrow} p$ - Laplace operator \eqref{elapl2}.

In the present paper we consider a problem involving a more general type of operator, that is,
\begin{equation}\label{eop}
\di\sum_{i=1}^N\di\partial_{x_i}a_i\left(x,\di\partial_{x_i}u \right).
\end{equation}
Let  $\Omega\subset\RR^N$ be a bounded open set. The applications $a_i:\Omega\times\RR\ri\RR$ are Carath\'{e}odory
 functions fulfilling the following hypotheses for all $i\in\{1,\dots,N\}$:
 \begin{description}
   \item[(A1)] There exists a positive constant $\bar c_{i}$ such
that $a_i$ satisfies the growth condition
$$|a_i(x,s)|\leq \bar c_{i}(d_i(x)+|s|^{p_i(x)-1}),$$
for all $x\in\Omega$ and $s\in\RR$, where $d_i\in L^{p'_i(\cdot)}(\Omega)$ (with $1/p_i(x)+1/p'_i(x)=1$), is a nonnegative function.
   \item[(A2)] If we denote by $A_i:\Omega\times\RR\ri\RR$,
$$A_i(x,s)=\int_{0}^{s}a_i(x,t)dt,$$
then the following inequalities hold:
$$ |s|^{p_i(x)}\leq a_i(x,s)s\leq p_i(x)\;A_i(x,s),$$
for all $x\in\Omega$ and $s\in\RR$.
   \item[(A3)] $a_i$ is fulfilling
$$(a_i(x,s)-a_i(x,t))(s-t)> 0,$$
for all $x\in\Omega$ and $s,\,t\in\RR$ with $s\neq t$;.
 \end{description}
The operator presented above is a $\overset{\rightarrow} p(\cdot)$ - Laplace type operator because when we take
\begin{equation*}\label{e1}
a_i(x,s)=|s|^{p_i(x)-2} s\quad\mbox{for all }
i\in\{1,...,N\},
\end{equation*}
 we have $A_i(x,s)=\frac{1}{p_i(x)}\,|s|^{p_i(x)}$
for all $i\in\{1,...,N\}$ and we get the $\overset{\rightarrow}
p(\cdot)$ - Laplace operator \eqref{elapl3}. Obviously, there are many other operators deriving from \eqref{eop}. Indeed, to give another interesting example, if we take
\begin{equation*}\label{e2}
a_i(x,s)=(1+|s|^2)^{(p_i(x)-2)/2}s\quad\mbox{for all }i\in\{1,...,N\},
\end{equation*}
 we
have $A_i(x,s)=\frac{1}{p_i(x)}[(1+|s|^2)^{p_i(x)/2}-1]$ for all
$i\in\{1,...,N\}$, and we obtain the anisotropic variable mean
curvature operator
\begin{equation*}\label{curv}
\di\sum_{i=1}^N\di\partial_{x_i}\left[\left(1+\left|\di\partial_{x_i}u
\right|^2\right)^{(p_i(x)-2)/2}\di\partial_{x_i}u \right].
\end{equation*}

Notice that the general operator given by \eqref{eop}  can admit degenerate and singular points. So, since there are a number of features standing in the favor of this $\overset{\rightarrow} p(\cdot)$ - Laplace type operator described by \eqref{eop} and (A1) - (A3), there is no surprise in the fact that there are already papers treating problems with this kind of operator. To give some examples, we refer to \cite{TJM, KOT, MihMor} where the authors were preoccupied with Dirichlet problems. We, on the other hand, are interested in a Neumann problem. One of the novelties of our work is that we consider a problem  with a nonlinear term on the boundary for the study of which it is necessary to introduce and prove a trace theorem. More exactly, we analyze the problem:
 \begin{equation}\label{P}
\left\{\begin{array}{ll}
-\di\sum_{i=1}^N\di\partial_{x_i}a_i\left(x,\di\partial_{x_i}u \right)+ b(x)|u|^{p_M(x)-2}u= f(x,u)
&\mbox{in }
\Omega,\\
u\geq 0 &\mbox{in }
\Omega,\\
\di\sum_{i=1}^N a_i\left(x,\di\partial_{x_i}u \right)\nu_i=g(x,u) &\mbox{on }\partial\Omega.
\end{array}\right.
\end{equation}


Here $a_i:\Omega\times\RR\ri\RR$, $i\in\{1,\dots,N\}$, are the Carath\'{e}odory
 functions characterized by (A1) - (A3), $\Omega\subset\RR^N$ is a bounded open set with smooth boundary and $\nu_i$, $i\in\{1,\dots,N\}$, are the components of the outer normal unit vector. As for the rest of the functions involved in \eqref{P}, we are going to enumerate their properties after we make some notation.

For any ${\Omega}\subset\RR^N$ we set
$$C_+(\overline {\Omega})=\{h\in
C(\overline {\Omega})\,:\,\inf_{x\in\Omega}h(x)>1\}$$  and
  we denote
$$h^+=\sup_{x\in{\Omega}}h(x)\qquad\mbox{and}\qquad h^-=
\inf_{x\in{\Omega}}h(x).$$
Moreover,

$$h^\partial(x)=\left\{\begin{array}{lll} {(N-1)h(x)}/[N-h(x)]
&\mbox{if}& h(x)<N,\\
\infty &\mbox{if}& h(x)\geq N.
\end{array}\right.$$


In our case, $\overset{\rightarrow}
p:\overline\Omega\ri\RR^N$, $\overset{\rightarrow} p(x)=(p_1(x),
p_2(x), ... p_N(x))$ with $p_i\in C_+(\overline\Omega)$,
$i\in\{1,...,N\}$, and for all $x\in\overline\Omega$ we put
$$p_M(x)=\max\{p_1(x),\dots,p_N(x)\}\quad\mbox{and}\quad
p_m(x)=\min\{p_1(x),\dots,p_N(x)\}.$$

In addition, for the Carath\'{e}odory
 functions $f:\Omega\times\RR\ri\RR$ and $g:\partial\Omega\times\RR\ri\RR$, we consider the antiderivatives
$ F:\Omega\times\RR\ri\RR$
$$F(x,s)=\int_{0}^{s}f(x,t)dt,$$
respectively
$ G:\partial\Omega\times\RR\ri\RR$
$$G(x,s)=\int_{0}^{s}g(x,t)dt.$$
Under the previous notation, the functions $\overset{\rightarrow} p$, $b$, $f$ and $g$ satisfy the conditions:
\begin{description}
  \item[(B)] $b\in L^\infty(\Omega)$ and there exists $b_0>0$ such that $b(x)\geq b_0$ for all $x\in\Omega$.
  \item[(F)] There exist a positive constant $k_1$ and
 $q\in L_+^\infty(\Omega)$ with $q^+<p_m^-$, such that
$$|f(x,s)|\leq k_1\left(1+|s|^{q(x)-1}\right),$$
for all $x\in\Omega$ and $s\in\RR$.
  \item[(G)] There exist a positive constant $k_2$ and
 $r\in C(\overline\Omega)$ with $r^+<\min_{x\in\partial\Omega}\{p_1^\partial(x),\dots, p_N^\partial(x)\}$ and $r^+<p_m^-$, such that
$$|g(x,s)|\leq k_2\left(1+|s|^{r(x)-1}\right),$$
for all $x\in\partial\Omega$ and $s\in\RR$.
\end{description}


