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

\AtBeginDocument{{\noindent\small
\emph{Electronic Journal of Differential Equations},
Vol. 2013 (2013), No. 158, pp. 1--12.\newline
ISSN: 1072-6691. URL: http://ejde.math.txstate.edu or http://ejde.math.unt.edu
\newline ftp ejde.math.txstate.edu}
\thanks{\copyright 2013 Texas State University - San Marcos.}
\vspace{9mm}}

\begin{document}
\title[\hfilneg EJDE-2013/158\hfil Existence of solutions]
{Existence of solutions for a Neumann  problem involving the $p(x)$-Laplacian}

\author[G. Barletta,  A. Chinn\`i \hfil EJDE-2013/158\hfilneg]
{Giuseppina Barletta, Antonia Chinn\`i}  % in alphabetical order

\address{Giuseppina Barletta \newline
Universit\`{a} degli Studi Mediterranea di Reggio Calabria,
MECMAT-Dipartimento di Meccanica e Materiali, Via Graziella,
Localit\`{a} Feo di Vito, 89100 Reggio Calabria, Italy}
\email{giuseppina.barletta@unirc.it}

\address{Antonia Chinn\`i \newline
Department of Civil, Information Technology, Construction,
Environmental Engineering and Applied Mathematics,
University of Messina, 98166 Messina, Italy}
\email{achinni@unime.it}


\thanks{Submitted March 29, 2013. Published July 10, 2013.}
\subjclass[2000]{35J60, 35J20}
\keywords{$p(x)$-Laplacian; variable exponent Sobolev spaces}

\begin{abstract}
 We study the existence and multiplicity of weak solutions
 for a parametric Neumann problem driven by the $p(x)$-Laplacian.
 Under a suitable condition on the behavior of the potential at $0^+$,
 we obtain an interval such that when a parameter $\lambda$ is in this interval,
 our problem admits at least one nontrivial  weak solution.
 We show the multiplicity of solutions for potentials
 satisfying also the Ambrosetti-Rabinowitz condition. Moreover,
 if the right-hand side $f$ satisfies the  Ambrosetti-Rabinowitz condition,  
 then we obtain the existence of two nontrivial weak solutions.
\end{abstract}

\maketitle
\numberwithin{equation}{section}
\newtheorem{theorem}{Theorem}[section]
\newtheorem{lemma}[theorem]{Lemma}
\newtheorem{proposition}[theorem]{Proposition}
\newtheorem{definition}[theorem]{Definition}
\newtheorem{example}[theorem]{Example}
\newtheorem{remark}[theorem]{Remark}
\allowdisplaybreaks

\section{Introduction}

In this article  we are interested in the multiplicity of weak solutions 
of the Neumann problem
\begin{equation} \label{eq:neumann}
\begin{gathered}
  -\Delta _{p(x)}u + a(x)|u|^{p(x)-2}u = \lambda f(x,u)\quad\text{in }\Omega\\
  \frac{\partial u}{\partial \nu}=0\quad\text{on } \partial\Omega
\end{gathered}
\end{equation}
where $\Omega\subset \mathbb{R}^{N}$ is an open bounded domain with smooth
boundary $\partial\Omega$, $p \in C(\bar{\Omega})$, 
$\Delta _{p(x)}u :=\operatorname{div} (|\nabla u|^{p(x)-2}\nabla u)$ denotes the 
$p(x)$-Laplace operator, $a$ belongs to $L^{\infty}(\Omega)$ and 
$a_{-}:=\operatorname{ess\,inf}_\Omega a(x)>0$, $\lambda$ is a positive 
parameter  and $\nu$ is the outward unit normal to $\partial\Omega$.
 In this context we assume that $p \in C(\bar{\Omega})$ satisfies 
the condition
\begin{equation}\label{eq:p(x)}
1 <p^{-}:=\inf_{x\in\Omega}p(x)\le p(x)\le p^{+}
:=\sup_{x\in \Omega} p(x)< +\infty,
\end{equation}
 and that $f:\Omega\times\mathbb{R}\to\mathbb{R}$ is a Carath\'{e}odory function satisfying
\begin{itemize}
\item[(F1)] there exist $a_1,a_{2}\in [0,+\infty[$ and
$q\in C(\bar{\Omega})$ with $1< q(x) < p^{*}(x)$ for each 
$x\in\bar{\Omega}$, such that
$$
|f(x,t)|\le a_1+ a_{2}|t|^{q(x)-1}
$$
for each $(x,t)\in \Omega\times\mathbb{R}$, where
\begin{equation}\label{eq:pstar}
p^{*}(x):= \begin{cases}
  \frac{Np(x)}{N-p(x)} & \text{if } p(x)<N\\
  \infty & \text{if } p(x)\ge N.
\end{cases}
\end{equation}
\end{itemize}

 In recent years there has been an increasing interest in the study of 
variational problems and elliptic equations with variable exponent. 
We refer to \cite{FAN5,KOVACIKRAKOSNIK,MUSIELAK} and references therein 
for general properties of the spaces $L^{p(x)}(\Omega)$ and 
$ W^{m,p(x)}(\Omega)$.
Many authors investigated the existence and multiplicity of solutions 
for problems involving the $p(x)$-Laplacian, with Neumann boundary conditions.
We refer to \cite{BOCHI4,FJ} for the existence of infinitely many solutions 
and to \cite{CCD6, ChaoJi2, ChiLi1, FAN2, LIU, MIHAILESCU, MOSCHETTO} for 
results concerning the existence of a finite number of them. Since in this 
paper we are interested in the latter case, we want to say something more
 about the results obtained in the last years. We observe that the solutions 
(three in most cases) are obtained as critical points of a suitable functional 
$I$ and the main tool for achieving the existence of such points is a 
critical point result due to Ricceri \cite{RICCERI1} or some variants of it.

