\documentclass[11pt]{amsart}
\newtheorem{theorem}{Theorem}[section]
\newtheorem{lemma}[theorem]{Lemma}
\newtheorem{corollary}[theorem]{Corollary}
\newtheorem{proposition}[theorem]{Proposition}
\newtheorem{definition}[theorem]{Definition}
\newtheorem{example}[theorem]{Example}
\newtheorem{remark}[theorem]{Remark}
\def\erre{{\mathbb R}}
\def\R{{\mathbb R}}
\def\enne{{\mathbb N}}
\def\meas{\mathop{\rm meas}}
\def\d{\mathop{\rm dist}}
\def\R{{\mathbb R}}
\def\R{{\mathbb R}}
\def\N{{\mathbb N}}
\def\di{\displaystyle}
\def\essinf{\mathop {\rm ess\,inf}}
\def\n{\noindent}
\def\phi{\varphi}
\def\div{\mathop{\rm div}}
\def\meas{\mathop{\rm meas}}
\def\H{\mathop{W^{1,p}(\Omega)}}
\def\xseg{\bar{x}}
\def\useg{\bar{u}}
\def\d{\mathop{\rm dist}}
\def\d{{\rm diam(\Omega)}}
%\date{}
%\title[On nonlocal type equations via linking methods]{On nonlocal
%type equations via linking methods}
\title[Nonlocal Neumann Problems versus Local Linking]{Applications of
Local Linking to Nonlocal Neumann Problems}
\author{Giovanni Molica Bisci and Vicen\c{t}iu D. R\u{a}dulescu}
\address[G. Molica Bisci]{Dipartimento P.A.U., Universit\`a  degli
Studi Mediterranea di Reggio Calabria, Salita Melissari - Feo di
Vito, 89100 Reggio Calabria, Italy}\email{gmolica@unirc.it}
%
\address[V. R\u{a}dulescu]{Institute of Mathematics ``Simion Stoilow"
of the Romanian Academy, 014700 Bucharest, Romania. Department of
Mathematics, University of Craiova,  200585 Craiova,
Romania}\email{vicentiu.radulescu@imar.ro}
%
\thanks{{\it 2010 Mathematics Subject Classification.} 35A15, 35J20,
35J62.}
%
\keywords{Variational Methods; Linking; Weak solutions; Nonlocal
equations.}
%
%\thanks{Work performed under the auspicies of G.N.A.M.P.A., of I.N.D.A.M.}
%21/03/11
\thanks{Typeset by \LaTeX}
\begin{document}

\begin{abstract}
We study a nonlocal Neumann problem driven by a nonhomogeneous
elliptic differential operator. The reaction term is a nonlinearity
function that exhibits $p$-superlinear growth but need not satisfy
the Ambrosetti-Rabinowitz condition. By using an abstract linking
theorem for smooth functionals,
we prove a multiplicity result on the existence of weak solutions for
such problems. An explicit example illustrates the main abstract
result of this paper.
\end{abstract}
\maketitle
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\section{Introduction}
Let $X$ be a real Banach space with a direct decomposition
$$
X=Y\oplus V,
$$
for certain linear subspaces $Y$ and $V$.

The condition of local linking was introduced in Li \& Liu
\cite{liuli}. We say that $\phi\in C^1(X,\R)$ has a {\it local
linking} if
\begin{equation}\label{link1}
\mbox{dim}\, Y<\infty
\end{equation}
 and there are $\alpha,\,\rho>0$ such that
\begin{equation}\label{link2}
\phi(u)\geq \alpha\qquad\forall\, u\in V,\,\|u\|= \rho.
\end{equation}

Under these assumptions, Li \& Liu \cite{liuli} established the
existence of nontrivial critical points for functionals in the
following classes:\\
(a) bounded from below;\\
(b) super-quadratic;\\
(c) asymptotically quadratic.

The weaker notion of {\it local linking at the origin} is due to
 Li \& Willem \cite{LW}. We say that the functional $\phi\in
C^1(X,\R)$ has a {\it local linking at the origin} if there exists a
positive constant $\rho$ such that
\begin{equation}\label{link3}
\phi(u)\leq 0\qquad\forall\, u\in Y,\,\|u\|\leq \rho
\end{equation}
and
\begin{equation}\label{link4}
\phi(u)\geq 0\qquad\forall\, u\in V,\,\|u\|\leq \rho.
\end{equation}

The concept of local
linking generalizes the notions of local minimum and local maximum.
When $0_X$ is a nondegenerate critical point of a functional $\phi$ of
class $C^2$
defined on a Hilbert space and $\phi (0_X)=0$, then $\phi$ has a local
linking at zero. Successively, Br\'{e}zis and Nirenberg, in their
quoted paper \cite{brenir}, proved the existence of a nontrivial
critical point
in case (a), assuming only \eqref{link1}, \eqref{link3},
\eqref{link4}, and the Palais-Smale condition.


\section{Statement of the Problem}
Let $\Omega$ be a bounded domain in $({\R}^{N},|\cdot|)$ with smooth
boundary $\partial \Omega$.  Assume that $p>1$ is a real number. Let
$\Delta_p u:=\div(|\nabla u|^{p-2}\nabla u)$ denote the $p$-Laplace
operator
and let $\displaystyle{\partial u}/{\partial\nu}$ be the outer
unit normal derivative.

In this paper we are interested in the existence of weak solutions to
the following nonlocal Neumann problem:
\begin{equation} \tag{$N_{M,f}^p$} \label{Nostro}
\left\{
\begin{array}{ll}
-\Big[M\Big(\displaystyle\int_\Omega|\nabla
u(x)|^pdx\Big)\Big]^{p-1}\Delta_pu
= f(x,u) & \rm in \quad \Omega
\\ \displaystyle\frac{\partial u}{\partial\nu}=0  & \rm on \,
\partial \Omega.
\end{array}
\right.
\end{equation}

In the sequel we will assume that $f:\Omega\times \erre\rightarrow
\erre$ is a Carath\'{e}odory function and $M:[0,+\infty)\to
[0,+\infty)$ is
continuous such that
\begin{itemize}
\item[$(C_M^1)$] $0< m_0\leq M(t)$, \textit{for every} $t\in
[0,+\infty)$.
 \end{itemize}
 Further, we require that
 \begin{itemize}
 \item[$(C_M^2)$] $
\displaystyle{\displaystyle\int_0^t [M(s)]^{p-1}ds}\geq t [M(t)]^{p-1},
$
\textit{for every} $t\in [0,+\infty)$.
\end{itemize}


\smallskip
Problem \eqref{Nostro} is called {\it nonlocal} because of the
presence of the term $M\Big(\displaystyle\int_\Omega|\nabla
u(x)|^pdx\Big)$, which implies that the quasilinear partial
differential equation in \eqref{Nostro} is no longer a pointwise
identity.
This phenomenon creates several mathematical difficulties in the
qualitative analysis of such equations.