Note that by adding the following assumptions
\begin{description}
  \item[(F0)] $f$ is fulfilling the monotonicity condition
$$(f(x,s)-f(x,t))(s-t)< 0,$$
for all $x\in\Omega$ and $s,\,t\in\RR$ with $s\neq t$.
  \item[(G0)] $g$ is fulfilling the monotonicity condition
$$(g(x,s)-g(x,t))(s-t)< 0,$$
for all $x\in\partial\Omega$ and $s,\,t\in\RR$ with $s\neq t$.
\end{description}
we can deduce the uniqueness of the solution.

The variable exponent spaces have various applications like those in electrorheological fluids \cite{hal, Sob6, rajruz, R, Sob5}, thermorheological fluids \cite{AnRo}, elastic mechanics \cite{z0} or  image restoration \cite{chen}. However, there are some nonhomogenous  materials that have different behavior on different space directions, hence the need for anisotropic spaces with variable exponent. The next section is dedicated to a brief presentation of the variable exponent spaces, putting into the light the properties that are going to help us establish our main results.









\section{Abstract framework and preliminary results}



In this section we will use the notation ${\Omega_0}$ for a generic set from $\RR^N$ that will be used to recall definitions and properties. For any
measurable subset ${\Omega_0}\subset\RR^N$, $N\geq 2$, with $0<|{\Omega_0}|<\infty$
(where $|{\Omega_0}|$ represents the Lebesgue measure of
the set ${\Omega_0}$), we consider
$p\in
C_+(\overline\Omega_0)$. The isotropic Lebesgue space with variable exponent is defined by
$$L^{p(\cdot)}({\Omega_0})=\{u\,:\, u \mbox{ is a
 measurable real--valued function such that }
\int_{\Omega_0}|u(x)|^{p(x)}\;dx<\infty\}$$ endowed with the
 {  Luxemburg norm}
 \begin{equation}\label{elnorm}
\|u\|_{L^{p(\cdot)}({\Omega_0})}=\inf\left\{\mu>0\;:\quad\;\int_{\Omega_0}\left|
\frac{u(x)}{\mu}\right|^{p(x)}\;dx\leq 1\right\}.
\end{equation}
Notice that the norm of the classical Lebesgue space $L^p({\Omega_0})$, that is,
$$\|u\|_{L^{p}({\Omega_0})}=\left(\int_{{\Omega_0}}|u(x)|^p\right)^{1/p},$$  is a particular case of \eqref{elnorm} for
$p$ constant. Let us recall some of the most important properties of the space
$\left(L^{p(\cdot)}({\Omega_0}),\, \|\cdot\|_{L^{p(\cdot)}({\Omega_0})}
\right)$. This space is a separable and reflexive Banach
space (see \cite[Theorem 2.5, Corollary 2.7]{KR}) and the inclusion
between spaces generalizes naturally.

\begin{theorem}(\cite[Theorem 2.8]{KR})\label{teor Leb inclusion} If $0 <|{\Omega_0}|<\infty$ and
$p_1$, $p_2\in C(\overline\Omega_0;\RR)$, $1<
p_i^-\leq p_i^+<\infty$ ($i=1,2$), are such that $p_1 \leq p_2$
in ${\Omega_0}$, then the embedding $L^{p_2(\cdot)}({\Omega_0})\hookrightarrow L^{p_1(\cdot)}({\Omega_0})$ is continuous.
\end{theorem}


Furthermore, the following H\"older-type
inequality
\begin{equation}\label{Hol}
\left|\int_{\Omega_0} u(x)v(x)\;dx\right|
\leq2\,\|u\|_{L^{p(\cdot)}({\Omega_0})}\|v\|_{L^{p'(\cdot)}({\Omega_0})}
\end{equation}
holds true for all $u\in L^{p(\cdot)}({\Omega_0})$ and $v\in
L^{p'(\cdot)}({\Omega_0})$ (see \cite[Theorem 2.1]{KR}), where we
denoted by $L^{p'(\cdot)}({\Omega_0})$ the conjugate space of
$L^{p(\cdot)}({\Omega_0})$, obtained by conjugating the exponent
pointwise, that is,  $1/p(x)+1/p'(x)=1$ (see \cite[Corollary
2.7]{KR}).

Moreover, the application $\rho_{{\Omega_0},\,p(\cdot)}:L^{p(\cdot)}({\Omega_0})\rightarrow\RR$,
$$\rho_{{\Omega_0},\,p(\cdot)}(u)=\int_{\Omega_0}|u(x)|^{p(x)}\;dx,$$
 called the {$p(\cdot)$-modular} of the
$L^{p(\cdot)}({\Omega_0})$ space, is very useful in handling these
 Lebesgue spaces with variable exponent. Indeed, we have the following properties (see for example \cite[Theorem 1.3, Theorem 1.4]{fanz}). If $u\in L^{p(\cdot)}({\Omega_0})$
 and $p<\infty$ then,
\begin{equation}\label{L40}
\|u\|_{L^{p(\cdot)}({\Omega_0})}<1\;(=1;\,>1)\;\;\;\Leftrightarrow\;\;\;\rho_{{\Omega_0},\,p(\cdot)}(u) <1\;(=1;\,>1)
\end{equation}
\begin{equation}\label{L4}
\|u\|_{L^{p(\cdot)}({\Omega_0})}>1\;\;\;\Rightarrow\;\;\;\|u\|_{L^{p(\cdot)}({\Omega_0})}^{p^-}\leq\rho_{{\Omega_0},\,p(\cdot)}(u)
\leq\|u\|_{L^{p(\cdot)}({\Omega_0})}^{p^+}
\end{equation}
\begin{equation}\label{L5}
\|u\|_{L^{p(\cdot)}({\Omega_0})}<1\;\;\;\Rightarrow\;\;\;\|u\|_{L^{p(\cdot)}({\Omega_0})}^{p^+}\leq \rho_{{\Omega_0},\,p(\cdot)}(u)\leq\|u\|_{L^{p(\cdot)}({\Omega_0})}^{p^-}
\end{equation}
\begin{equation}\label{L06}
\|u\|_{L^{p(\cdot)}({\Omega_0})}\rightarrow
0\;(\rightarrow\infty)\;\;\;\Leftrightarrow\;\;\;\rho_{{\Omega_0},\,p(\cdot)} (u)\rightarrow 0\;(\rightarrow\infty).
\end{equation}
If, in addition, $(u_n)_n\subset L^{p(\cdot)}({\Omega_0})$, then
\begin{equation*}\label{L6}
\lim_{n\ri\infty}\|u_n-u\|_{L^{p(\cdot)}({\Omega_0})}=
0\;\;\;\Leftrightarrow\;\;\;\lim_{n\ri\infty}\rho_{{\Omega_0},\,p(\cdot)} (u_n-u)= 0\;\;\;\Leftrightarrow
\end{equation*}
$$\Leftrightarrow \;\;\;(u_n)_n \mbox{ converges to } u \mbox{ in measure and }\lim_{n\ri\infty}\rho_{{\Omega_0},\,p(\cdot)}
(u_n)=\rho_{{\Omega_0},\,p(\cdot)}(u).$$
\smallskip