One of the first paper devoted to this topic is \cite{MIHAILESCU}, 
where $f(x,t)=|t|^{q(x)-2}t-t$, with $2<q(x)< p^-$ and $p(x)>N$. 
Later, Xiayang Shi and Xuanhao Ding in \cite{SD} extend the results 
of \cite{MIHAILESCU} to Carath\'{e}odory functions $f$ satisfying a 
growth condition of type (F1), but with $1<q(x)\leq q^+<p^-$ and once
again $p^->N$.

A two parameter problem was studied first in \cite{LIU} and then in 
\cite{ChaoJi2}, where $f$ and $g$ are continuous and satisfy our condition 
(F1) but with a more restrictive assumption for the variable exponents $q$ 
and $p$. However, we emphasize that in both papers the authors need some 
additional hypotheses on the potentials $F$ and $G$. For instance, 
in \cite{LIU} we have a growth $r$ for $F$ and $G$, with $1<r^-<r^+< p^-$.
 Furthermore, to obtain their results they strengthen the hypotheses on $F$, 
for which they need  sign assumptions. Also in \cite{CCD6}, the authors
 have two parameters rather than one, but they deal with $p^->N\geq 2$ 
(we do not have such restriction).
In \cite{FAN2} the nonlinear term is $f+\lambda g$ with $f$ and $g$ 
continuous functions verifying our growth condition (F1) with respect 
to the second variable, but with the restrictions $p^+<p^*(x)$ and $p^+<q^-$.

Finally,  Liu  \cite{DLIU} takes $\lambda =1$. Under a regularity assumption 
on $f(x,\cdot)$ and standard growth conditions on $f_u(x,u)$, he shows 
the existence of three nontrivial solutions: one positive, one negative 
and the third is nodal.
In this paper we obtain multiplicity results for \eqref{eq:neumann}
 weakening the assumptions present in most of the papers cited above. 
In fact we deal with a Carath\'{e}odory function and we avoid the 
restriction $p^{-}>N$ for the exponent $p$. Furthermore, we have no 
relation between $q$ and $p$ except for the standard $q(x)<p^{*}(x)$. 
We point out that the elliptic case has been investigated in \cite{BOCHI5}.
 The paper is arranged as follows: in Section 2 we list some auxiliary 
results that we need to prove our main theorems that are exposed in Section 3.
 Finally, in Section 4 we give some examples of functions verifying assumptions
 requested in our main results.


\section{Preliminaries}

Here and in the sequel, we assume that $p\in C(\bar{\Omega})$ satisfies 
condition \eqref{eq:p(x)}. The variable exponent Lebesgue 
space $L^{p(x)}(\Omega )$ is defined as
$$
L^{p(x)}(\Omega )=\{ u:\Omega \to \mathbb{R}: u \text{ is measurable and }
 \rho_{p}(u):=\int_{\Omega}|u(x)|^{p(x)}dx < +\infty \}\,.
$$
On $L^{p(x)}(\Omega )$ we consider the norm
$$
\| u\|_{L^{p(x)}(\Omega )}= \inf\Big\{ \lambda >0:\int_{\Omega} 
\big|\frac{u(x)}{\lambda}\big|^{p(x)}dx \le 1\Big\}.
$$
The generalized Lebesgue-Sobolev space $W^{1,p(x)}(\Omega)$ is defined as
$$
W^{1,p(x)}(\Omega):=\big\{ u\in  L^{p(x)}(\Omega):|\nabla u |\in   
L^{p(x)}(\Omega) \big\}
$$
with the norm
\begin{equation}\label{eq:norm}
\| u\|_{W^{1,p(x)}(\Omega)}:= \| u\|_{L^{p(x)}(\Omega )} 
+ \| |\nabla u |\|_{L^{p(x)}(\Omega )}\cdot	
\end{equation}
With such norms, $L^{p(x)}(\Omega)$ and $W^{1,p(x)}(\Omega)$ are separable, 
reflexive and uniformly convex Banach spaces.

 The following result generalizes the well-known Sobolev embedding theorem.

\begin{theorem}[{\cite[Proposition 2.5]{FAN3}}]  \label{Imbedding}
 Assume that $p\in C(\bar{\Omega})$ with $p(x)> 1$ for each $x\in \bar{\Omega}$. 
If $r \in C(\bar{\Omega})$ and $1 < r(x) <p^{*}(x)$  for all $x\in\Omega$, 
then there exists a continuous and compact embedding 
$W^{1,p(x)}(\Omega)\hookrightarrow L^{r(x)}(\Omega)$ where $p^{*}$ 
is the critical exponent related to $p$ defined in \eqref{eq:pstar}.
\end{theorem}

 In the sequel, we will denote by $k_{r}$ the best constant for which one has
\begin{equation}\label{embeddingconstant}
	\| u\|_{L^{r(x)}(\Omega)} \le k_{r}\| u \|
\end{equation}
for all $u \in W^{1,p(x)}(\Omega)$.

