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

\AtBeginDocument{{\noindent\small
2007 Conference on Variational and Topological Methods: Theory, Applications,
Numerical Simulations, and Open Problems.
{\em Electronic Journal of Differential Equations},
Conference 18 (2010),  pp. 57-66.\newline
ISSN: 1072-6691. URL: http://ejde.math.txstate.edu or http://ejde.math.unt.edu
\newline ftp ejde.math.txstate.edu}
\thanks{\copyright 2010 Texas State University - San Marcos.}
\vspace{9mm}}

\begin{document} \setcounter{page}{57}
\title[\hfilneg EJDE-2010/Conf/18/\hfil
Solutions for the noncoercive Neumann p-Laplacian]
{Existence and multiplicity of solutions for the noncoercive
Neumann p-Laplacian}

\author[N. S. Papageorgiou, E. M. Rocha\hfil EJDE/Conf/18 \hfilneg]
{Nikolaos S. Papageorgiou, Eugenio M. Rocha}  % in alphabetical order

\address{Nikolaos S. Papageorgiou \newline
Department of Mathematics, National Technical University,
Zografou Campus, Athens 15780, Greece}
\email{npapg@math.ntua.gr}

\address{Eugenio M. Rocha \newline
Department of Mathematics,
Campus de Santiago, University of Aveiro, 3810-193 Aveiro, Portugal}
\email{eugenio@ua.pt}

\thanks{Published July 10, 2010.}
\thanks{E. M. Rocha was partially supported by the grant SFRH/BPD/38436/2007
from FCT, \hfill\break\indent
 and the research unit Mathematics and Applications.}
\subjclass[2000]{35J25, 35J80}
\keywords{Locally Lipschitz function; generalized subdifferential;
\hfill\break\indent second deformation theorem; Palais-Smale condition}

\begin{abstract}
 We consider a nonlinear Neumann problem driven by the p-La\-pla\-cian
 differential operator with a nonsmooth potential
 (hemivariational inequality). Using variational techniques
 based on the smooth critical point theory and the second
 deformation theorem, we prove an existence theorem and a
 multiplicity theorem, under hypothesis that in general do not
 imply the coercivity of the Euler functional.
\end{abstract}

\maketitle
\numberwithin{equation}{section}
\newtheorem{theorem}{Theorem}[section]
\newtheorem{proposition}[theorem]{Proposition}
\newtheorem{example}[theorem]{Example}

\section{Introduction}

Let $Z\subseteq\mathbb{R}^N$ be a bounded domain with
$C^2$ boundary, $\partial Z$.
This article  concerns the existence and multiplicity of
nontrivial solutions for the following nonlinear Neumann problem
with a nonsmooth potential (hemivariational inequality):
\begin{equation}\label{pr:prob1}
 \begin{gathered}
  -\Delta_{p}\, x(z)\in \partial j(z,x(z))\quad \text{a.e. in } Z,\\
  \frac{\partial x}{\partial n}=0 \quad \text{on }\partial Z.
 \end{gathered}
\end{equation}
Here $\Delta_{p}\, x=\mathop{\rm div}(\|Dx\|_{\mathbb{R}^N}^{p-2}Dx)$
($1<p<\infty$) is the $p$-Laplacian differential operator, $j$
is a measurable potential function, and for almost all $z\in Z$
the function $x\mapsto j(z,x)$ is locally Lipschitz and in
general nonsmooth. By $\partial j(z,x)$ we denote the generalized
(Clarke) subdifferential of the locally Lipschitz function
$x\mapsto j(z,x)$. Our aim, in this work, is to prove existence
and multiplicity results for  \eqref{pr:prob1},
under hypotheses which do not guarantee the coercivity of the
Euler functional.