Let us introduce now the definition of the isotropic Sobolev space with variable exponent, $W^{1,p(\cdot)}({\Omega_0})$. We set
$$W^{1,p(\cdot)}({\Omega_0})=\left\{u\in L^{p(\cdot)}({\Omega_0}) : \,|\nabla u|\in L^{p(\cdot)}({\Omega_0}) \right\}$$
endowed with the norm
\begin{equation}\nonumber
\|u\|=\|u\|_{L^{p(\cdot)}({\Omega_0})}+\|\nabla u\|_{L^{p(\cdot)}({\Omega_0})},
\end{equation}
where by $\|\nabla u\|_{L^{p(\cdot)}({\Omega_0})}$ we understand
$\|\,|\nabla u|\,\|_{L^{p(\cdot)}({\Omega_0})}$. The space
$\left(W^{1,p(\cdot)}({\Omega_0}),\,\|\cdot\|\right)$ is a separable and reflexive Banach space
(see \cite[Theorem 1.3]{KR}) and we have the following trace theorem.

\begin{theorem}(\cite[Corollary 2.4]{fan0})\label{turma}
Let ${\Omega_0}\subset\RR^N$, $N\geq 2$, be a bounded open set with smooth
boundary. Suppose that $p\in C_+(\overline{{\Omega_0}})$ and $r\in
C(\overline{{\Omega_0}})$ satisfy the condition
\begin{equation}\nonumber
1\leq r(x) < p^\partial(x),\qquad\forall\, x\in \partial{\Omega_0}.
\end{equation}
Then there is a compact boundary trace embedding $
W^{1,p(\cdot)}({\Omega_0}) \hookrightarrow L^{r(\cdot)}
(\partial{\Omega_0}).$
\end{theorem}

We refer to \cite{fan0} for more details regarding the extension
of the classical trace to Lebesgue-Sobolev spaces with variable
exponent. Everywhere below, when we refer to the trace of $u$ we will
 write $u$ instead of $u|_{\partial{\Omega_0}}$ or $\gamma u$.

Next, we will introduce the space $W^{1,\overset{\rightarrow} p(\cdot)}(\Omega_0)$, where
$\overset{\rightarrow} p:\overline\Omega_0\rightarrow\RR^N$ is the
vectorial function $$\overset{\rightarrow}
p(\cdot)=\left(p_1(\cdot), ...,p_N(\cdot)\right)$$
and $p_i\in
C_+(\overline\Omega_0)$ for all $i\in\{1,\dots,N\}$. The
anisotropic variable exponent Sobolev
space is defined by
\begin{eqnarray*}
W^{1,\overset{\rightarrow} p(\cdot)}(\Omega_0)&=&\{u\in L^{p_M(\cdot)}(\Omega_0):\quad \partial_{x_i}u\in L^{p_i(\cdot)}(\Omega_0)\mbox{ for all } i\in\{1,\dots,N\}\}\\
&=&\{u\in L^{1}_{loc}(\Omega_0):\quad u\in L^{p_i(\cdot)}(\Omega_0),\,\partial_{x_i}u\in L^{p_i(\cdot)}(\Omega_0)\mbox{ for all } i\in\{1,\dots,N\}\}
\end{eqnarray*}
endowed with the norm
$$\|u\|_{W^{1,\overset{\rightarrow} p(\cdot)}(\Omega_0)}=\|u\|_{L^{p_M(\cdot)}(\Omega_0)}+\sum_{i=1}^N\left\|\partial_{x_i}u \right\|_{L^{p_i(\cdot)}(\Omega_0)}.$$
The space $\left(W^{1,\overset{\rightarrow} p(\cdot)}(\Omega_0),\|\cdot\|_{W^{1,\overset{\rightarrow} p(\cdot)}(\Omega_0)}\right)$ is a reflexive Banach space (see \cite[Theorem 2.1, Theorem 2.2]{fananis}) and the following embedding theorem takes place.

\begin{theorem}(\cite[Corollary 2.1]{fananis})\label{th anis embedding}
Let $\Omega_0\subset\RR^N$ be a bounded open set and for all $i\in\{1,\dots,N\}$, $p_i\in L^\infty(\Omega_0)$, let $p_i(x)\geq 1$ a.e. in $\Omega_0$. Then for any $q\in L^\infty(\Omega_0)$ with $q(x)\geq 1$ a.e. in $\Omega_0$ such that
$${\rm ess}\inf_{x\in\Omega_0}(p_M(x)-q(x))>0$$
we have the compact embedding
$$W^{1,\overset{\rightarrow} p(\cdot)}(\Omega_0)\hookrightarrow L^{q(\cdot)}(\Omega_0).$$
\end{theorem}

As we announced in the introduction, a trace theorem is also needed and we will provide one in the following section. Keep in mind that, since $p_i^->1$,
$$W^{1,\overset{\rightarrow} p(\cdot)}(\Omega_0)\hookrightarrow W^{1,1}(\Omega_0)\quad\mbox{continously}$$
and by the Gagliardo trace theorem
$$W^{1,1}(\Omega_0)\hookrightarrow L^{1}(\partial\Omega_0)\quad\mbox{compactly},$$
with $\Omega_0\subset\RR^N$ being a bounded open set with smooth boundary.
Hence for $u\in W^{1,\overset{\rightarrow} p(\cdot)}(\Omega_0)$ the trace has definite meaning.

Meanwhile, let us recall two other
important results that are necessary for the proofs from Section 3.


\begin{theorem}(\cite[Theorem 6.2.1.]{KZ})\label{tconv}

Let $X$ be a reflexive Banach space, and let $f : M\subseteq X\ri\RR$ be
G\^{a}teaux differentiable over the closed, convex set $M$. Then the following conditions
are equivalent:


(i) f is convex over $M$.

(ii) We have
$$f(u)-f(v)\geq \langle f'(v), u-v\rangle_{X^\star\times X}\quad \forall\, u,\,v\in M,$$
where $X^\star$ denotes the dual of the space $X$.