 If we assume that
\begin{itemize}
\item[(H1)]  $a\in L^\infty(\Omega)$, with 
$a_{-}:=\operatorname{ess\,inf}_\Omega a(x)>0$,
\end{itemize}
then on $W^{1,p(x)}(\Omega)$ it is possible to consider the  norm
$$
\|u\|_a= \inf\Bigl\{\sigma >0:\int_{\Omega}
\Bigl (\bigl|\frac{\nabla u(x)}{\sigma}\bigr|^{p(x)}
+a(x)\bigl|\frac{u(x)}{\sigma}\bigr|^{p(x)}\Bigr)\, dx \leq 1 \Bigr\} \,,
$$
which is equivalent to that introduced in \eqref{eq:norm}
(see \cite{CCD6}).
In particular, if for $\alpha>0$ and $h\in C(\bar{\Omega})$ with $1<h^{-}$,
 we put
\begin{gather*}
[\alpha]^{h}:=\max\{\alpha^{h^-},\alpha^{h^+}\}\,\\
[\alpha]_{h}:=\min\{\alpha^{h^-},\alpha^{h^+}\},
\end{gather*}
then it is easy to verify that
$$
[\alpha]^{1/h}=\max\{\alpha^{1/h^-},\alpha^{1/h^+}\},\quad
[\alpha]_{\frac{1}{h}}=\min\{\alpha^{1/h^-},\alpha^{1/h^+}\}\,.
$$
 Now, starting from the definition of $\|\cdot\|_{a}$ and 
$\|\cdot \|_{L^{p(x)}(\Omega)}$ and using standard arguments, 
the following estimate is obtained
\begin{equation}\label{equivalenza}
\frac{[ a_{-}]_{1/p}}{1+[ a_{-}]_{1/p}}\|u\|_{W^{1,p(x)}(\Omega)}
\le \|u\|_{a}\le \left(1+\|a\|_{\infty}\right)^{1/p^-}\|u\|_{W^{1,p(x)}(\Omega)}
\end{equation}
 for each $u\in W^{1,p(x)}(\Omega)$.

\begin{remark} \label{stima di kr}\rm
If $\Omega$ is an open convex subset of $\mathbb{R}^{N}$ and the variable exponents
$r$ and $p$ verify conditions $r^{+}<{p^{-}}^{*}$ and $p^{-}\neq N$, 
then it is possible to provide an upper estimate for the constant 
$k_{r}$ in \eqref{embeddingconstant}. We recall that in \cite{BonMolRad} 
(see Remark \ref{due soluzioni non nulle}), an upper bound for the constant 
of the  embedding $W^{1,h}(\Omega)\hookrightarrow L^{q}(\Omega)$ with 
$q\in [1, h^{*}[$ has been obtained when $\Omega$ is an open convex 
set of $\mathbb{R}^{N}$ and $h\neq N$. Precisely, denoted by $\tilde{k}_{h,q}$
such constant, one has
\begin{equation}\label{stimanelcasocostante}
	\|u\|_{L^{q}(\Omega)}\le \tilde{k}_{h,q} \|u\|_{a,W^{1,h}(\Omega)}
\end{equation}
 for each $u\in W^{1,h}(\Omega)$ where
$$
\|u\|_{a,W^{1,h}(\Omega)}= \Big(\int_{\Omega}|\nabla u(x)|^{h}\,dx
+\int_{\Omega}a(x)|u(x)|^{h}\,dx\Big)^{1/h}
$$
 and $\tilde{k}_{h,q}$ depends on the diameter of $\Omega$, on the measure 
of $\Omega$ and on $a_{-}$. Now, if $p^{-}\neq N$ and $r^{+}<{p^{-}}^{*}$, 
starting from \eqref{stimanelcasocostante} with $q=r^{+}$ and $h=p^{-}$, 
for each $u\in W^{1,p(x)}(\Omega)$, it results
\begin{equation}\label{normar}
\|u\|_{L^{r^{+}}(\Omega)}\le \tilde{k}_{p^{-},r^{+}}\|u\|_{a,W^{1,p^{-}}
(\Omega)}\,.
\end{equation}
Taking into account that (see for instance \cite[Theorem 2.8]{KOVACIKRAKOSNIK}) 
$L^{p(x)}(\Omega)\hookrightarrow L^{p^{-}}(\Omega)$ and 
$L^{r^{+}}(\Omega)\hookrightarrow L^{r(x)}(\Omega)$ with continuous embeddings 
and that the constants of such embeddings do not exceed $1+|\Omega|$, one has
\begin{align*}
\|u\|_{a,W^{1,p^{-}}(\Omega)}^{p^-}
&\le \|\nabla u\|_{L^{p^{-}}(\Omega)}^{p^-}
 + \|a\|_{\infty}\| u\|_{L^{p^{-}}(\Omega)}^{p^-}\\
&\leq (1+|\Omega|)^{p^-}\|\nabla u\|_{L^{p(x)}(\Omega)}^{p^-}
 + \|a\|_{\infty}(1+|\Omega|)^{p^-}\| u\|_{L^{p(x)}(\Omega)}^{p^-}\\
&\leq (1+|\Omega|)^{p^-}(1+\|a\|_{\infty})\|u\|_{W^{1,p(x)}(\Omega)}^{p^-}
\end{align*}
and so
\begin{equation}\label{normaa}
	\|u\|_{a,W^{1,p^{-}}(\Omega)}\le 
(1+|\Omega|)(1+\|a\|_{\infty})^{1/p^-}\|u\|_{W^{1,p(x)}(\Omega)}\,.
\end{equation}
 On the other hand, one has
\begin{equation}\label{normarx}
	\| u\|_{L^{r(x)}(\Omega)}\le (1+|\Omega|)\| u\|_{L^{r^{+}}(\Omega)}\,.
\end{equation}
Starting from conditions \eqref{normar}, \eqref{normaa}, \eqref{normarx} and 
\eqref{equivalenza}, we obtain
 $$
\|u\|_{L^{r(x)}(\Omega)}\le \tilde{k}_{p^{-},r^{+}}(1+|\Omega|)^{2}
(1+\|a\|_{\infty})^{1/p^-}\frac{1+[ a_{-}]_{1/p}}{[ a_{-}]_{1/p}}
\|u\|_{a}\,,
$$
 and so
\begin{equation}\label{stima kr}
k_{r}\le\tilde{k}_{p^{-},r^{+}}(1+|\Omega|)^{2}
(1+\|a\|_{\infty})^{1/p^-}\frac{1+[ a_{-}]_{1/p}}{[ a_{-}]_{1/p}}\,.
\end{equation}
\end{remark}