The question of existence of multiple nontrivial solutions has
been studied extensively in the context of Dirichlet problems driven
by the $p$-Laplacian and there are several such papers in the literature,
using a variety of hypotheses and techniques. In contrast,
the Neumann $p$-Laplacian case, in some sense, is lagging behind.
Recently there have been some multiplicity results within the Neumann
setting. We mention the works  \cite{R1,R4,R13,R17}.
In these works, the authors establish the existence
of infinitely many solutions for certain nonlinear elliptic Neumann
problems, by imposing a kind of oscillatory behavior on the nonlinear
term. In all these works, it is crucial the assumption that $p>N$
(low dimensional problems). It is well-known that this dimensionality
condition implies that the Sobolev space $ W^{1,p}(Z)$ is
embedded compactly in $C(\bar{Z})$ and this fact is used extensively
by the authors, in the aforementioned works. The works \cite{R3} and \cite{R9},
consider nonlinear Neumann eigenvalue problems and prove a
``three solutions theorem", using an abstract multiplicity result
of Ricceri \cite{R16}. Again the condition $p>N$ is present in these works,
as it is in the recent work of Wu-Tan \cite{R18}, but their approach
is based on minimax techniques from critical point theory.
In all the aforementioned works, the potential function is smooth;
 i.e., $j(z,\cdot)\in C^1(\mathbb{R})$. Neumann problems involving
the $p$-Laplacian and a nonsmooth potential were investigated in
\cite{R2,R10,R14}. In these works, the
assumptions on the potential $j$ imply that the Euler functional,
or a suitable truncation of it, is coercive. In \cite{R2}, it is
assumed that $p\geq 2$ and the approach is degree theoretic. In
\cite{R10} and   \cite{R14}, the approach is variational
based on the nonsmooth critical point theorem (e.g., see \cite{R5,R11,R15}). 
Here, our hypotheses on the nonsmooth
potential $j$ do not necessarily imply the coercivity of the Euler
functional and our method of proof is based on the
nonsmooth second deformation theorem, due to Corvellec \cite{R8}.

This paper is organized as follows. In Section~2, we recall
various notions and results which will be used later. In
Section~3, we prove an existence theorem for a generalized version
of \eqref{pr:prob1}. Finally, in Section~4,  by
strengthening our hypotheses on $j$, we establish a multiplicity
result for \eqref{pr:prob1}.


\section{Mathematical Background}

The nonsmooth critical point theory, which we will use in the
variational arguments of  this paper, is based mainly on the
subdifferential theory for locally Lipschitz  functions.
So, we start by recalling some basic notions from this theory.
Details can be found in \cite{R7}.

Let $X$ be a Banach space. By $X^\ast$ we denote the topological
dual of $X$ and by ${{\langle {\cdot,\cdot}\rangle}}$ we denote
the duality brackets for the pair $(X^\ast,X)$.
If $\varphi:X\to\mathbb{R}$ is a locally Lipschitz function,
then the \emph{generalized directional derivative}
$\varphi^0(x;h)$ of $\varphi$ at $x\in X$, in the direction of
$h\in X$, is defined by
 $$
\varphi^0(x;h)=\limsup_{x'\to x,\, \lambda\downarrow 0}
\frac{\varphi(x'+\lambda h)-\varphi(x')}{\lambda}.
$$
It is easy to see that $h\mapsto\varphi^0(x;h)$ is sublinear
continuous and so, it is the support function of a nonempty,
convex and $w^\ast$-compact set $\partial\varphi(x)\subset X^\ast$
defined by
 $$
\partial\varphi(x)=\{x^\ast\in X^\ast: {{\langle {x^\ast,h}\rangle}}
\leq \varphi^0(x;h)\text{ for all }h\in X\}.
$$
If $\varphi\in C^1(X)$, then $\varphi$ is locally Lipschitz and
$\partial\varphi(x)=\{\varphi'(x)\}$. Similarly, if
$\varphi:X\to\mathbb{R}$ is continuous convex, then $\varphi$
is locally Lipschitz and the generalized subdifferential of $\varphi$,
coincides with the subdifferential in the sense of convex analysis,
given by
 $$
\partial_c\varphi(x)=\{x^\ast\in X^\ast : {{\langle {x^\ast,y-x}\rangle}}
\leq\varphi(y)-\varphi(x)\text{ for all } y\in X\}.
$$
We say that $x\in X$ is a \emph{critical point} of the locally
 Lipschitz function $\varphi:X\to\mathbb{R}$, if $0\in\partial\varphi(x)$. In this case, $c=\varphi(x)$ is a \emph{critical value} of $\varphi$. It is easy to check that, if $x\in X$ is a local extremum of $\varphi$ (i.e., $x\in X$ is a local minimum or a local maximum), then $x\in X$ is a critical point of $\varphi$.