(iii) The first G\^ateaux derivative is monotone, that is,

$$ \langle f'(u)- f'(v), u-v\rangle_{X^\star\times X}\geq 0,\quad \forall\, u,\,v\in M.$$

(iv) The second G\^ateaux derivative of $f$ exists and it is positive, that is,

$$ \langle f''(u)\circ v, v\rangle_{X^\star\times X}\geq 0,\quad \forall \,v\in M.$$
\end{theorem}



\begin{theorem}(\cite[1.2 Theorem]{Struwe})\label{tinf}
Suppose $X$ is a reflexive Banach space with norm $\|\cdot\|_X$
and let $M\subset X$ be a weakly closed subset of $X$. Suppose
$\Phi:M\ri\RR\cup\{\infty\}$ is coercive and (sequentially) weakly
lower semi-continuous on $M$ with respect to $X$, that is, suppose
the following conditions are fulfilled:


(i) $\Phi(u)\ri\infty$ as $\|u\|_X\ri\infty$, $u\in M.$

(ii) For any $u\in M$, any subsequence $(u_m)_m$ in $M$ such
that $u_m\rightharpoonup u$ weakly in $X$ there holds
$$\Phi(u)\leq\liminf_{m\ri\infty} \Phi(u_m).$$




Then $\Phi$ is bounded from below on $M$ and attains its infimum in
$M$.
\end{theorem}












\section{Main results}

Before analyzing the existence of solutions to problem \eqref{P} we introduce a trace theorem.

\begin{theorem}\label{turmanoua}
Let ${\Omega}\subset\RR^N$ ($N\geq 2$) be a bounded open set with smooth
boundary and let $\overset{\rightarrow} p\in \left(C_+(\overline{{\Omega}})\right)^N$, $r\in
C(\overline{{\Omega}})$ satisfy the condition
\begin{equation}\nonumber
1\leq r(x) < \min_{x\in\partial\Omega}\{p_1^\partial(x),\dots, p_N^\partial(x)\},\qquad\forall\, x\in \partial{\Omega}.
\end{equation}
Then there is a compact boundary trace embedding
$$W^{1,\overset{\rightarrow} p(\cdot)}(\Omega) \hookrightarrow L^{r(\cdot)}
(\partial{\Omega}).$$
\end{theorem}



\begin{proof}

For clarity, for any subset $M\subset\overline\Omega$ and any function $h$, we will use the notation
$$h^-(M)=\inf_{x\in M}h(x)\qquad\mbox{and}\qquad h^+(M)=\sup_{x\in M}h(x).$$
We know that $r(x) < \min_{x\in\partial\Omega}\{p_1^\partial(x),\dots, p_N^\partial(x)\},$ for all $x\in \partial{\Omega}$. In particular, for any $x\in \partial{\Omega}$, we have that $r(x) < p_i^\partial(x)$, for all $i\in \{1,\dots,N\}$. Then for an arbitrarily fixed $i$ and for any given $x\in \partial\Omega$, there exists a relatively open neighborhood $\Omega_x$ of $x$ in $\overline\Omega$ such that
$$r^+(\partial\Omega\cap \Omega_x)<p_i^\partial(x).$$
Thus, by Theorem \ref{turma} there exists $\tilde{c}_i>0$ such that
$$\|u\|_{L^{r^+(\partial\Omega\cap \Omega_x)}(\partial\Omega\cap \Omega_x)}\leq \tilde{c}_i \|u\|_{W^{1, p_i(\cdot)}(\Omega_x)},$$
that is,
\begin{equation}\label{eeq1}
\|u\|_{L^{r^+(\partial\Omega\cap \Omega_x)}(\partial\Omega\cap \Omega_x)}\leq \tilde{c}_i\left( \|u\|_{L^{ p_i(\cdot)}(\Omega_x)}+\|\partial_{x_i}u\|_{L^{ p_i(\cdot)}(\Omega_x)}\right).
\end{equation}
Due to the fact that $p_i(x)\leq p_M(x)$ for all $x\in\overline\Omega$, by Theorem \ref{teor Leb inclusion} there exists $\tilde{c}_0>0$ such that
\begin{equation}\label{eeq2}
\|u\|_{L^{ p_i(\cdot)}(\Omega_x)}\leq \tilde{c}_0 \|u\|_{L^{ p_M(\cdot)}(\Omega_x)}.
\end{equation}
Using \eq{eeq1} and \eq{eeq2} we get
$$\|u\|_{L^{r^+(\partial\Omega\cap \Omega_x)}(\partial\Omega\cap \Omega_x)}\leq \tilde{c} \|u\|_{W^{1,\overset{\rightarrow} p(\cdot)}(\Omega_x)},$$
where $\tilde{c}=\frac{\max\{\tilde c_0,1\}}{N}\sum_{i=1}^{N}c_i>0.$ Hence
\begin{equation}\label{eeq3}
W^{1,\overset{\rightarrow} p(\cdot)}(\Omega_x) \hookrightarrow L^{r^+(\partial\Omega\cap \Omega_x)}
(\partial{\Omega}\cap \Omega_x)\quad\mbox{continuously}.
\end{equation}
On the other hand, the fact that $p_i(x)>1$ for all $x\in\overline\Omega$ implies that
\begin{equation}\label{eeq4}
W^{1,\overset{\rightarrow} p(\cdot)}(\Omega_x) \hookrightarrow W^{1,1} (\Omega_x)\quad\mbox{continuously}.
\end{equation}
Moreover, by the Gagliardo trace theorem,
\begin{equation}\label{eeq5}
W^{1,1} (\Omega_x) \hookrightarrow L^{1}
(\partial{\Omega}\cap \Omega_x)\quad\mbox{compactly}.
\end{equation}
Combining \eq{eeq4} and \eq{eeq5} we obtain that
\begin{equation}\label{eeq6}
W^{1,\overset{\rightarrow} p(\cdot)}(\Omega_x) \hookrightarrow L^{1}
(\partial{\Omega}\cap \Omega_x)\quad\mbox{compactly}.
\end{equation}
By \eq{eeq3}, \eq{eeq6} and by interpolation between $L^{1}
(\partial{\Omega}\cap \Omega_x)$ and $L^{r^+(\partial\Omega\cap \Omega_x)}
(\partial{\Omega}\cap \Omega_x)$ we arrive at
$$W^{1,\overset{\rightarrow} p(\cdot)}(\Omega_x) \hookrightarrow L^{r^+(\partial\Omega\cap \Omega_x)}
(\partial{\Omega}\cap \Omega_x)\quad\mbox{compactly}.$$
Since $r(x)<r^+(\partial\Omega\cap \Omega_x)$ we use again Theorem \ref{teor Leb inclusion} and by the above relation
$$W^{1,\overset{\rightarrow} p(\cdot)}(\Omega_x) \hookrightarrow L^{r(\cdot)}
(\partial{\Omega}\cap \Omega_x)\quad\mbox{compactly}.$$
Taking into account the finite covering theorem for the compact set $\partial\Omega$ we can deduce that
$$W^{1,\overset{\rightarrow} p(\cdot)}(\Omega) \hookrightarrow L^{r(\cdot)}
(\partial{\Omega})\quad\mbox{compactly}.$$

\end{proof}

We now have all the necessary tools to prove the existence of a weak solution to \eq{P}. Before starting the discussion, we point out that everywhere below we will work under the following hypotheses.