\begin{remark} \label{stima di k1}\rm
Arguing as in the previous remark, if we denote by $k_1$ the best constant 
of the embedding $W^{1,p(x)}(\Omega)\hookrightarrow L^{1}(\Omega)$, 
then we obtain
\begin{equation}\label{stima k1}
k_1\le\tilde{k}_{p^{-},1}(1+|\Omega|)(1+\|a\|_{\infty})^{1/p^-}
\frac{1+[ a_{-}]_{1/p}}{[ a_{-}]_{1/p}}\,.
\end{equation}
\end{remark}

 We recall that, fixed $\lambda >0$, a point $u\in W^{1,p(x)}(\Omega)$ 
is a weak solution to  \eqref{eq:neumann} if
$$
\int_{\Omega} \Big(|\nabla u(x)|^{p(x)-2}\nabla u(x)\nabla v(x)
+a(x)|u|^{p(x)-2} uv\Big)\, dx
=\lambda \int_{\Omega}f(x,u(x))v(x)dx
$$
holds for each $v \in W^{1,p(x)}(\Omega)$. To obtain one or more solutions 
to \eqref{eq:neumann}, fixed $\lambda >0$, we denote by $I_{\lambda}$ 
the energy functional
$$
I_{\lambda}(\cdot):=\Phi(\cdot)-\lambda\Psi(\cdot),
$$
 where $\Phi, \Psi:W^{1,p(x)}(\Omega)\to\mathbb{R}$ are defined as follows
\begin{gather*}
\Phi(u)=\int_{\Omega}\frac{1}{p(x)}\Bigl(|\nabla u|^{p(x)} 
+a(x)|u|^{p(x)} \Bigr) \, dx,\\
\Psi(u)=\int_{\Omega}F(x,u(x))dx
\end{gather*}
 for each $u\in W^{1,p(x)}(\Omega)$ and
$$
F(x,\xi):=\int_{0}^{\xi}f(x,t)\,dt
$$
 for each $(x,\xi )\in \Omega\times\mathbb{R}$. When $I_{\lambda}$ is $C^1$
its critical points are weak solutions to \eqref{eq:neumann}.
 Similar arguments to those used in \cite{MIHAILESCU} and in 
\cite{FAN2} imply that $\Phi$ is sequentially weakly lower semi-continuous 
and is a $C^1$ functional in $W^{1,p(x)}(\Omega)$, with the derivative 
given by
$$
\langle \Phi'(u),v\rangle=\int_{\Omega}
\Big(|\nabla u(x)|^{p(x)-2}\nabla u(x) \nabla v(x) + a(x)|u|^{p(x)-2}uv\Big)\,dx ,
$$
for any $u, v \in W^{1,p(x)}(\Omega)$. 
Moreover (see \cite[Lemma 3.1]{CCD6}), $\Phi '$ is an homeomorphism. 
Finally we recall that in \cite[Proposition 2.2]{CCD6} it was shown  
that $\Phi$ is in close relation with the norm $\|\cdot\|_a$. 
In fact, we have the following result.


\begin{proposition} \label{teo:limitazioni2}
Let $u \in W^{1,p(x)}(\Omega)$. Then
\begin{itemize}
\item[(j)]  If $\|u\|_a < 1$ then 
  $\frac{1}{p^+}\|u\|_a^{p^+}\leq \Phi(u) \leq \frac{1}{p^-}\|u\|_a^{p^-}$.
\item[(jj)] If $\|u\|_a > 1$ then 
  $\frac{1}{p^+}\|u\|_a^{p^-}\leq \Phi(u) \leq \frac{1}{p^-}\|u\|_a^{p^+}$.
\end{itemize}
\end{proposition}

 We observe that $\Psi$ can be defined in the space $L^{q(x)}(\Omega)$. 
In fact, from \cite[Theorems  4.1 and 4.2]{KOVACIKRAKOSNIK},
 we know that the growth condition (F1) imposed on $f$ guarantees 
that the Nemytsky operator $N_{f}$ defined by 
$N_{f}(u)=f(\cdot ,u(\cdot))$ maps $L^{q(x)}(\Omega)$ in $L^{q'(x)}(\Omega)$ 
where $q'(x)=\frac{q(x)}{q(x)-1}$ and that is continuous and bounded.
 Before studying the regularity properties of $\Psi$, we introduce the functional 
$J:L^{q'(x)}(\Omega)\to (L^{q(x)}(\Omega))^{*}$ defined as
$$
J(h)(w):=\int_{\Omega}h(x)w(x)\,dx
$$
for each $h\in L^{q'(x)}(\Omega)$, $w\in L^{q(x)}(\Omega)$. 
From \cite[Theorem 3.4.6]{Diening}, we know that $J$ is an isomorphism 
from $L^{q'(x)}(\Omega)$ to $(L^{q(x)}(\Omega))^{*}$.