Given a locally Lipschitz function $\varphi:X\to\mathbb{R}$,
we say that $\varphi$ satisfies the \emph{Palais-Smale condition}
at the level $c\in\mathbb{R}$ (the ``$PS_c$-condition" for short),
if every sequence $\{x_n\}_{n\geq 1}\subseteq X$ such that
$\varphi(x_n)\to c$ and $m(x_n)\to 0$ as $n\to+\infty$,
with $m(x_n)=\inf\{\|x^\ast\|: x^\ast\in\partial\varphi(x_n)\}$,
has a strongly convergent subsequence. We say that $\varphi$
satisfies the ``$PS$-condition", if it satisfies the $PS_c$-condition
at every level $c\in\mathbb{R}$.

For a locally Lipschitz function $\varphi:X\to\mathbb{R}$ and
$c\in\mathbb{R}$, we define the sets:
\begin{gather*}
\dot{\varphi}^c=\{x\in X: \varphi(x)<c\},\quad
\varphi^c=\{x\in X:\varphi(x)\leq c\},\\
K_c=\{x\in X: 0\in\partial\varphi(x),\, \varphi(x)=c\}.
\end{gather*}

The next theorem, due to Corvellec \cite{R8}, is a partial extension to a nonsmooth setting
of the so-called ``second deformation theorem"
(see \cite[p.23]{R6} and \cite[p.628]{R12})
 In fact, the result of Corvellec is formulated in a more general
framework, namely, for continuous (or even lower semicontinuous)
functions on a metric space, using the notion of weak slope
(see \cite{R8}, \cite[Section~1.3.5]{R12}, and \cite[Section~2.3]{R15}).
 For our purposes, the following particular version of the result
suffices.

\begin{theorem}\label{theo1}
If $X$ is a Banach space, $\varphi:X\to\mathbb{R}$ is
locally Lipschitz, $-\infty<a<b<+\infty$, $\varphi$ satisfies
the $PS_c$-condition for every $c\in[a,b)$, $\varphi$ has no critical
values in $[a,b)$, and $K_a$ is a finite set consisting of local
minima, then there exists a continuous deformation
$h:[0,1]\times\dot{\varphi}^b\to\dot{\varphi}^b$ such that:
 \begin{itemize}
  \item[(a)] $h(t,\cdot)\big|_{K_a}=id\big|_{K_a}$ for all $t\in[0,1]$;
  \item[(b)] $h(1,\dot{\varphi}^b)\subseteq \dot{\varphi}^a\cup K_a$;
  \item[(c)] $\varphi(h(t,x))\leq \varphi(x)$ for all $t\in[0,1]$ and all $x\in\dot{\varphi}^b$.
 \end{itemize}
\end{theorem}

In particular, this theorem implies that the set
$\dot{\varphi}^a\cup K_a$ is a weak deformation retract of
$\dot{\varphi}^b$. In the smooth version of the second deformation
theorem, the conclusion is that $\varphi^a$ is a strong deformation
retract of $\varphi^b\backslash K_b$.

\section{Existence Theorem}

We shall prove an existence theorem, for the following more general
version of problem~\eqref{pr:prob1},
\begin{equation}\label{pr:prob2}
\begin{gathered}
  -\Delta_{p}\, x(z)\in \partial j(z,x(z))+h(z)\quad\text{ a.e. in } Z,\\
  \frac{\partial x}{\partial n}=0 \quad\text{ on }\partial Z,
 \end{gathered}
\end{equation}
where $h\in L^\infty(Z)$ satisfies
\begin{equation}
\label{pr:eq3}
  \int_Z h(z)\,dz=0.
\end{equation}
To prove an existence theorem for \eqref{pr:prob2},
we will need the following hypotheses on the nonsmooth potential $j$:

\begin{itemize}
\item[(H1)] $j:Z\times\mathbb{R}\to\mathbb{R}$ is a function such
that $j(z,0)=0$ a.e. on $Z$ and
\begin{itemize}
  \item[(i)] for all $x\in\mathbb{R}$, $z\mapsto j(z,x)$ is measurable;
  \item[(ii)] for almost all $z\in Z$, $x\mapsto j(z,x)$ is
 locally Lipschitz;
 \item[(iii)] for almost all $z\in Z$, all $x\in\mathbb{R}$
 and all $u\in\partial j(z,x)$, we have
 $$
 |u|\leq a(z)+c |x|^{r-1}
 $$
 with $a\in L^\infty(Z)_+$, $c>0$ and $1\leq r < p^\ast$;
 \item[(iv)] there exists $\xi\in L^1(Z)_+$ such that
 $j(z,x)\leq \xi(z)$ for a.a. $z\in Z$ and all $x\in\mathbb{R}$;
 \item[(v)] there exists $c_0\in\mathbb{R}\backslash\{0\}$
 such that $\int_Z j(z,c_0)\,dz>0$.
\end{itemize}

\end{itemize}
Here, $p^\ast$ is the usual Sobolev critical exponent
$$
p^\ast=\begin{cases}
  \frac{Np}{N-p} & \text{if }N>p,\\
  +\infty & \text{if }N\leq p.
 \end{cases}
$$

\begin{example} \label{exa1} \rm
 The following potential function $j$ satisfies hypotheses (H1),
where for the sake of simplicity we drop the $z$-dependence,
$$
j(x)=\begin{cases}
   s(x) c\left(|x|^r-|x|^q\right) & \text{if }|x|\leq 1,\\
  s(x)\left(\frac{1}{x^2}-\ln|x|-1\right) & \text{if }|x|>1,
 \end{cases}
$$
where $s(x)\equiv 1$ or $s(x)=sign(x)+2$, $c>0$ and $1\leq r\leq p < q$.
In the latter case, where $s$ is nonconstant, $j$ has no symmetry properties.
Moreover, if $1<r$ and $c=\frac{3}{q-r}>0$,
then $j\in C^1(\mathbb{R})$.
\end{example}

\begin{example} \label{exa2}  \rm
The following potential function $j$ satisfies hypotheses (H1),
where again for the sake of simplicity we drop the $z$-dependence,
$$
j(x)=\begin{cases}
   |x|^r & \text{if }|x|\leq 1,\\
  \frac{1}{x^2}\ln(|x|)+1 & \text{if }|x|>1,
 \end{cases}
$$
where $1\leq r\leq p$. Note that the corresponding Euler
functional is noncoercive.
\end{example}

In what follows, we set
$$
\beta=\int_Z \limsup_{|x|\to\infty} j(z,x)\,dz.
$$
By hypothesis (H1)(iv), we have
$\beta\in\mathbb{R}\cup\{-\infty\}$.

It is worth pointing out, that hypotheses (H1) incorporate,
in our framework of analysis, problems which are strongly resonant
with respect to the principal eigenvalue $\lambda_0=0$ of
the Neumann $p$-Laplacian. For this reason, we do not expect
the $PS$-condition to be satisfied globally (i.e., at all levels).
This will be confirmed in the sequel (see Proposition \ref{prop4}).
But to be able to reach that result, we shall need some preparation.

So, we consider the following auxiliary Neumann problem:
\begin{equation}\label{pr:prob4}
\begin{gathered}
  -\Delta_{p} x(z)=h(z)\quad \text{a.e. in } Z,\\
  \frac{\partial x}{\partial n}=0 \quad \text{on }\partial Z,
 \end{gathered}
\end{equation}
with a $h\in L^\infty(Z)$ that satisfies \eqref{pr:eq3}.
We consider the direct sum decomposition
\begin{equation}
\label{pr:eq5}
   W^{1,p}(Z)=\mathbb{R}\oplus V\quad
\text{with }V=\big\{\hat{x}\in W^{1,p}(Z):
\int_Z \hat{x}(z)\,dz=0\big\}.
\end{equation}
Then, we have the following simple result.