 Problem \eqref{Nostro} is related to the stationary analogue of the
hyperbolic equation
\begin{equation}\label{laprima}
u_{tt}-\Big(a+b\displaystyle\int_{\Omega}|\nabla
u(x)|^{2}dx\Big)\Delta u=f(x,u)\quad \rm in \quad
\Omega,
\end{equation}
where $\Delta$ is the usual Laplace operator. Equation \eqref{laprima}
is a general version of the Kirchhoff equation
 \begin{equation}\label{eqk}
 \rho\,
u_{tt}-\left(\frac{P_0}{h}+\frac{E}{2L}\int_0^Lu_x^2\,dx\right)u_{xx}=0,\end{equation}
presented by Kirchhoff \cite{Kirchhoff}.\par
 This relation is as an extension of the classical
d'Alembert's wave equation by considering the effects of changes in
the length of the strings during the vibrations. The parameters in
equation \eqref{eqk} have the following meanings: $E$ is the Young
modulus of the material, $\rho$ is the mass density, $L$ is the length
of the string, $h$ is the area of cross-section, and $P_0$ is the
initial tension.

 The Kirchhoff's model takes
into account the length changes of the string produced by transverse
vibrations. The early
classical studies dedicated to Kirchhoff equations were given by
Bernstein \cite{Ber} and Pohozaev \cite{Poh}.
However, equation \eqref{laprima} received much attention only after
the paper by Lions \cite{Lions}, where an abstract
framework to the problem was proposed. Some related results can be
found, for example,
in \cite{AP,CFL,DS}. We also point out that Arosio and Panizzi
\cite{AP} studied the Cauchy-Dirichlet problem related to
\eqref{laprima} in the Hadamard sense as a special case of an abstract
second order Cauchy problem in a Hilbert space. D'Ancona and Spagnolo
\cite{DS} considered Kirchhoff's equation as a quasi-linear hyperbolic
Cauchy problem that describes the transverse oscillations of a
stretched string. Perera and Zhang \cite{perera} obtained a nontrivial
solution for nonlocal equations via Yang index and critical groups.
Chipot and Lovat \cite{10} pointed out that this kind of problems
models several physical and biological systems, where $u$ describes a
process depending on the average of itself (for example, population
density).
Nonlocal problems have been studied in these last years by variational
arguments; see the papers \cite{Anello2,Raffy3,YZ} and
references therein.
For completeness we refer the reader to some recent interesting
results obtained by Autuori and Pucci in \cite{AP1,AP2,AP3}
studying Kirchhoff equations by using different approaches.

\indent Recently, there have been results
for nonlinear Neumann problems driven by the $p$-Laplacian
differential operator. See, for instance, the works \cite{APS, Anello,
faraci, FGP} and \cite{MP,Ri,WT}. With
the exceptions of \cite{APS,MP}, in all the cited works, it is assumed
that $p > N$ (low dimensional problems) and the authors exploit the fact
that in this setting, the Sobolev space $W^{1,p}(\Omega)$ is compactly
embedded  in $C^0(\bar \Omega)$. Further, in \cite{FGP, MP}, the Euler
energy
functional is assumed to be coercive in $W^{1,p}(\Omega)$.\par


In this paper, motivated by this large interest on nonlocal Neumann
equations and exploiting a slight variant of
a result of Li and Willem \cite{LW} for functionals having a local
linking at zero, we prove the existence of at least one
nontrivial weak solution for problem \eqref{Nostro}.
 In contrast with the above cited papers, in our case the Euler
functional is indefinite.\par
More precisely, we prove an existence theorem (see Theorem
\ref{Theorem1} below) for problem \eqref{Nostro} when the right-hand
 side nonlinearity $f$ is $p$-superlinear. In the context of the
Dirichlet problems,  this setting was investigated by Liu \cite{liu},
who employed the
Ambrosetti-Rabinowitz (simply $(\rm AR)$) condition. His approach uses
Morse theory. In particular,
the $(\rm AR)$ condition was crucial in the computation of the
critical groups of the Euler functional at infinity.
In Theorem \ref{Theorem1} we do not assume the $(\rm AR)$ condition on
$f$ and so the
approach of \cite{liu} cannot be adopted.

In the present paper we use some ideas developed in \cite{GP}, where
the authors consider nonlinear Neumann problems driven by
$p$-Laplacian type operators
which are not necessarily homogeneous.

\indent The plan of the paper is as follows. Section 3 is devoted to
our abstract framework, while Section 4 is dedicated to the main
results.
A concrete example illustrates the main abstract result of this paper
(see Example \ref{esempio}). We refer to Br\'{e}zis \cite{brezis} for
basic
analytic preliminaries in relationship with partial differential
equations.
We also cite the monographs \cite{GPbook,k2} as general references on
the variational setting adopted in this paper.