\begin{lemma} \label{derivata-continua-e-compatta}
Under assumption {\rm (F1)} $\Psi$ is a continuously G\^ateaux differentiable 
functional with
$$
\Psi '(u)(v)=\int_{\Omega}f(x,u(x))v(x)dx
$$
 for each $u, v \in W^{1,p(x)}(\Omega)$ and $\Psi'$ is a compact operator.
\end{lemma}

\begin{proof}
 In a standard way we obtain
$$
\Psi '(u)(v)=\int_{\Omega}f(x,u(x))v(x)dx
$$
 from each $u, v \in W^{1,p(x)}(\Omega)$. If $\left\{u_{n}\right\}\rightarrow u$ 
in $W^{1,p(x)}(\Omega)$ then, for Theorem \ref{Imbedding}, $\left\{u_{n}\right\}\rightarrow u$ in $L^{q(x)}(\Omega)$. Thanks to the properties of the Nemytsky operator, one has $\left\{N_{f}(u_{n})\right\}\rightarrow N_{f}(u)$ in $L^{q'(x)}(\Omega)$ and so $\left\{J(N_{f}(u_{n}))\right\}\rightarrow J(N_{f}(u))$ in $(L^{q(x)}(\Omega))^{*}$. This condition leads to $\left\{J(N_{f}(u_{n}))\right\}\rightarrow J(N_{f}(u))$ in $(W^{1,p(x)}(\Omega))^{*}$ and, taking into account that
$$
J(N_{f}(u))(\cdot )=\Psi'(u)(\cdot )
$$
for each $u\in W^{1,p(x)}(\Omega)$, we obtain the continuity of $\Psi '$.
 If we suppose that $\left\{u_{n}\right\}\rightharpoonup u$ in 
$W^{1,p(x)}(\Omega)$, then, thanks to the compactness of the embedding 
$W^{1,p(x)}(\Omega)\hookrightarrow L^{q(x)}(\Omega)$,
 $\left\{u_{n}\right\}\rightarrow u$ in $L^{q(x)}(\Omega)$ 
(up to a subsequence). This ensures the continuity of $\Psi '$ on 
$L^{q(x)}(\Omega)$ and so its compactness. 
\end{proof}

 To conclude this section we introduce two abstract results obtained 
by Bonanno in \cite{BONANNO} and \cite{BONANNO2} that will allow us to 
obtain multiple solutions to \eqref{eq:neumann}. Before to recall them we 
give the following definition.

\begin{definition} \label{def:PS} \rm
Let $\Phi$ and $\Psi$ be two continuously G\^ateaux differentiable functionals 
defined on a real Banach space $X$ and fix $r\in\mathbb{R}$. The functional
$I=\Phi-\Psi$ is said to verify the Palais-Smale condition cut off upper 
at $r$ (in short $(P.S.)^{[r]}$) if any sequence $\{u_{n}\}$ 
in $X$ such that
\begin{itemize}
	\item [$(\alpha)$] $\left\{I(u_n)\right\}$ is bounded;
	\item [$(\beta)$] $\lim_{n\to +\infty}\|I'(u_n )\|_{X^*}=0$;
	\item [$(\gamma)$] $\Phi(u_n )<r$ for each $n\in\mathbb{N}$;
\end{itemize}
 has a convergent subsequence.
\end{definition}

 The following abstract result is a particular case of
\cite[Theorem 5.1]{BONANNO}.

\begin{theorem}[\cite{BONANNO2}] \label{teo:bon}
Let $X$ be a real Banach space, $\Phi, \Psi:X\to\mathbb{R}$ be two continuously
 G\^ateaux differentiable functionals such that 
$\inf_{x\in X}\Phi(x)=\Phi (0)=\Psi (0)=0$. Assume that there exist
 $r>0$ and $\bar{x}\in X$, with $0<\Phi(\bar{x})<r$, such that:
\begin{itemize}
\item[(A1)] $\frac{\sup_{\Phi(x)\le r}\Psi(x)}{r}
< \frac{\Psi(\bar{x})}{\Phi(\bar{x})}$,

\item[(A2)] for each 
$\lambda\in ] \frac{\Phi(\bar{x})}{\Psi(\bar{x})},
 \frac{r}{\sup_{\Phi(x)\le r}\Psi(x)} [$, the functional 
$I_{\lambda}:=\Phi-\lambda\Psi$ satisfies the $(P.S.)^{[r]}$ condition.
\end{itemize}
Then, for each $\lambda \in \Lambda_{r}:= 
] \frac{\Phi(\bar{x})}{\Psi(\bar{x})},\frac{r}{\sup_{\Phi(x)\le r}\Psi(x)} [ $, 
there is $x_{0,\lambda}\in \Phi^{-1}(]0,r[)$ such that 
$I_{\lambda}'(x_{0,\lambda})\equiv\vartheta_{X^*}$ and 
$I_{\lambda}(x_{0,\lambda})\le I_{\lambda}(x)$ for all $x\in \Phi^{-1}(]0,r[)$.
\end{theorem}

\begin{remark} \label{condizione PS} \rm
\cite[Proposition 2.1]{BONANNO} guarantees that if $\Phi$ is a sequentially 
weakly lower semicontinuous, coercive, continuously G\^ateaux differentiable 
function whose G\^ateaux derivative admits a continuous inverse and  
$\Psi$ is a G\^ateaux differentiable function whose G\^ateaux derivative 
is compact then the functional $\Phi - \Psi$ satisfies the $(P.S.)^{[r]}$ 
condition for each $r\in\mathbb{R}$.
\end{remark}

The last abstract result that we will use in this paper is the following.

\begin{theorem}[{\cite[Theorem 3.2]{BONANNO2}}] \label{teo:bon2}
Let $X$ be a real Banach space, $\Phi, \Psi:X\to\mathbb{R}$ be two continuously
G\^ateaux differentiable functionals such that $\Phi$ is bounded from 
below and $\Phi(0)=\Psi (0)=0$. Fix $r>0$ and assume that, for each
$$
\lambda\in ]0,\frac{r}{\sup_{u\in \Phi^{-1}(]-\infty, r [)}\Psi(u)} [,
$$
 the functional $I_{\lambda}:=\Phi-\lambda\Psi$ satisfies $(P.S.)$ 
condition and it is unbounded from below. Then, for each
$$
\lambda\in ]0,\frac{r}{\sup_{u\in \Phi^{-1}(]-\infty, r [)}\Psi(u)} [,
$$
 the functional $I_{\lambda}$ admits two distinct critical points.
\end{theorem}