\bigskip

\textbf{{Hypotheses (H).}}
\emph{We consider $\Omega\subset\RR^N$, $N\geq 2$, to be a bounded open set with smooth boundary and $\overset{\rightarrow} p\in\left(C_+(\overline\Omega)\right)^N$. We assume that for all $i\in\{1,\dots,N\}$, the applications $a_i,\,f:\Omega\times\RR\ri\RR$, $g:\partial\Omega\times\RR\ri\RR$ are Carath\'{e}odory functions satisfying (A1) - (A3), (F), respectively (G), and $b:\Omega\ri\RR$ satisfies (B).}

\bigskip

For simplicity, we denote
$$W^{1,\overset{\rightarrow} p(\cdot)}(\Omega)=E\quad\mbox{and}\quad \|\cdot\|_{W^{1,\overset{\rightarrow} p(\cdot)}(\Omega)}=\|\cdot\|.$$

Now let us define the notion of weak solution.


\begin{definition}
By a weak solution to problem \eq{P} we understand a function $u\in
E$ such that
\begin{equation}\label{eq weak sol}
\int_{\Omega}\sum_{i=1}^Na_i\left(x,\di\partial_{x_i}u
\right)\,\partial_{x_i}\phi\,dx+
\int_{\Omega}{b(x)}|u|^{p_M(x)-2}u\phi\,dx - \int_{\Omega}f(x,u)\phi\,dx- \int_{\partial\Omega}g(x,u)\phi\,dS=0,
\end{equation}
for all $\phi\in E$.
\end{definition}

To problem \eq{P} we associate an energetic functional
$I:E\rightarrow\RR$, defined by


\begin{equation}\nonumber
I(u)=\int_{\Omega}\sum_{i=1}^N\,A_i\left(x,\di\partial_{x_i}u
\right)\,dx+ \int_{\Omega}\frac{b(x)}{p_M(x)}|u|^{p_M(x)}\,dx- \int_{\Omega}F(x,u_+)\,dx-\int_{\partial\Omega}G(x,u_+)\,dx,
\end{equation}
where $u_+(x)=\max\{u(x),0\}$.


To easy our writing, we denote by $\Lambda,\, J:E\rightarrow\RR$
the functionals
$$\Lambda(u)=\int_{\Omega}\sum_{i=1}^N\,A_i\left(x,\di\partial_{x_i}u
\right)dx$$
and
$$J(u)=\int_{\Omega}\sum_{i=1}^N\,A_i\left(x,\di\partial_{x_i}u
\right)\,dx+ \int_{\Omega}\frac{b(x)}{p_M(x)}|u|^{p_M(x)}\,dx=\Lambda(u)+ \int_{\Omega}\frac{b(x)}{p_M(x)}|u|^{p_M(x)}\,dx.$$
We remind the following result.

\begin{lemma} \label{l1} (see \cite[Lemma 3.4]{KOT})
The functional $\Lambda$ is well-defined on $E$. In addition,
 the functional  $\Lambda$ is of class
$C^{1}(E,\mathbb{R})$ and
\[
\langle \Lambda '(u),\varphi\rangle
=\int_{\Omega}\sum_{i=1}^N\,a_{i}(x,\di\partial_{x_i}u) \di\partial_{x_i} \varphi\,
dx,
\]
 for all $u,\varphi\in E$.
\end{lemma}

\begin{remark}
The study from \cite{KOT} is conducted for $E=W_0^{1,\overset{\rightarrow} p(\cdot)}(\Omega)$, that is, the closure of $C_0^\infty(\Omega)$ in $W_0^{1,\overset{\rightarrow} p(\cdot)}(\Omega)$
 with respect to the norm
$$\|u\|_{W_0^{1,\overset{\rightarrow} p(\cdot)}(\Omega)}=\sum_{i=1}^N\left\|\partial_{x_i}u \right\|_{L^{p_i(\cdot)}(\Omega)}.$$
Since we want to avoid repeating the same arguments and the calculus is almost
identical, we omit the proof.
\end{remark}



Due to Lemma \ref{l1}, a standard calculus leads to the fact that $I$ is well-defined on
$E$ and $I\in C^{1}(E,\mathbb{R})$ with the derivative given by
\[
\langle I'(u),\varphi\rangle=\int_{\Omega}\sum_{i=1}^Na_i\left(x,\di\partial_{x_i}u
\right)\,\partial_{x_i}\phi\,dx+
\int_{\Omega}{b(x)}|u|^{p_M(x)-2}u\phi\,dx - \int_{\Omega}f(x,u)\phi\,dx- \int_{\partial\Omega}g(x,u)\phi\,dS,
\]
for all $u,\varphi\in E$. Obviously, the critical points
of $I$ are weak solutions to \eqref{P}, so, by means of Theorem \ref{tinf}, we intend to establish the existence of critical points in order to deduce the existence of weak solutions. Our second main result is the following.

\begin{theorem}\label{teor exist}
If hypotheses (H) are fulfilled, then there exists a weak solution to problem \eq{P}.
\end{theorem}


As suggested above, for the proof of Theorem \ref{teor exist} we show that our energetic functional $I$ fulfils the hypotheses of Theorem \ref{tinf}. To this end, we proceed with the following lemma.

\begin{lemma}\label{lema coerciv}
If hypotheses (H) are fulfilled, then the functional $I$ is coercive.
\end{lemma}

\begin{proof}

Let $u\in E$ be such that $\|u\|\ri\infty$. Using (A2) we deduce
\begin{equation}\label{cluj002}
\Lambda(u)\geq\frac{1}{p_M^+}\sum_{i=1}^N\int_{\Omega}|\di\partial_{x_i}u|^{p_i(x)}.
\end{equation}

We make the following notation:

$${\cal I}_{1}= \{i\in\{1,\dots, N\} : \quad\|\partial_{x_i}u\|_{L^{p_i(\cdot)}}\leq1\}$$
and
$${\cal I}_{2}= \{i\in\{1,\dots, N\} : \quad\|\partial_{x_i}u\|_{L^{p_i(\cdot)}}>1\}.$$