\section{Abstract Framework}
 Let $W^{1,p}(\Omega)$  be the
usual Sobolev space, equipped with the norm
$$
\|u\|:=\Big(\int_{\Omega}(|\nabla
u(x)|^{p}+|u(x)|^{p})dx\Big)^{1/p}.
$$

Let $\langle\cdot,\cdot\rangle$ denote
 the duality pairing between $(W^{1,p}(\Omega))^*$ and
$W^{1,p}(\Omega)$.
We denote by $p^*$ the critical exponent of the Sobolev embedding
$\H\hookrightarrow L^q(\Omega)$.
 Recall that if $p<N$ then $p^*=Np/(N-p)$ and for every $q\in [1,p^*]$
there exists a positive constant $c_q$ such that
\begin{equation}\label{Poi}
\|u\|_{L^q(\Omega)}\leq c_q \|u\|,\qquad\forall\, u\in \H.
\end{equation}
 Moreover, when $p\geq N$, this inequality holds for all $q\in
[1,+\infty[$, since $p^*=+\infty$.\par
We also recall the classical Poincar\'{e}-Wirtinger inequality (see
Br\'{e}zis \cite[Chapter 9]{brezis}): there exists $\eta>0$ such that
\begin{equation}\label{PW}
\displaystyle\int_\Omega|
u(x)|^pdx \leq \eta\displaystyle\int_\Omega|\nabla
u(x)|^pdx,\quad\forall\, u\in  W^{1,p}(\Omega)\,\,{\rm s.t.}\,
\int_\Omega u(x)dx=0.
\end{equation}



For the sake of completeness, we recall that a $C^1$-functional
$\phi:X\to\R$ satisfies the Cerami condition at
level $\mu\in\R$, (briefly $\textrm{(C)}_{\mu}$) if
\begin{itemize}
\item[$\textrm{(C)}_{\mu}$] {\it Every sequence $\{u_n\}$ in $X$ such
that
$$
\phi(u_n)\to \mu \quad{\rm and}\quad
(1+\|u_n\|)\|\phi'(u_n)\|_{X^*}\to0,
$$
as $n\rightarrow \infty$, possesses a convergent subsequence.}
\end{itemize}
We say that $\phi$ satisfies the Cerami condition (in short
$\textrm{(C)}$) if $\textrm{(C)}_{\mu}$ holds for every $\mu\in
\erre$.\par

A basic tool in the present paper is the following abstract local
linking theorem due to Li \& Willem \cite{LW}. We reformulate this
result
in the special case given in \cite[Theorem 2.1]{GP}.

\begin{theorem}\label{LW}
Let $X$ be a Banach space such that $X=Y\oplus V$ with $\dim\,
Y<+\infty$. Assume that $\phi\in C^1(X)$ satisfies the following
conditions:
\begin{itemize}
\item[$i)$] $\phi$ has a local linking at zero$;$
\item[$ii)$] $\phi$ satisfies the $(\rm C)$ condition$;$
\item[$iii)$]$\phi$ maps bounded sets into bounded sets$;$
\item[$iv)$] for every finite dimensional subspace $E\subseteq V$, we
have
$$
\phi(u)\rightarrow -\infty\qquad
\mbox{
as $\|u\|\rightarrow +\infty$ and $u\in Y\oplus E$}.$$
\end{itemize}

\noindent Then $\phi$ admits at least one nontrivial critical point.
\end{theorem}

\section{Main Result}
Let $\Phi: W^{1,p}(\Omega)\rightarrow \erre$ be the smooth functional
defined by
\begin{equation}\label{E29}
\Phi(u):=\frac1p\widehat{M}\left(\displaystyle\int_\Omega|\nabla
u(x)|^pdx\right),
\end{equation}
 where
\begin{gather*}
\widehat{M}(t):=\int_0^t [M(s)]^{p-1}ds,\qquad\forall\, t\in [0,\infty).
\end{gather*}
Then the
Fr\'echet
derivative of $\Phi$ is $\Phi':W^{1,p}(\Omega)\rightarrow
(W^{1,p}(\Omega))^*$ defined by
\begin{eqnarray*}
\langle\Phi'(u),v\rangle =\left[M\left(\displaystyle\int_\Omega|\nabla
u(x)|^pdx\right)\right]^{p-1}\int_\Omega|\nabla
u(x)|^{p-2}\nabla u(x)\cdot \nabla v(x)dx,
\end{eqnarray*}
for all $u,v\in W^{1,p}(\Omega)$.\par
\indent Moreover, it is well-known that the operator
$A:W^{1,p}(\Omega)\rightarrow W^{-1,p}(\Omega)$ given by
\begin{eqnarray*}
\langle A(u),v\rangle =\int_\Omega|\nabla
u(x)|^{p-2}\nabla u(x)\cdot \nabla v(x)dx,
\end{eqnarray*}
satisfies the $(S_+)$ property.\par
 This means that for every sequence $\{u_n\}\subset W^{1,p}(\Omega)$
such that $u_n\rightharpoonup u$ (weakly) in $W^{1,p}(\Omega)$ and
\begin{equation}\label{S}
\limsup_{n\to\infty} \langle A(u_n),u_n-u\rangle \leq0,
\end{equation}
 it follws that $u_n\rightarrow u$ (strongly) in $W^{1,p}(\Omega)$.\par