\begin{proposition}\label{prop2}
Problem~\eqref{pr:prob4} has a unique solution
$\hat{x}_0\in C^1(\bar{Z})\cap V$.
\end{proposition}

\begin{proof}
 Let $ n: W^{1,p}(Z)\to\mathbb{R}$ be the $C^1$-functional
defined by
 $$
 \eta(z)=\frac{1}{p}\|Dx\|_p^p-\int_Z hx\,dz
$$
for all $x\in W^{1,p}(Z)$. Every $x\in W^{1,p}(Z)$
can be written in a unique way as
$$
x=\bar{x}+\hat{x}
$$
with $\bar{x}\in\mathbb{R}$ and $\hat{x}\in V$ (see \eqref{pr:eq5}).

Because of \eqref{pr:eq3}, we see that $\eta|_{\mathbb{R}}=0$.
Let $\hat{\eta}=\eta|_V$ (i.e., $\hat{\eta}$ is the restriction
of $\eta$ on $V$). By virtue of the Poincar\'e-Wirtinger inequality,
we see that $\hat{\eta}$ is coercive on $V$. Moreover,
it is clear that $\hat{\eta}$ is sequentially weakly lower semicontinuous on $V$. So, by the Weierstrass theorem,
we can find $\hat{x}_0\in V$ such that
$-\infty<\hat{m}_0=\hat{\eta}(\hat{x}_0)=\inf_V \hat{\eta}$,
which implies
\begin{equation}
\label{pr:eq6}
  \hat{\eta}'(\hat{x}_0)=0\quad\text{ in }V^\ast.
\end{equation}
Let $p_V: W^{1,p}(Z)\to V$ be the projection operator onto $V$.
It exists since $V$ is finite codimensional. Using the chain rule,
we have
\begin{equation}
\label{pr:eq7}
\eta'(x)=p^\ast_V\hat{\eta}'(p_V(x))=p^\ast_V\hat{\eta}'(\hat{x})
\quad \text{for all }x\in W^{1,p}(Z).
\end{equation}
In what follows, by ${{\langle {\cdot,\cdot}\rangle}}_V$ we denote
the duality brackets for the pair $(V^\ast,V)$.
Then for any $x,y\in W^{1,p}(Z)$, we have
 \begin{align*}
  {{\langle {\eta'(x),y}\rangle}}
&={{\langle {p^\ast_V\hat{\eta}'(p_V(x)),y}\rangle}}\quad\text{(see \eqref{pr:eq7})}\\
& ={{\langle {\hat{\eta}'(p_V(x)),p_V(y)}\rangle}}_{V}
 \end{align*}
which implies
\begin{equation}
\label{pr:eq8}
  {{\langle {\eta'(\hat{x}_0),y}\rangle}}
 ={{\langle {\eta'(\hat{x}_0),p_V(y)}\rangle}}_V=0.
\end{equation}
Because $y\in W^{1,p}(Z)$ was arbitrary, from \eqref{pr:eq8}
it follows that $\eta'(\hat{x}_0)=0$ in $ W^{1,p}(Z)$, so
\begin{equation}
\label{pr:eq9}
  A(\hat{x}_0)=h,
\end{equation}
where $A: W^{1,p}(Z)\to W^{1,p}(Z)^\ast$ is
the nonlinear operator defined by
\begin{equation}
\label{pr:eq10}
  {{\langle {A(x),y}\rangle}}
=\int_Z \|Dx\|_{\mathbb{R}^N}^{p-2}(Dx,Dy)_{\mathbb{R}^N}\,dz
\end{equation}
for all $x,y\in W^{1,p}(Z)$.
Evidently, $A$ is strictly monotone (strongly monotone, if $p\geq 2$)
and continuous.  From \eqref{pr:eq9}, using the nonlinear Green's
identity and the nonlinear regularity theory (e.g., see \cite{R12}),
we infer that $\hat{x}_0\in C^1(\bar{Z})$ and it solves
\eqref{pr:prob4}. Moreover, the strict monotonicity of
$A|_V$ implies that $\hat{x}_0\in V$ is unique in $V$.
\end{proof}