By \eqref{L40}, \eqref{L4}, \eqref{L5} and \eqref{cluj002} we find that
\begin{eqnarray*}
\Lambda(u)&\geq&\frac{1}{p_M^+}\left( \di\sum_{i\in{\cal
I}_{1}}\left\|\di\partial_{x_i}u\right\|^{p_M^+}_{L^{p_i(\cdot)}}+\di\sum_{i\in{\cal I}_{2}}\left\|\di\partial_{x_i}u\right\|^{p_m^-}_{L^{p_i(\cdot)}}\right)\\
&\geq&\frac{1}{p_M^+}\left(\sum\limits_{i=1}^N\left\|\di\partial_{x_i}u
\right\|^{p_m^-}_{L^{p_i(\cdot)}}-\di\sum_{i\in{\cal
I}_{1}}\left\|\di\partial_{x_i}u \right\|_{L^{p_i(\cdot)}}^{p_m^-}\right)\\
&\geq&\frac{1}{p_M^+}\left(\sum\limits_{i=1}^N\left\|\di\partial_{x_i}u
\right\|^{p_m^-}_{L^{p_i(\cdot)}}-N\right).
\end{eqnarray*}


By the generalized mean inequality or the Jensen inequality applied to the convex function
$a:\RR^+\ri \RR^+$, $a(t)=t^{p_m^-}$,
$p_m^-> 1$  we get
\begin{equation}\label{ineg J}
\Lambda(u)\geq
\frac{1}{p_M^+}\left[\di\frac{1}{N^{p_m^--1}}\left(\sum\limits_{i=1}^N\left\|\di\partial_{x_i}u
\right\|_{L^{p_i(\cdot)}}\right)^{p_m^-}-N\right].
\end{equation}

\medskip

We analyze now the two cases corresponding to the values of $\|u\|_{L^{p_M(\cdot)}}$.

\smallskip

\emph{Case 1:} $\|u\|_{L^{p_M(\cdot)}}\geq 1.$

By \eq{ineg J} and (B) we have
$$J(u)\geq \frac{1}{p_M^+}\left[\di\frac{1}{N^{p_m^--1}}\left(\sum\limits_{i=1}^N\left\|\di\partial_{x_i}u
\right\|_{L^{p_i(\cdot)}}\right)^{p_m^-}-N\right]+\frac{b_0}{p_M^+}\|u\|^{p_m^-}_{L^{p_M(\cdot)}},$$
thus
\begin{equation}\label{case 1}
J(u)\geq
\frac{1}{2^{p_m^-}p_M^+}\min\left\{\frac{1}{N^{p_m^--1}},b_0\right\}\|u\|^{p_m^-} -\frac{1}{p_M^+N^{p_m^--2}}.
\end{equation}

\bigskip

\emph{Case 2:} $\|u\|_{L^{p_M(\cdot)}}< 1.$ Then
\begin{eqnarray*}
J(u)&\geq& \frac{1}{p_M^+}\left[\di\frac{1}{N^{p_m^--1}}\left(\sum\limits_{i=1}^N\left\|\di\partial_{x_i}u
\right\|_{L^{p_i(\cdot)}}\right)^{p_m^-}-N\right]\\
&\geq& \frac{1}{p_M^+}\left[\di\frac{1}{N^{p_m^--1}}\left(\sum\limits_{i=1}^N\left\|\di\partial_{x_i}u
\right\|_{L^{p_i(\cdot)}}\right)^{p_m^-}+\|u\|^{p_m^-}_{L^{p_M(\cdot)}}-N-1\right]
\end{eqnarray*}
and we obtain
\begin{equation}\label{case 2}
J(u)\geq
\frac{1}{2^{p_m^-}p_M^+}\min\left\{\frac{1}{N^{p_m^--1}},1\right\}\|u\|^{p_m^-} -\frac{N+1}{p_M^+N^{p_m^--1}}.
\end{equation}
 In conclusion, by \eq{case 1} and \eq{case 2} we deduce that there exists $\tilde k_0,\,\tilde k_3>0$ such that
\begin{equation}\label{cases 1 and 2}
J(u)\geq
\tilde k_0\|u\|^{p_m^-} -\tilde k_3.
\end{equation}

\medskip

Let us evaluate now the other two terms from the formula of $I$. By (F),

$$\int_{\Omega}F(x,u_+)\,dx\leq k_1 \|u\|_{L^1(\Omega)} + \frac{k_1}{q^-}\int_{\Omega}u_+^{q(x)}dx.$$
Since

$$(u_+(x))^{q(x)}\leq|u(x)|^{q^-}+|u(x)|^{q^+}\quad \forall
x\in\overline{\Omega}$$
and
$$L^{q^+}(\Omega)\hookrightarrow L^{q^-}(\Omega)\hookrightarrow L^{1}(\Omega)\quad \mbox{continuously}$$
we infer that there exists $\bar k_1>0$ such that
\begin{equation}\label{ani2}
\int_{\Omega}F(x,u_+)dx\,\leq
\bar k_1\left(\|u\|_{L^{q^+}(\Omega)}^{q^-}+\|u\|_{L^{q^+}(\Omega)}^{q^+}\right).
\end{equation}
Theorem \ref{th anis embedding} yields the continuous embedding
$$E\hookrightarrow L^{q^+}(\Omega)$$
because $q^+<p_m^-$.
Thus, using \eq{ani2} and the fact that $\|u\|\geq 1$, we deduce that there exists $\tilde k_1>0$ such that
\begin{equation}\label{ani3}
\int_{\Omega}F(x,u_+)\,dx\leq \tilde k_1\|u\|^{q^+}.
\end{equation}

Using hypothesis (G) instead of (F) and Theorem \ref{turmanoua} instead of Theorem \ref{th anis embedding}, by similar arguments we obtain the existence of a positive constant $\tilde k_2$ such that
\begin{equation}\label{ani4}
\int_{\Omega}G(x,u_+)\,dx\leq \tilde k_2\|u\|^{r^+},
\end{equation}
where $r$ is given by condition (G). Putting together relations \eqref{cases 1 and 2}, \eq{ani3} and \eq{ani4}, we arrive at
$$I(u)\geq \tilde k_0\|u\|^{p_m^-}-\tilde k_1\|u\|^{q^+} -\tilde k_2\|u\|^{r^+}-\tilde k_3.$$
Keeping in mind the fact that $q^+,r^+<p_m^-$, we find that $I(u)\ri\infty$ when $\|u\|\ri\infty$, hence $I$ is coercive.