\subsection{Remarks on our Assumptions}
The validity of the next lemma will be crucial in the sequel.
 \begin{lemma}\label{lemmaSer}
 Suppose that conditions $(C_M^1)$ and $(C_M^2)$ are fulfilled. Then
there are positive constants $m_1$ and $m_2$ such that
 \begin{equation}\label{cresce}
 \widehat{M}(t)\leq m_1t +m_2,
 \end{equation}
 for every $t\in [0,+\infty)$.
\end{lemma}
 \begin{proof}
 Let $t_1>0$. By our assumptions we have
 $$
 \frac{[M(t)]^{p-1}}{\widehat{M}(t)}\leq \frac{1}{t},
 $$
 for every $t\in ]t_1,+\infty)$. Integrating this inequality we obtain
 $$
\int_{t_1}^{t}\frac{[M(s)]^{p-1}}{\widehat{M}(s)}ds= \log
\frac{\widehat{M}(t)}{{\widehat{M}}(t_1)}\leq \log \frac{t}{t_1},
 $$
 for every $t\in]t_1,+\infty)$. Therefore
 $$
 \widehat{M}(t)\leq \frac{\widehat{M}(t_1)}{t_1}\,t,
 $$
 for every $t\in ]t_1,+\infty)$. Hence the growth condition
\eqref{cresce} holds taking, for instance,
$m_1:=\displaystyle\frac{\widehat{M}(t_1)}{t_1}$ and
$m_2:=\displaystyle\max_{t\in [0,t_1]}\widehat{M}(t)$.
 \end{proof}
Owing to conditions $(C_M^1)$ and $(C_M^2)$, by Lemma \ref{lemmaSer}
we deduce the following inequalities:
 \begin{itemize}
 \item[$(\widehat{C}_M)$]
$\displaystyle\frac{m_0^{p-1}}{p}\displaystyle\int_\Omega|\nabla
u(x)|^pdx\leq \Phi(u)\leq {\frac{m_1}{p}\|u\|^{p} +\frac{m_2}{p}},$
 \end{itemize}
 for every $u\in W^{1,p}(\Omega)$.\par
\indent Moreover, from now on, we assume that the nonlinearity
$f:\Omega\times \erre\rightarrow \erre$ satisfies the following
hypotheses $\textbf{H}(f)$:
\begin{itemize}\it{
\item [$\textrm{h}_1)$]for every $t\in\erre$, the function $x\mapsto
f(x,t)$ is measurable$;$
\item [$\textrm{h}_2)$]for every $x\in\Omega$, the function $t\mapsto
f(x,t)$ is continuous$;$
\item [$\textrm{h}_3)$]there exist $a_1\in L^{\infty}(\Omega)_+$,
$a_2>0$ and $r\in (p,p^*)$, such that for almost every $x\in \Omega$
and $t\in \erre$, we have
    $$
    |f(x,t)|\leq a_1(x)+a_2|t|^{r-1};
    $$
\item [$\textrm{h}_4)$]if $F(x,\xi):=\displaystyle \int_0^\xi
f(x,t)dt$ $($for all $\xi\in \erre$$)$ then
$$
\displaystyle \lim_{|\xi|\rightarrow
\infty}\frac{F(x,\xi)}{|\xi|^p}=+\infty,
$$
uniformly with respect to a.e. $x\in \Omega$$;$
\item [$\textrm{h}_5)$]there exists $$\displaystyle \mu\in
\left((r-p)\max\left\{1,\frac{N}{p}\right\},r\right]$$ such that
$$
\displaystyle \liminf_{|\zeta|\rightarrow
\infty}\frac{f(x,\zeta)\xi-pF(x,\zeta)}{|\zeta|^\mu}>0,
$$
uniformly for almost every $x\in \Omega$$;$
\item [$\textrm{h}_6)$]we have
$$
\lim_{t\rightarrow 0}\frac{f(x,t)}{|t|^{p-2}t}=0,
$$
uniformly for almost every $x\in \Omega$$;$
\item [$\textrm{h}_7)$] there exists $\delta>0$ such that
$F(x,\xi)\geq 0$ for almost every $x\in \Omega$ and $|\xi|\leq
\delta$.}
\end{itemize}

\begin{remark}\label{fe}\rm{
A typical example when the above conditions hold is given by the real
function
\begin{enumerate}
\item[] $\displaystyle
f(t):=|t|^{p-2}t\log(1+t^2)+\frac{2t|t|^{p}}{p(1+t^2)},\qquad\forall\,
t\in\erre.$
\end{enumerate}
\noindent  A direct computation ensures that the $(\rm AR)$ condition
fails in this special case.}
\end{remark}

For the sake of completeness we recall that a \textit{weak solution}
of problem \eqref{Nostro} is a function $u\in W^{1,p}(\Omega)$ such
that
\begin{eqnarray*}
\langle\Phi'(u),v\rangle =\int_\Omega f(x,u(x))v(x)dx,\qquad\forall\,
v\in W^{1,p}(\Omega).
\end{eqnarray*}

The main result in this paper is the following.

\begin{theorem}\label{Theorem1}
Assume that hypotheses $\textbf{H}(f)$ are fulfilled. Then problem
\eqref{Nostro} has at least one nontrivial weak solution.
\end{theorem}

\subsection{Auxiliary Properties} We first establish the boundedness
of Cerami sequences for a related energy functional.
\begin{lemma}\label{lemma1}
 Suppose that conditions $\textbf{H}(f)$ are fulfilled. Then every
Cerami sequence for the functional
 $$
 \phi(u):=\Phi(u)-\int_\Omega F(x,u(x))dx,\qquad \forall\,u\in
W^{1,p}(\Omega)
 $$
 is bounded in $W^{1,p}(\Omega)$.
\end{lemma}

\begin{proof}
Let $\{u_n\}\subset W^{1,p}(\Omega)$ be a Cerami sequence, i.e.:
\begin{equation}\label{E1}
|\phi(u_n)|\leq M_1,\quad \forall\, n\in\enne
\end{equation}
for some positive constant $M_1$ and
\begin{equation}\label{E2}
\quad (1+\|u_n\|)\|\phi'(u_n)\|_{(W^{1,p}(\Omega))^*}\to0,
\end{equation}
as $n\rightarrow \infty$.\par
\noindent We proceed arguing by contradiction. So, suppose that the
conclusion is not true. Passing to a subsequence if necessary, we may
assume that
$$
\|u_n\|\rightarrow +\infty\qquad\mbox{as}\  n\rightarrow \infty.
$$
By relations \eqref{E1} and \eqref{E2}, we have
 \begin{equation}\label{E3}
 |\langle\phi'(u_n),v\rangle|=\left|\langle\Phi'(u_n),v\rangle-\int_\Omega
f(x,u_n(x))v(x)dx\right|\leq \frac{\varepsilon_n\|v\|}{1+\|u_n\|},
 \end{equation}
 for every $u\in W^{1,p}(\Omega)$, with $\varepsilon_n \searrow
0^+$.\par
\noindent Choosing as test function $v:=u_n$ in \eqref{E3}, we easily
obtain
 \begin{equation}\label{E4}
 -\langle\Phi'(u_n),u_n\rangle+\int_\Omega f(x,u_n(x))u_n(x)dx\leq
\varepsilon_n,\quad\forall\, n\in\enne.
 \end{equation}
 Inequality \eqref{E1} yields
  \begin{equation}\label{E5}
 \widehat{M}\left(\displaystyle\int_\Omega|\nabla
u_n(x)|^pdx\right)-p\int_\Omega F(x,u_n(x))dx\leq pM_1,\quad \forall\,
n\in\enne.
  \end{equation}