\section{Main results}

The first result guarantees the existence of one non trivial solution 
to  problem \eqref{eq:neumann}.


\begin{theorem} \label{teo:ris1}
Let $f :\Omega\times\mathbb{R} \to\mathbb{R}$ be a Carath\'{e}odory function satisfying
{\rm (F1)} and
\begin{itemize}
\item [(F2)]
$$
\limsup_{t\to 0^{+}}\frac{\int_{\Omega}F(x,t)\,dx}{t^{p^{-}}}=+\infty.
$$
\end{itemize}
Put $\lambda^{*}=\frac{1}{a_1k_1(p^{+})^{1/p^-}
+\frac{a_{2}}{q^{-}}[k_{q}]^{q}(p^{+})^{\frac{q^{+}}{p^{-}}}}$, 
where $k_1$ and $k_{q}$ are given by \eqref{embeddingconstant}.
Then for each $\lambda\in ]0,\lambda^{*}[$,  problem \eqref{eq:neumann} 
admits at least one nontrivial weak solution.
\end{theorem}

\begin{proof}
 Put $X:=W^{1,p(x)}(\Omega)$ equipped by norm $\|\cdot\|_{a}$.
We consider the functional 
$$
I_{\lambda}(\cdot):=\Phi(\cdot)-\lambda\Psi(\cdot)
$$
introduced in the previous section and note that 
$\Phi$ and $\Psi$ satisfy the regularity assumptions required in 
Theorem \ref{teo:bon} as well as the condition (A2) for all $r, \lambda >0$ 
(see Lemma \ref{derivata-continua-e-compatta} and Remark \ref{condizione PS}).
 Fixed $\lambda\in ]0,\lambda^{*}[$, we choose $r=1$ and verify condition 
(A1) of Theorem \ref{teo:bon}.
 By (F2) there exists
\begin{equation}\label{eq:xi}
0<\xi_{\lambda}<\min \Big\{1,\Big(\frac{p^-}{\|a\|_{\infty}|\Omega|}
\Big)^{1/p^-}\Big\}
\end{equation}
 such that
\begin{equation}\label{eq:magg}
\frac{p^{-}\int_{\Omega}F(x,\xi_{\lambda})\,dx}{\xi_{\lambda}^{p^{-}}
\|a\|_{\infty}|\Omega|}> \frac{1}{\lambda}\,. 	
\end{equation}
 We denote by $u_{\lambda}$ the function of $X$ defined by
$u_{\lambda}(x)=\xi_{\lambda}$
for each $x\in\Omega$ and observe that
 \begin{equation}\label{eq:phi}
	\Phi(u_{\lambda})\le\frac{1}{p^-}\|a\|_{\infty}|\Omega|[\xi_{\lambda}]^{p}<1
\end{equation}
 and
$$
\Psi(u_{\lambda})=\int_{\Omega}F(x,\xi_{\lambda})\,dx\,.
$$
 We observe that condition (F1) implies
$$
|F(x,t)|\le a_1|t|+ \frac{a_{2}}{q(x)}|t|^{q(x)}
$$
 for each $(x,t)\in \Omega\times \mathbb{R}$. 
For each $u\in \Phi^{-1} (]-\infty ,1])$ it results
 $$
\Psi (u)\le a_1\int_{\Omega}|u(x)|\,dx
+ \frac{a_2}{q^-}\int_{\Omega}|u(x)|^{q(x)}\,dx
=a_1\|u\|_{L^{1}(\Omega)}+ \frac{a_2}{q^-}\rho_{q}(u)\,.
$$
\cite[Theorem 1.3]{FAN5} and the embeddings 
$W^{1,p(x)}(\Omega)\hookrightarrow L^{1}(\Omega)$ and 
$W^{1,p(x)}(\Omega)\hookrightarrow L^{q(x)}(\Omega)$ ensure
\begin{equation}\label{eq:Psi}
 a_1\|u\|_{L^{1}(\Omega)}+ \frac{a_2}{q^-}\rho_{q}(u)
\le a_1\|u\|_{L^{1}(\Omega)}+ \frac{a_2}{q^-}[\|u\|_{L^{q(x)}(\Omega)}]^{q}
\le a_1k_1\|u\|_{a}+ \frac{a_2}{q^-}[k_{q}\|u\|_{a}]^{q}
\end{equation}
  Taking into account that for each $u\in \Phi^{-1} (]-\infty ,1])$, 
thanks to Proposition \ref{teo:limitazioni2}, one has
$$
\|u\|_{a}\le (p^{+})^{1/p^-},
$$
 conditions \eqref{eq:magg} and \eqref{eq:Psi} lead to
 \begin{equation}\label{eq:sup1}
\begin{split}
\sup_{\Phi (u)\le 1}\Psi (u)
&\le a_1k_1(p^{+})^{1/p^-}
+\frac{a_2}{q^-}[k_{q}]^{q}(p^{+})^{\frac{q^+}{p^-}}\\
&=\frac{1}{\lambda^{*}}<\frac{1}{\lambda}\\
&<\frac{p^{-}\int_{\Omega}F(x,\xi_{\lambda})\,dx}{\xi_{\lambda}^{p^{-}}
\|a\|_{\infty}|\Omega|}<\frac{\Psi (u_{\lambda})}{\Phi (u_{\lambda})}
\end{split}
\end{equation}
 and so condition (A1) of Theorem \ref{teo:bon} is verified.
 Since $\lambda\in \big]  \frac{\Phi(u_{\lambda})}{\Psi(u_{\lambda})},
\frac{1}{\sup_{\Phi(u)\le 1}\Psi(u)} \big [$,
 Theorem \ref{teo:bon} guarantees the existence of a local minimum point
 $\bar{u}$ for the functional $I_{\lambda}$ such that
$$
0<\Phi(\bar{u})<1
$$
and so $\bar{u}$ is a non-trivial weak solution of problem \eqref{eq:neumann}.
\end{proof}