\end{proof}


\bigskip

\begin{lemma}\label{lemma semicont}
If hypotheses (H) are fulfilled, then the functional $I$ is weakly lower semi-continuous.
\end{lemma}

\begin{proof}
We start by showing that $J$ is weakly lower semi-continuous.
By \cite[Corollary III.8]{brezis}, it is enough to prove that
$J$ is lower semi-continuous. To this end,
fix $u\in E$ and $\epsilon>0$. By (A3) and Theorem \ref{tconv} we deduce that for any $v\in E$, the following
inequality holds
$$
J(v)
\geq J(u)
+\sum_{i=1}^{N}\int_{\Omega}a_{i}(x,\di\partial_{x_i}u)
(\di\partial_{x_i}v-\di\partial_{x_i}u)\,dx+
\int_{\Omega}{b(x)}|u|^{p_M(x)-2}u(v-u)\,dx.
$$
Using (A1), (B) and the H\"{o}lder-type inequality \eqref{Hol} we infer
\begin{eqnarray*}
J(v)
&\geq&J(u)
-\max\{\bar c_1,\dots, \bar c_N\}\sum_{i=1}^{N}\int_{\Omega}d_{i}(x)|\partial_{x_i} v-\partial_{x_i}u|\,dx
\\
& & -\max\{\bar c_1,\dots, \bar c_N\}\sum_{i=1}^{N}\int_{\Omega}|\di\partial_{x_i}u|^{p_{i}(x)-1}
|\frac{\partial}{\partial x_{i}} (v-u)|\,dx\\
& &-\|b\|_{L^\infty(\Omega)}\int_{\Omega}\left||u|^{p_M(x)-2}u\right||v-u|\,dx
\\
&\geq&J(u)
 -2\max\{\bar c_1,\dots, \bar c_N\}\sum_{i=1}^{N}\|d_{i}\|_{L^{p_{i}'(\cdot)}(\Omega)}\|\partial_{x_i}
 v-\partial_{x_i}u\|_{L^{p_{i}(\cdot)}(\Omega)}
\\
& &-2\max\{\bar c_1,\dots, \bar c_N\}\sum_{i=1}^{N}\||\partial_{x_i}
 u|^{p_{i}(x)-1}\|_{L^{p_{i}'(\cdot)}(\Omega)}\|\partial_{x_i}
 v-\partial_{x_i}u\|_{L^{p_{i}(\cdot)}(\Omega)}
\\
& &-\|b\|_{L^\infty(\Omega)}
 \||u|^{p_{M}(x)-1}\|_{L^{p_{M}'(\cdot)}(\Omega)}\|
 v-u\|_{L^{p_{M}(\cdot)}(\Omega)}\\
\end{eqnarray*}
The above inequality and relation \eq{L06} imply that there exists $C>0$ such that
$$J(v)\geq J(u)
 -C\|v-u\|
\geq J(u)-\epsilon,
$$
for all $v\in E$ with
$\|v-u\|<\delta=\epsilon/C$.
Therefore $J$ is weakly lower semi-continuous.

\medskip
Next, we denote
 $$ h_1(u)=\int_{\Omega}F(x,u)\,dx\quad\mbox{and}\quad h_2(u)=\int_{\partial\Omega}G(x,u)\,dx.$$
Then $h'_1,\,h'_2:E\to E^{\star}$
are completely continuous, that is,
if $u_{n}\rightharpoonup u$, then $h'_1(u_{n})\to h'_1(u)$ and $h'_2(u_{n})\to h'_2(u)$.
Hence the functionals $h'_1,\,h'_2$ are weakly continuous and, since $J$ is weakly lower semi-continuous, we conclude that $I$ is weakly lower semi-continuous.
\end{proof}

\bigskip
\textbf{Proof of Theorem \ref{teor exist}.}
The proof follows directly from Lemma \ref{lema coerciv}, Lemma \ref{lemma semicont} and Theorem \ref{tinf}.
\endproof


Since we have an existence result, we are concerned with the uniqueness of solution.

\begin{theorem}\label{t uniqueness}
If, in addition to the hypotheses (H), the conditions (F0), (G0) are fulfilled, then the weak solution to problem \eqref{P} is unique.
\end{theorem}


\begin{proof}
We suppose there exist two weak solutions to problem
\eqref{P}, that is, $u_1$ and $u_2$. We replace the solution $u$ by
$u_1$ in \eqref{eq weak sol} and we choose $\varphi=u_1-u_2$. Then
$$
\int_{\Omega}\sum_{i=1}^Na_i\left(x,\di\partial_{x_i}u_1
\right)\,\partial_{x_i}(u_1-u_2)\,dx + \int_{\Omega}b(x) |u_1|^{p_M(x)-2}u_1(u_1-u_2)\,dx$$$$-\int_{\Omega} f(x,u_1)(u_1-u_2)\,dx-\int_{\partial\Omega} g(x,u_1)(u_1-u_2)\,dx=0.
$$
Next, we replace the solution $u$ by $u_2$ in \eqref{eq weak sol} and we
choose $\varphi=u_2-u_1$. We have
$$
\int_{\Omega}\sum_{i=1}^Na_i\left(x,\di\partial_{x_i}u_2
\right)\,\partial_{x_i}(u_2-u_1)\,dx + \int_{\Omega}b(x) |u_2|^{p_M(x)-2}u_2(u_2-u_1)\,dx$$$$- \int_{\Omega}f(x,u_2)(u_2-u_1)\,dx-\int_{\partial\Omega} g(x,u_2)(u_2-u_1)\,dx=0.
$$


By combining the previous two equalities we obtain
\begin{eqnarray*}
\int_{\Omega}\left\{\sum_{i=1}^N\left[a_i\left(x,\di\partial_{x_i}u_1\right)
-a_i\left(x,\di\partial_{x_i}u_2
\right)\right]\,(\partial_{x_i}u_1-\partial_{x_i}u_2)\right\}\,dx \\ + \int_{\Omega} b(x) \left[|u_1|^{p_M(x)-2}u_1-|u_2|^{p_M(x)-2}u_2\right](u_1-u_2)\,
 dx \\- \int_{\Omega} \left[f(x,u_1)-f(x,u_2)\right](u_1-u_2)\,dx\\- \int_{\partial\Omega} \left[g(x,u_1)-g(x,u_2)\right](u_1-u_2)\,dx=0.
\end{eqnarray*}

By (A3), (F0) and (G0), all the terms in the above equality are
positive unless $u_1=u_2$, and this yields the uniqueness of the
solution.
\end{proof}