  \noindent Adding \eqref{E4} and \eqref{E5} we also have that
  \begin{eqnarray}\label{E6}
\qquad\,\widehat{M}\left(\displaystyle\int_\Omega|\nabla
u_n(x)|^pdx\right) &-& \langle \Phi'(u_n),u_n\rangle\\\nonumber
               &+&
\int_{\Omega}[f(x,u_n(x))u_n(x)-pF(x,u_n(x))]dx\\\nonumber&\leq&
M_2,\quad \forall\, n\in\enne,
\end{eqnarray}
for some $M_2>0$.\par
\noindent Next, we observe that, by condition $(C_M^2)$, one has
$$
\frac{{\widehat{M}\left(\displaystyle\int_\Omega|\nabla
u_n(x)|^pdx\right)}}{\left[M\left(\displaystyle\int_\Omega|\nabla
u_n(x)|^pdx\right)\right]^{p-1}}\geq \displaystyle\int_\Omega|\nabla
u_n(x)|^pdx,
$$
for every $n\in \enne$.
Therefore
\begin{eqnarray}\label{E7}
\widehat{M}\left(\displaystyle\int_\Omega|\nabla
u_n(x)|^pdx\right)-\langle \Phi'(u_n),u_n\rangle\geq 0,\quad
\forall\,n\in\enne.
\end{eqnarray}

\noindent From now on, arguing as in \cite{GP}, by virtue of
hypothesis $\rm h_5)$, we find two positive constants, denoted by
$\beta$ and $M_{3,\beta}$, such that
\begin{eqnarray}\label{E8}
0<\beta |\zeta|^\mu\leq f(x,\zeta)\zeta-pF(x,\zeta),
\end{eqnarray}
for almost every $x\in \Omega$ and every $|\zeta|\geq M_{3,\beta}$.
Further, condition $\rm h_3)$ implies that there exists a positive
constant $M_4$ such that
 \begin{eqnarray}\label{E9}
 \left|f(x,\zeta)\zeta-pF(x,\zeta)\right|\leq M_4,
 \end{eqnarray}
 for almost every $x\in \Omega$ and every $|\zeta|\geq M_{3,\beta}$ .
 Hence, through \eqref{E8} and \eqref{E9}, we obtain
\begin{eqnarray}\label{E10}
\beta|\zeta|^\mu-(M_4+\beta M_{3,\beta}^\beta)\leq f(x,\zeta)\zeta-
pF(x,\zeta),
\end{eqnarray}
for almost every $x\in \Omega$ and every $\zeta\in\erre$. Exploiting
relation \eqref{E6}, by using \eqref{E7} and \eqref{E10}, we have
$$
\int_\Omega |u_n(x)|^\mu dx\leq M_5,\quad \forall\, n\in\enne
$$
for some $M_5>0$.
 Thus, it follows that the sequence $\{u_n\}$ is bounded in
$L^\mu(\Omega)$. Now, since $\mu\leq r<p^*$, let $\sigma\in [0,1)$ be
such that
 $$
 \displaystyle\frac{1}{r}=\frac{1-\sigma}{\mu}+\frac{t}{p^*}\,.
 $$
 Thus, we have
 $$
 \|u_n\|_{L^r(\Omega)}\leq
\|u_n\|_{L^\mu(\Omega)}^{1-\sigma}\|u_n\|_{L^{p^*}(\Omega)}^{\sigma},
 $$
 for every $n\in\enne$; see, for completeness \cite[p. 905]{GPbook}.\par
\noindent Then
 \begin{eqnarray}\label{E11}
 \|u_n\|_{L^r(\Omega)}\leq M_6 \|u_n\|_{L^{p^*}(\Omega)}^{\sigma r},
 \end{eqnarray}
 for every $n\in\enne$ and for some $M_6>0$.\par
 \noindent From \eqref{E3}, with $v=u_n$, we have
 \begin{equation}\label{E12}
 \left|\langle\Phi'(u_n),u_n\rangle-\int_\Omega
f(x,u_n(x))u_n(x)dx\right|\leq {\varepsilon_n},
 \end{equation}
 for every $n\in\enne$.\par
 \noindent Hypotheses $\rm h_3)$ and $\rm h_6)$ imply that for a given
$\varepsilon>0$, we can find $c_\varepsilon>0$ such that
 \begin{equation}\label{E13}
 |f(x,t)t|\leq \varepsilon |t|^p+c_\varepsilon |t|^r,
 \end{equation}
 for almost every $x\in \Omega$ and every $t\in\erre$.
 Using \eqref{E13} and \eqref{E12}, we obtain
 $$
 \langle\Phi'(u_n),u_n\rangle\leq \varepsilon_n +\varepsilon
\|u_n\|_{L^p(\Omega)}^{p}+c_\varepsilon \|u_n\|_{L^r(\Omega)}^{r},
 $$
 for every $n\in\enne$.
 Hence
  \begin{equation}\label{E14}
  m_0^{p-1}\int_\Omega|\nabla
u_n(x)|^{p}dx\leq \varepsilon_n +\varepsilon
\|u_n\|_{L^p(\Omega)}^{p}+c_\varepsilon M_6
\|u_n\|_{L^{p^*}(\Omega)}^{\sigma r},
   \end{equation}
    for every $n\in\enne$.\par
    \noindent Let us denote $\displaystyle y_n:=\frac{u_n}{\|u_n\|}$.
Then $\|y_n\|=1$ and so we may assume that
    \begin{equation}\label{E15}
    y_n\rightharpoonup y\quad {\rm in}\quad W^{1,p}(\Omega),
    \end{equation}
    and
     \begin{equation}\label{E16}
    y_n\rightarrow y\quad {\rm in}\quad L^p(\Omega).
    \end{equation}

