\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
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 $1N$.
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 $1N\geq 2$
(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 $p^+N$ for the exponent $p$. Furthermore, we have no
relation between $q$ and $p$ except for the standard $q(x)
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\}
$$
with the norm
\begin{equation}\label{eq:norm}
\| u\|_{W^{1,p(x)}(\Omega)}:= \| u\|_{L^{p(x)}(\Omega )}
+ \| |\nabla u |\|_{L^{p(x)}(\Omega )}\cdot
\end{equation}
With such norms, $L^{p(x)}(\Omega)$ and $W^{1,p(x)}(\Omega)$ are separable,
reflexive and uniformly convex Banach spaces.
The following result generalizes the well-known Sobolev embedding theorem.
\begin{theorem}[{\cite[Proposition 2.5]{FAN3}}] \label{Imbedding}
Assume that $p\in C(\bar{\Omega})$ with $p(x)> 1$ for each $x\in \bar{\Omega}$.
If $r \in C(\bar{\Omega})$ and $1 < r(x)
0$,
\end{itemize}
then on $W^{1,p(x)}(\Omega)$ it is possible to consider the norm
$$
\|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\} \,,
$$
which is equivalent to that introduced in \eqref{eq:norm}
(see \cite{CCD6}).
In particular, if for $\alpha>0$ and $h\in C(\bar{\Omega})$ with $10$, a point $u\in W^{1,p(x)}(\Omega)$
is a weak solution to \eqref{eq:neumann} if
$$
\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 )
$$
for each $u\in W^{1,p(x)}(\Omega)$, we obtain the continuity of $\Psi '$.
If we suppose that $\left\{u_{n}\right\}\rightharpoonup u$ in
$W^{1,p(x)}(\Omega)$, then, thanks to the compactness of the embedding
$W^{1,p(x)}(\Omega)\hookrightarrow L^{q(x)}(\Omega)$,
$\left\{u_{n}\right\}\rightarrow u$ in $L^{q(x)}(\Omega)$
(up to a subsequence). This ensures the continuity of $\Psi '$ on
$L^{q(x)}(\Omega)$ and so its compactness.
\end{proof}
To conclude this section we introduce two abstract results obtained
by Bonanno in \cite{BONANNO} and \cite{BONANNO2} that will allow us to
obtain multiple solutions to \eqref{eq:neumann}. Before to recall them we
give the following definition.
\begin{definition} \label{def:PS} \rm
Let $\Phi$ and $\Psi$ be two continuously G\^ateaux differentiable functionals
defined on a real Banach space $X$ and fix $r\in\mathbb{R}$. The functional
$I=\Phi-\Psi$ is said to verify the Palais-Smale condition cut off upper
at $r$ (in short $(P.S.)^{[r]}$) if any sequence $\{u_{n}\}$
in $X$ such that
\begin{itemize}
\item [$(\alpha)$] $\left\{I(u_n)\right\}$ is bounded;
\item [$(\beta)$] $\lim_{n\to +\infty}\|I'(u_n )\|_{X^*}=0$;
\item [$(\gamma)$] $\Phi(u_n )0$ and $\bar{x}\in X$, with $0<\Phi(\bar{x})0$ and assume that, for each
$$
\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^{+}\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}