\section{Additional comments}

One of the questions that the reader might ask him/herself is the following: what happens if we change the order of the exponents $q^+$ and $p_m^-$, or $r^+$ and $p_m^-$? Does problem \eqref{P} still have solution? Two of the most well known variational tools are the mountain pass theorem of Ambrosetti and Rabinowitz and the Ekeland variational principle (see for example \cite{PRad}). Are they suitable for our problem? We consider the following assumptions.
\begin{description}
  \item[(F1)] There exist a positive constant $k_1$ and
 $q\in L_+^\infty(\Omega)$ with $1<p_M^+<q^-$, such that
$$|f(x,s)|\leq k_1\left(1+|s|^{q(x)-1}\right),$$
for all $x\in\Omega$ and $s\in\RR$.
  \item[(F2)] $f$ verifies an Ambrosetti-Rabinowitz type
condition: there exists a constant $\alpha_1>p_M^+$ such that for every
$x\in\Omega$
$$0<\alpha_1 F(x,s)\leq sf(x,s),\quad\forall s>0.$$
  \item[(G1)] There exist a positive constant $k_2$ and
 $r\in C(\overline\Omega)$ with $1< p_M^+<r^-$ and $r(x)<\min\{p_1^\partial(x),\dots, p_N^\partial(x)\}$ for all $x\in\partial\Omega$, such that
$$|g(x,s)|\leq k_2\left(1+|s|^{r(x)-1}\right),$$
for all $x\in\Omega$ and $s\in\RR$.
  \item[(G2)] $g$ verifies an Ambrosetti-Rabinowitz type
condition: there exists a constant $\alpha_2>p_M^+$ such that for every
$x\in\partial\Omega$
$$0<\alpha_1 G(x,s)\leq sg(x,s),\quad\forall s>0.$$
\end{description}


If we replace conditions (F) and (G) by (F1), respectively (G1),
and we add
conditions (F2), (G2)
 to the hypotheses (H), we can expect to obtain the existence of solutions to problem \eq{P} by using the mountain pass theorem instead of Theorem \ref{tinf}. However, a first remark is that we need a better embedding theorem because when $q^->p_M^+$ we can not apply Theorem \ref{th anis embedding}. For $h,\,p_i\in C_+(\overline {\Omega})$, $i\in\{1,\dots,N\}$, we make the notation
$$h^\star(x)=\left\{\begin{array}{lll} {Nh(x)}/[N-h(x)]
&\mbox{if}& h(x)<N,\\
\infty &\mbox{if}& h(x)\geq N,
\end{array}\right.$$
and
$$\bar p(x)=\frac{N}{\sum_{i=1}^{N}{1}/{p_i(x)}}.$$
Furthermore, we say that a set $\Omega\subset\RR^N$ is a rectangular-like domain if it is a union
of finitely many rectangular domains with edges parallel to the
coordinate axes in $\RR^N$. Recently, the following result was proved.

\begin{theorem}(see \cite[Theorem 2.5]{fananis})\label{t rectang}
Let $\Omega\subset\RR^N$ be a rectangular-like domain and $p_i\in C_+(\overline {\Omega})$ for all $i\in\{1,\dots,N\}$. If  $q\in C_+(\overline {\Omega})$ and
$$q(x)<\max\{\bar p^\star(x), p_M(x)\}\quad\mbox{for all }x\in\overline\Omega,$$
then we have the compact embedding
$$W^{1,\overset{\rightarrow} p(\cdot)}(\Omega)\hookrightarrow L^{q(\cdot)}(\Omega).$$
\end{theorem}
Notice that, in the statement of Theorem \ref{t rectang}, $\Omega$ is not a set with smooth boundary, hence we can not define the outer normal. Moreover, since $W^{1,\overset{\rightarrow} p(\cdot)}(\Omega)$ can be considered a generalization of the space $W^{1,\overset{\rightarrow} p}(\Omega)$, with the exponent $\overset{\rightarrow} p$ being a constant vector, there are reasons to believe that Theorem \ref{t rectang} fails for domains that are not of the rectangular type, see \cite{monatsh}. Therefore, at least for the time being, we must remain faithful to Theorem \ref{th anis embedding} and we can not apply the mountain pass theorem.

\smallskip

The next question arrives: do we have an existence result if in the hypotheses (H) we replace (G) by (G1), we keep (F) as it is and we add (F2), (G2)? Unfortunately, there is another impediment in our way: we do not know if $\Lambda$ is of type (S+). We recall that $\Lambda$ is said to be of type (S+) if any sequence $(u_n)_n\subset E$ that is weakly convergent to
$u\in E$ such that
\begin{equation}\nonumber
\limsup\limits_{n\rightarrow\infty}\langle J^{'}(u_n),u_n-u\rangle
\leq 0
\end{equation}
converges strongly to $u$ in $E$. Due to \cite[Lemma 2]{TJM}, we know that $\Lambda$ is of type (S+) when $E=W_0^{1,\overset{\rightarrow} p(\cdot)}(\Omega)$. To follow the idea from the proof of \cite[Lemma 2]{TJM} and to apply it to the case $E=W^{1,\overset{\rightarrow} p(\cdot)}(\Omega)$ we need the following embedding:
\begin{equation}\label{bad emb}
W^{1,\overset{\rightarrow} p(\cdot)}(\Omega)\hookrightarrow L^{p_M(\cdot)}(\Omega)\quad\mbox{compactly}.
\end{equation}
This embedding is not provided by Theorem \ref{th anis embedding}, and, as we previously discussed, we do not have an improved embedding theorem for open bounded sets with smooth boundary. Adding a uniform convexity assumption to (A1) - (A3), the authors of \cite{MihMor} obtained a result similar to \cite[Lemma 2]{TJM}. However, the problem persists: the proof can be adapted to the case of $W^{1,\overset{\rightarrow} p(\cdot)}(\Omega)$ from the case $W_0^{1,\overset{\rightarrow} p(\cdot)}(\Omega)$ if \eq{bad emb} holds. Therefore, until this difficulty is surpassed, we can use neither the mountain pass theorem, nor the Ekeland principle.

\bigskip

And finally, we make our last comment. There exists a weak solution to the problem


\begin{equation}\label{P2}
\left\{\begin{array}{ll}
-\di\sum_{i=1}^N\di\partial_{x_i}a_i\left(x,\di\partial_{x_i}u \right)+ b_1(x)|u|^{p_M(x)-2}u= f(x,u)+b_2(x)
&\mbox{in }
\Omega,\\
u\geq 0 &\mbox{in }
\Omega,\\
\di\sum_{i=1}^N a_i\left(x,\di\partial_{x_i}u \right)\nu_i=g(x,u) &\mbox{on }\partial\Omega,
\end{array}\right.
\end{equation}
where $\Omega,\,a_i,\,b_1,\,\overset{\rightarrow} p,\,f,\,g$ satisfy hypotheses (H) and $b_2\in L^\infty(\Omega)$. Obviously the proof will follow the same steps as those presented in Section 3 and the last term will not raise any problems when showing that the functional attached to problem \eq{P2} is coercive and weakly lower semi-continuous. We bring into attention problem \eq{P2} only because, by adding the function $b_2$, the function $u\equiv 0$ can not be a solution, independently of the form of $f$ and $g$.


\bigskip

\noindent{\bf Acknowledgment.}  \noindent The authors acknowledge the support through
 Grant CNCSIS--PNII-ID-PCE-2011-3-0195.


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\end{document} 