    \noindent Moreover, by \eqref{E16}, we also have
    \begin{equation}\label{G1}
     y_n(x)\rightarrow y(x)\quad {\rm a.e.\,\,\, in}\,\,\, \Omega,
    \end{equation}
    and there exists $h\in L^p(\Omega)$ such that
    \begin{equation}\label{G2}
     |y_n(x)|\leq h(x),
    \end{equation}
    for almost all $x\in \Omega$.\par
    \noindent Dividing by $\|u_n\|^p$ relation \eqref{E14} we obtain
    \begin{equation}\label{E17}
  m_0^{p-1}\int_\Omega|\nabla
y_n(x)|^{p}dx\leq \frac{\varepsilon_n}{\|u_n\|^p} +\varepsilon
\|y_n\|_{L^p(\Omega)}^{p}+\frac{c_\varepsilon M_6}{\|u_n\|^{p-\sigma
r}} \|y_n\|_{L^{p^*}(\Omega)}^{\sigma r},
   \end{equation}
   for every $n\in\enne$.\par
   \noindent Now, we observe that condition
   $$
   \mu>(r-p)\max\left\{1,\frac{N}{p}\right\}
   $$
   is equivalent to  $\sigma r<p.$

  \noindent Passing to the limit as $n\rightarrow +\infty$ in
\eqref{E17} and using \eqref{E15}, we have
   \begin{equation}\label{E18}
  m_0^{p-1}\int_\Omega|\nabla
y(x)|^{p}dx\leq \varepsilon \|y\|_{L^p(\Omega)}^{p}\leq
\varepsilon\|y\|^{p}\leq \varepsilon,
   \end{equation}
   But
   $$
   \|y\|\leq\displaystyle \liminf_{n\rightarrow \infty}\|y_n\|=1.
   $$
   Now, since $\varepsilon >0$ is arbitrary, we deduce that
$y=\kappa\in \erre$. If $y=0$, we have $\nabla y_n\rightarrow 0$
in
$L^p(\Omega;\erre^N)$ and so $y_n\rightarrow 0$ in $W^{1,p}(\Omega)$.
This fact is a contradiction since $\|y_n\|=1$, for all $n\in\enne$.
 Consequently one has $y\neq 0$. This implies that
 $$
 |u_n(x)|\rightarrow +\infty,\,\,{\rm for\, every\,}\, x\in \Omega.
 $$
 By \eqref{E1} we have
  \begin{equation}\label{E19}
 \left|\frac{\Phi(u_n)}{\|u_n\|^p}-\int_\Omega
\frac{F(x,u_n(x))}{\|u_n\|^p}dx\right|\leq \frac{M_1}{\|u_n\|^p},
 \end{equation}
 for every $n\in\enne$.
 Therefore, the right-hand side of condition $(\widehat{C}_M)$ implies
that
 \begin{equation}\label{E20}
 \frac{\Phi(u_n)}{\|u_n\|^p}\leq \frac{m_1}{p}+\frac{m_2}{p\|u_n\|^p},
 \end{equation}
 for every $n\in\enne$.
 Hence, by \eqref{E19} and \eqref{E20} we can write
 \begin{equation}\label{E21}
 \int_\Omega \frac{F(x,u_n(x))}{\|u_n\|^p}dx\leq
\frac{M_1}{\|u_n\|^p}+\frac{m_1}{p}+\frac{m_2}{p\|u_n\|^p},
 \end{equation}
 for every $n\in\enne$.\par
 \noindent On the other hand, condition $\rm h_4)$ shows that for a
given $\eta>0$, there exists $M_{7,\eta}>0$ such that
\begin{equation}\label{E22}
\displaystyle\frac{F(x,\xi)}{|\xi|^p}\geq \eta>0,
 \end{equation}
 for almost every $x\in \Omega$ and every $|\xi|\geq M_{7,\eta}$. Set
 $$
 M_{7,\eta}^{+}:=\{x\in\Omega:|u_n(x)|\geq M_{7,\eta}\}\qquad
 M_{7,\eta}^{-}:=\{x\in\Omega:|u_n(x)|< M_{7,\eta}\}.
 $$

 \noindent By using \eqref{E22} and $\rm h_3)$ we can write
 \begin{eqnarray}\label{E23}
\displaystyle\int_\Omega\frac{F(x,u_n(x))}{\|u_n\|^p}dx &=&
\displaystyle\int_{M_{7,\eta}^{+}}\frac{F(x,u_n(x))}{|u_n(x)|^p}|y_n(x)|^pdx\\\nonumber
               &+&
\int_{M_{7,\eta}^{-}}\displaystyle\frac{F(x,u_n(x))}{\|u_n\|^p}dx
\\\nonumber
               &\geq& \displaystyle\int_{M_{7,\eta}^{+}}\eta
|y_n(x)|^pdx\\\nonumber
               &+&
\int_{M_{7,\eta}^{-}}\displaystyle\frac{F(x,u_n(x))}{\|u_n\|^p}dx
\\\nonumber
               &\geq&
\eta\displaystyle\int_{M_{7,\eta}^{+}}|y_n(x)|^pdx-\frac{M_8}{\|u_n\|^p},\nonumber
\end{eqnarray}
for some $M_8>0$ and for every $n\in\enne$.
On the other hand, observing that $|u_n(x)|\rightarrow +\infty$ for
almost every $x\in \Omega$ and bearing in mind \eqref{G1}, we have
$$
\displaystyle\chi_{M_{7,\eta}^{+}}(x)y_n(x)\rightarrow\chi_{\Omega}(x)\kappa,
$$
for almost every $x\in \Omega$. Moreover, by \eqref{G2},
$$
\chi_{M_{7,\eta}^{+}}(x)|y_n(x)|^p\in L^1(\Omega).
$$
Thus, as $n\rightarrow \infty$,
 \begin{eqnarray}\label{E24}
\displaystyle\int_{M_{7,\eta}^{+}}|y_n(x)|^pdx=\displaystyle\int_{\Omega}\chi_{M_{7,\eta}^{+}}(x)|y_n(x)|^pdx\rightarrow
|\kappa|^p\meas(\Omega).
\end{eqnarray}
So, passing to the limit as $n\rightarrow \infty$ and using
\eqref{E24}, we obtain
$$
\liminf_{n\rightarrow
\infty}\displaystyle\int_\Omega\frac{F(x,u_n(x))}{\|u_n\|^p}dx\geq
\eta |\kappa|^p\meas(\Omega).
$$
Since $\eta>0$ is arbitrary and $\kappa\neq 0$, it follows that
\begin{eqnarray}\label{E25}
\liminf_{n\rightarrow
\infty}\displaystyle\int_\Omega\frac{F(x,u_n(x))}{\|u_n\|^p}dx=+\infty.
\end{eqnarray}
Comparing relations \eqref{E25} and \eqref{E21} we obtain a
contradiction. The proof of Lemma \ref{lemma1} is now complete.
\end{proof}