To establish the existence of two solutions to problem \eqref{eq:neumann}, 
we assume that the nonlinear term $f$ satisfies this Ambrosetti-Rabinowitz-type 
condition
\begin{itemize}
\item [(F3)] there exist $\mu >p^{+}$ and $\beta>0$ such that
$$
0<\mu F(x,\xi)\le \xi f(x,\xi)
$$ 
for each $x\in\Omega$ and for $|\xi| \ge \beta$.
\end{itemize}

\begin{lemma}\label{PS under AR}
Let $f :\Omega\times\mathbb{R} \to\mathbb{R}$ be a Carath\'{e}odory function
satisfying {\rm (F1)} and {\rm (F3)}. Then, for each $\lambda >0$, 
$I_\lambda$ satisfies the $(PS)$-condition.
\end{lemma}

\begin{proof}
Let $\{u_n\}$ be a $(PS)$ sequence for $I_\lambda$. Then:
\begin{gather}\label{9}
    |I_\lambda(u_n)|\leq M\quad\text{for some $M>0$ and all } n\geq 1\,,\\
\label{10}
    I_\lambda'(u_n)\to 0\quad \text{in } W^{1,p(x)}(\Omega)^*,\text{ as }
 n\to\infty\,.
\end{gather}
Due to \eqref{10} we can find $\overline n\in N$, such that
\begin{equation}\label{11}
\begin{aligned}
    -I_\lambda'(u_n)(u_n)
&=-\int_{\Omega}\Bigl(|\nabla u_n|^{p(x)} +a(x)|u_n|^{p(x)} \Bigr) \, dx
 +\lambda\int_{\Omega}f(x,u_n(x))u_n(x)\, dx\\
& \leq\|u_n\|_a\quad \text{for all } n\ge \overline n\,.
\end{aligned}
\end{equation}
We argue by contradiction and we assume that $\{u_n\}$ is unbounded, 
so we can choose $\overline n$ such that $\|u_n\|_a>1$ for any
 $n\geq \overline n$. Our assumptions on $f$ guarantee that we 
can find a number $A(\beta)>0$ such that for any $n\in \mathbb{N}$ one has:
\begin{equation}\label{12}
   \int_{\{x\in\Omega: |u_n(x)|\leq\beta\}}\left(f(x,u_n(x))u_n(x)
-\mu F(x,u_n(x))\right)\,dx
   \ge -A(\beta).
\end{equation}
Gathering \eqref{9}, \eqref{11}, \eqref{12} and taking into account $(jj)$ 
of Proposition \ref{teo:limitazioni2}, for $n$ large enough we obtain
\begin{equation}\label{14}
\begin{aligned}
&\mu\cdot M+\|u_n\|_a\\
&\geq \mu I_\lambda(u_n)-I_\lambda'(u_n)(u_n)\\
&=\int_{\Omega}\frac{\mu-p(x)}{p(x)}\Bigl(|\nabla u_n|^{p(x)} 
 +a(x)|u_n|^{p(x)} \Bigr) \, dx \\
&\quad +\lambda\int_{\{x\in\Omega: |u_n(x)|\leq\beta\}}
 \left(f(x,u_n(x))u_n(x)-\mu F(x,u_n(x))\right)\,dx \\
&\quad +\lambda\int_{\{x\in\Omega:|u_n(x)|\geq\beta\}}
 \left(f(x,u_n(x))u_n(x)-\mu F(x,u_n(x))\right)\,dx \\
&\geq (\mu-p^+)\Phi (u_n)-\lambda A(\beta)\\
&\geq \frac{(\mu-p^+)}{p^+}\|u_n\|_a^{p^-}-\lambda A(\beta)\,,
\end{aligned}
\end{equation}
which contradicts the unboundedness of $\{u_n\}$, since $p^->1$. 
So $\{u_n\}$ is bounded, so, taking into account that $\Psi '$ 
is compact, we obtain the existence of a convergent subsequence.
\end{proof}

\begin{theorem} \label{teo:ris2}
Let $f :\Omega\times\mathbb{R} \to\mathbb{R}$ be a Carath\'{e}odory function
satisfying {\rm (F1)} and {\rm (F3)}. 
Then, for each $\lambda\in ]0,\lambda^{*}[$, where $\lambda^{*}$ 
is the constant introduced in the statement of Theorem \ref{teo:ris1}, 
 problem \eqref{eq:neumann} admits at least two distinct weak solutions.
\end{theorem}