The next result shows that the functional
$\phi$ defined in Lemma \ref{lemma1} satisfies the Cerami condition.
\begin{lemma}\label{lemma2}
Assume that conditions $\textbf{H}(f)$ hold. Then  $\phi$
satisfies the compactness condition $\rm{(C)}$.
\end{lemma}

\begin{proof}
 Let $\{u_{n}\}\subset W^{1,p}(\Omega)$ be a Cerami
sequence. Thus, by Lemma \ref{lemma1}, the sequence $\{u_n\}$ is
bounded in $W^{1,p}(\Omega)$. Since $W^{1,p}(\Omega)$ is reflexive,
we can suppose that, up to a subsequence,
$u_{n}\rightharpoonup u$ in $W^{1,p}(\Omega)$.
 We prove that, in fact, $\{u_{n}\}$ strongly converges to $u\in
W^{1,p}(\Omega)$.
We first observe that
$$
\langle\Phi'(u_n),u_n-u\rangle = \langle \phi'(u_n),u_n-u\rangle
               + \int_{\Omega}f(x,u_n(x))(u_n-u)(x)dx.
$$
Since $\|\phi'(u_n)\|_{(W^{1,p}(\Omega))^*}\to0$ and the sequence
$\{u_n-u\}$ is bounded in
$W^{1,p}(\Omega)$,  we obtain
\[
\langle \phi'(u_n),u_n-u\rangle\to0,
\]
as $n\rightarrow \infty$.\par
\noindent Next, by $\textrm{h}_3)$ and taking into account that
$u_n\rightarrow u$ in $L^r(\Omega)$ for all $r<p^*$, we deduce that
\[
\int_{\Omega}|f(x,u_n(x))||u_n(x)-u(x)|dx\to0,
\]
as $n\rightarrow \infty$. Therefore, by $(C_M^1)$, one has
\[
\langle A(u_n),u_n-u\rangle
\rightarrow 0,
\]
as $n\rightarrow \infty$.\par
\noindent Since $A$ has the $(S_+)$ property, we conclude that
$u_{n}\to u$ strongly in $W^{1,p}(\Omega)$.
The proof of Lemma \ref{lemma2} is now complete.
\end{proof}

The next result establishes that $\phi$ has a local linking behaviour.
\begin{proposition}\label{Prop1}
Assume that hypotheses $\textbf{H}(f)$ hold. Then $\phi$ has a local
linking at the origin.
\end{proposition}

\begin{proof}
Let $\delta>0$ as in condition $\rm h_7)$. Fix $\gamma\in\erre$ such
that $|\gamma|\leq \delta$. Then
$$
\|\gamma\|=\delta\meas\,(\Omega)^{1/p}.
$$
Set $\rho_1:=\delta\meas(\Omega)^{1/p}$. Using  $\rm h_7)$, since
$\Phi(0_{W^{1,p}(\Omega)})=0$, we deduce that
\begin{eqnarray}\label{C1}
\phi(u)\leq 0\qquad\forall\, u\in W^{1,p}(\Omega),\ \|\gamma\|\leq
\rho_1.
\end{eqnarray}

\noindent Next, combining the Mean Value Theorem with \eqref{E13} we
obtain that
\begin{equation}\label{Y2}
 |F(x,\xi)|\leq \varepsilon |\xi|^p+{c_\varepsilon} |\xi|^r,
 \end{equation}
 for almost every $x\in \Omega$ and every $\xi\in\erre$. Let us consider
 $$
 V:=\left\{u\in W^{1,p}(\Omega):\int_\Omega u(x)dx=0\right\}.
 $$
 Then, by \eqref{Y2} and the Poincar\'{e}-Wirtinger inequality
\eqref{PW}, we have that
 \begin{eqnarray}\label{Y6}
\phi(u) &=& \Phi(u)-\int_\Omega F(x,u(x))dx\\\nonumber
               &\geq &
\displaystyle\frac{m_0^{p-1}}{p}\displaystyle\int_\Omega|\nabla
u(x)|^pdx -\varepsilon\|u\|^p_{L^p(\Omega)}-{c_\varepsilon}
\|u\|^r_{L^r(\Omega)}\\\nonumber
&\geq & \left(\displaystyle\frac{m_0^{p-1}}{p}-\varepsilon
\eta\right)\displaystyle\int_\Omega|\nabla
u(x)|^pdx-{c_\varepsilon} \|u\|^r_{L^r(\Omega)}.
\end{eqnarray}
Fix $$\varepsilon \in \left(0,\frac{m_0^{p-1}}{\eta p}\right).$$

\noindent Since $p<r<p^*$, the space $W^{1,p}(\Omega)$ is continuously
embedded in $L^r(\Omega)$. Thus, by \eqref{Poi} and \eqref{PW}, it
follows that
 \begin{eqnarray}\label{giov1}
\|u\|_{L^r(\Omega)}\leq
c_r(1+\eta^p)^{1/p}\left(\displaystyle\int_\Omega|\nabla
u(x)|^pdx\right)^{1/p},\quad\forall\, u\in V.
 \end{eqnarray}