\begin{proof}
 We choose $r=1$, $X=W^{1,p(x)}(\Omega)$ and apply Theorem \ref{teo:bon2} 
to the functionals $\Phi$ and $\Psi$ introduced before.
 Clearly, $\Phi$ is bounded from below and $\Phi(0)=\Psi(0)=0$. 
From Lemma \ref{PS under AR} we know that our functional 
$I_{\lambda}(\cdot):=\Phi(\cdot)-\lambda \Psi (\cdot)$ satisfies the $(P.S.)$ 
condition for each $\lambda >0$.
 By integrating condition (F3), we can find $a_{3}>0$ such that
$$
F(x, \xi)\ge a_{3}\xi^{\mu}
$$
 for each $|\xi|\ge\beta_1>\beta$. Fixed $k>\max\left\{ \beta_1, 1\right\}$,
 we consider the function $\bar{u}\equiv k \in X$ and we observe that, for each
 $t>1$ it results
$$
I_{\lambda}(t\bar{u})\le \frac{1}{p^-}\|a\|_{\infty}|\Omega|t^{p^+}k^{p^+}
-\lambda a_{3}|\Omega|t^{\mu}k^{\mu}\,.
$$
 Since $\mu >p^+$, this condition implies that $I_{\lambda}$ is unbounded 
from below. Finally, fixed $\lambda\in ]0,\lambda^{*}[$ and taking into 
account \eqref{eq:sup1}, we have
$$ 
0<\lambda <\frac{1}{\sup_{u\in \Phi^{-1}(]-\infty, 1 [)}\Psi(u)} 
$$
 and so, the functional $I_{\lambda}$ admits two distinct critical points 
that are weak solutions to problem \eqref{eq:neumann}.
\end{proof}

\begin{remark} \label{due soluzioni non nulle} \rm
We observe that, if $f(x,0)\neq 0$, then Theorem \ref{teo:ris2}
ensures the existence of two non trivial weak solutions for problem 
\eqref{eq:neumann}. 
\end{remark}

\begin{remark} \label{stima di lambda}\rm
Taking into account Remark \ref{stima di kr} and Remark \ref{stima di k1}, 
if $\Omega$ is an open convex subset of $\mathbb{R}^{N}$ and the variable exponents
$q$ and $p$ verify conditions $q^{+}<{p^{-}}^{*}$ and $p^{-}\neq N$, 
then it is possible to obtain a precise estimate of parameter $\lambda^{*}$ 
in Theorems \ref{teo:ris1} and \ref{teo:ris2}.
\end{remark}


\section{Examples}

 Now we give some applications of the previous results.

\begin{example} \label{es1}\rm
Let $a_1$ and $a_{2}$ in $L^{\infty}(\Omega)$, with 
$\operatorname{ess\,inf}_{x\in \Omega}a_1(x)>0$. We consider
$$
f(x,t)=a_1(x)+a_{2}(x)|t|^{q(x)-1}
$$
 for each $(x,t)\in\Omega\times \mathbb{R}$ where $q\in C(\bar{\Omega})$ 
with $1< q(x) < p^{*}(x)$ for each $x\in\bar{\Omega}$.
 We observe that condition (F1) of Theorem \ref{teo:ris1} is easily verified. 
Moreover, by integration we obtain
$$
F(x,t)=a_1(x)t+\frac{a_{2}(x)}{q(x)}t^{q(x)}
$$
 for each $x\in\Omega$ and $t>0$. This implies that
$$
\lim_{t\to 0^{+}}\frac{ \operatorname{ess\,inf}_{x\in \Omega}F(x,t)}{t^{p^{-}}}
=+\infty
$$
and so condition (F2) of Theorem \ref{teo:ris1} is satisfied.
\end{example}

Finally, we present an application of Theorem \ref{teo:ris2}.

\begin{example} \label{duesoluzioni}\rm
We take the function $f$ defined by
\[
f(x,t)= a+b q(x) |t|^{q(x)-2}t\quad\text{for  } (x,t)\in \Omega\times\mathbb{R}\,,
\]
where $a$ and $b$ are two positive constants and $p,\,q\in C(\bar{\Omega})$ 
satisfy the inequalities $1< p^{+}<q^{-}\le q(x) < p^{*}(x)$ for any 
$x\in\bar{\Omega}$.
Fixed $p^{+}<\mu <q^{-}$ and
$$
\beta>\max\Big\{\big[\frac{a(\mu -1)}{b(q^{-}-\mu)}\big]^{\frac{1}{q^--1}},\,
 \big(\frac{a}{b}\big)^{\frac{1}{q^--1}},\, 1\Big\}\,.
$$
 We prove that $f$ fulfills the assumptions requested in Theorem \ref{teo:ris2}. 
Condition (F1) of Theorem \ref{teo:ris2} is easily verified. 
Taking into account that
\begin{equation}
F(x,t)=  at+b |t|^{q(x)}\quad\text{for } (x,t)\in \Omega\times\mathbb{R}\,,
\end{equation}
and  $\beta > (\frac{a}{b})^{\frac{1}{q^--1}}$, one has
$$
F(x,t)\ge -a |t| +b|t|^{q(x)}=|t|(-a+b|t|^{q(x)-1})>0
$$
 for each $x\in\Omega$ and for $|t| \ge \beta$. 
Moreover, the assumption 
$\beta>[\frac{a(\mu -1)}{b(q^{-}-\mu)}]^{\frac{1}{q^--1}}$ 
leads to the following inequality
$$
b(q(x)-\mu)|t|^{q(x)-1}\ge b(q^--\mu)\beta^{q^--1}\ge a(\mu-1)
$$
 for each $x\in\Omega$ and $t\ge\beta$. This implies that
$$
\mu F(x,t)\le t f(x,t)
$$
holds for each $x\in\Omega$ and $|t|\ge \beta$ and so condition (F3) is verified.
\end{example}

\begin{remark} \label{stima della costante nell'esempio} \rm
We observe that the function $f$ in Example \ref{duesoluzioni} 
satisfies the condition $f(x,0)\neq 0$. This implies that 
problem \eqref{eq:neumann} admits at least two non trivial distinct 
solutions (see Remark \ref{due soluzioni non nulle}).
\end{remark}

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\end{document}