\noindent Set
 \begin{eqnarray}\label{giov2}
M_{9}:=c_{\varepsilon}\left(c_r(1+\eta^p)^{1/p}\right)^{r}.
\end{eqnarray}
\noindent By \eqref{Y6}, relations \eqref{giov1} and \eqref{giov2} yield
$$
\phi(u) \geq  \left(\displaystyle\frac{m_0^{p-1}}{p}-\varepsilon
\eta\right)\displaystyle\int_\Omega|\nabla
u(x)|^pdx-M_{9}\left(\displaystyle\int_\Omega|\nabla
u(x)|^pdx\right)^{r/p},
$$
 for every $u\in V$.\par
 \noindent Bearing in mind the choice of $\varepsilon$ and taking into
account that $p<r$, the above inequality implies that there is
$\rho_2>0$ sufficiently
small such that for all $u\in V$ with $\|u\|\leq \rho_2$,
 \begin{eqnarray}\label{B8}
 \phi(u)\geq 0.
 \end{eqnarray}
 \noindent In conclusion, if $\rho\leq \min\{\rho_1,\rho_2\}$,
relations \eqref{B8} and \eqref{C1} imply that the energy functional
$\phi$ has a local linking at the origin.
\end{proof}


\begin{proposition}\label{Prop2}
Assume that hypotheses $\textbf{H}(f)$ hold and $E\subseteq V$ is a
finite dimensional linear subspace. Then
$$
\phi(u)\rightarrow -\infty\qquad\mbox{as $\|u\|\rightarrow +\infty$,
$u\in \erre\oplus E$.}
$$
\end{proposition}

\begin{proof}
We will prove that the restriction of the functional $\phi$ to
$\erre\oplus E$ is anticoercive. For our goal, exploiting hypothesis
$\rm h_4)$, for a given $\theta>0$,
we find a positive constant $M_{10,\theta}$ such that
\begin{equation}\label{F1}
{F(x,\xi)}\geq \theta |\xi|^p,
 \end{equation}
 for almost every $x\in\Omega$ and every $|\xi|\geq M_{10,\theta}$.
  Moreover, owing to $\rm h_3)$, there exists $M_{11}>0$ such that
 \begin{equation}\label{F2}
|{F(x,\xi)}|\leq M_{11},
 \end{equation}
 for almost every $x\in\Omega$ and every $|\xi|<M_{11}$. Thus, thanks
to \eqref{F2} and \eqref{F1}, there exists $M_{12}>0$ such that
 \begin{equation}\label{F3}
{F(x,\xi)}\geq \theta|\xi|^p-M_{12},
 \end{equation}
 for almost every $x\in\Omega$ and every $\xi\in\erre$.\par
 \noindent By using the right-hand side of $(\widehat{C}_M)$ and
\eqref{F3}, we have for every $u\in \erre\oplus E$
\begin{equation}\label{k1}
\phi(u)=\Phi(u)-\int_\Omega F(x,u(x))dx\leq \displaystyle
\frac{m_1}{p}\|u\|^{p}-\theta \|u\|^p_{L^p(\Omega)}+ M_{13},
\end{equation}
for some $M_{13}>0$.
Now, since the space $\erre\oplus E$ is finite dimensional, all the
norms are equivalent and we have
\begin{equation}\label{k2}
\phi(u)\leq \left(\frac{m_1}{p}-\theta M_{14}\right)\|u\|^p+M_{13},
\end{equation}
for some $M_{14}>0$.
By relation \eqref{k2} and bearing in mind that $\theta$ was
arbitrary, we conclude that
$
\phi(u)\rightarrow -\infty$ as $\|u\|\rightarrow +\infty$, $u\in
\erre\oplus E$.
\end{proof}



\subsection{Proof of Theorem \ref{Theorem1} concluded}
We apply Theorem \ref{LW} for $Y:=\erre$,
$$
 V:=\left\{u\in W^{1,p}(\Omega):\int_\Omega u(x)dx=0\right\},
 $$
 and
 $$
 \phi(u):=\frac1p\widehat{M}\left(\displaystyle\int_\Omega|\nabla
u(x)|^pdx\right)-\int_\Omega\left(\int_0^{u(x)}f(x,t)dt\right)dx,\quad\forall\,
u\in W^{1,p}(\Omega).
 $$

\noindent The definition of $\phi$ shows that it maps bounded sets
into bounded sets. Moreover, Lemma \ref{lemma2} and Propositions
\ref{Prop1} and \ref{Prop2}
allow the application of Theorem \ref{LW}.
 Thus, we can find a  weak solution $u\in
W^{1,p}(\Omega)\setminus\{0\}$  of problem \eqref{Nostro}. The proof
is now complete.
\qed


\subsection{Example}
We conclude this paper with an illustration of Theorem \ref{Theorem1}
for an elliptic nonlocal Neumann problem involving the Laplace
operator.
We first remark that for $p=2$, the function
$$
M(t):=1+\frac{\cos t}{1+t^2}
$$
satisfies conditions $(C_M^1)$ and $(C_M^2)$.

\begin{example}\label{esempio}\rm
 Consider the nonlinear problem
\begin{equation} \tag{$N_f$} \label{Nostro2}
\left\{
\begin{array}{ll}
-M\Big(\displaystyle\int_\Omega|\nabla
u(x)|^2dx\Big)\Delta u= f(u) & \rm in \quad \Omega
\\ \displaystyle\frac{\partial u}{\partial \nu}=0  & \rm on \,
\partial \Omega,\\
\end{array}
\right.
\end{equation}
where $f$ is given in Remark \ref{fe} and
$$
M\Big(\displaystyle\int_\Omega|\nabla
u(x)|^2dx\Big):=1+\frac{\cos \Big(\displaystyle\int_\Omega|\nabla
u(x)|^2dx\Big)}{1+\Big(\displaystyle\int_\Omega|\nabla
u(x)|^2dx\Big)^2}.
$$
Thus, by Theorem \ref{Theorem1}, problem \eqref{Nostro2} admits at
least one nontrivial weak solution.
\end{example}

\begin{remark}\rm{
We notice that there are several multiplicity results for nonlinear
Neumann problems driven by $p$-Laplacian type operators.
For instance, Motreanu and Papageorgiou \cite{MP2} studied a nonlinear
Neumann problem driven by a nonhomogeneous
quasilinear degenerate elliptic differential operator. In \cite{MP2}
the reaction term is a Carath\'{e}odory
function that exhibits subcritical growth in the second variable. The
authors, using variational methods
based on the mountain pass and  deformation theorems, together
with truncation and minimization techniques, showed that the problem has
three nontrivial smooth solutions, two of which have constant sign
(one positive,
the other negative).}
\end{remark}

\medskip
  \indent \textbf{Acknowledgements.} G. Molica Bisci is supported by
the GNAMPA Project 2012: {\it Esistenza e molteplicit\`{a} di
soluzioni per problemi differenziali non lineari}.  V. R\u adulescu
acknowledges the support through Grant CNCS PCE-47/2011.

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\end{document} 