 From \cite[Proposition~12]{R14}, we have
the following useful fact about the nonlinear map
$A: W^{1,p}(Z)\to W^{1,p}(Z)^\ast$ defined by
\eqref{pr:eq10}.

\begin{proposition}\label{prop3}
If $A: W^{1,p}(Z)\to W^{1,p}(Z)^\ast$ is defined
by \eqref{pr:eq10}, then $A$ is maximal monotone and of type
$(S)_+$; i.e., if $x_n\stackrel{w}{\to} x$ in $ W^{1,p}(Z)$ and
$$
\limsup_{n\to+\infty} {{\langle {A(x_n),x_n-x}\rangle}}\leq 0,
$$
then $x_n\to x$ in $ W^{1,p}(Z)$.
\end{proposition}

The next proposition illustrates the failure of the global
$PS$-condition already mentioned earlier. So, let
$\varphi_1: W^{1,p}(Z)\to\mathbb{R}$ be the Euler functional
for \eqref{pr:prob2}, defined by
$$
\varphi_1(x)=\frac{1}{p}\|Dx\|_p^p-\int_Z j(z,x(z))\,dz
-\int_Z h(z)x(z)\,dz
$$
for all $x\in W^{1,p}(Z)$.  From \cite[p.83]{R7},
 we know that $\varphi_1$ is Lipschitz continuous on bounded sets,
hence it is locally Lipschitz.

\begin{proposition}\label{prop4}
If hypotheses {\rm (H1)} hold and $c<\eta(\hat{x}_0)-\beta$,
then $\varphi_1$ satisfies the $PS_c$-condition.
\end{proposition}

\begin{proof}
Consider a sequence $\{x_n\}_{n\geq1}\subseteq W^{1,p}(Z)$ such that
\begin{gather}
\label{pr:eq11}
  \varphi_1(x_n)\to c\text{ as }n\to+\infty\quad
\text{with }c<\eta(\hat{x}_0)-\beta,
\\
\label{pr:eq12}
 m_1(x_n)=\inf\{\|x^\ast\|: x^\ast\in\partial\varphi_1(x_n)\}\to 0
\quad\text{as }n\to\infty.
\end{gather}
Because $\partial\varphi_1(x_n)\subseteq W^{1,p}(Z)^\ast$ is
$w$-compact and the norm functional in a Banach space is weakly
lower semicontinuous, we can find $x^\ast_n\in\partial\varphi_1(x_n)$
such that $m_1(x_n)=\|x^\ast_n\|$. We know that
\begin{equation}
\label{pr:eq13}
   x^\ast_n=A(x_n)-u_n-h,
\end{equation}
with $u_n\in N(x_n)=\{u\in L^{r'}(Z): u(z)\in\partial j(z,x_n(z))
\text{ a.e. on }Z\}$ and $\frac{1}{r}+\frac{1}{r'}=1$
(see \cite[p. 83]{R7}). Also, we have $x_n=\bar{x}_n+\hat{x}_n$
with $\bar{x}_n\in\mathbb{R}$ and $\hat{x}_n\in V$.
 From \eqref{pr:eq11} and \eqref{pr:eq3}, we can find $M_1>0$ such that
\begin{equation} \label{pr:eq14}
\begin{aligned}
  M_1\geq \varphi_1(x_n)&=\frac{1}{p}\|D\hat{x}_n\|_p^p
 -\int_Zj(z,x(z))\,dz-\int_Z h(z)\hat{x}_n(z)\,dz \\
   &\geq\frac{1}{p}\|D\hat{x}_n\|_p^p-\|\xi\|_1-c_1\|D\hat{x}_n\|_p
\end{aligned}
\end{equation}
for some $c_1>0$ and all $n\geq 1$. Here, we have used the
Poincar\'e-Wirtinger inequality and hypothesis (H1)(iv).
 From \eqref{pr:eq14} and the Poincar\'e-Wirtinger inequality again,
we infer that
\begin{equation}
\label{pr:eq15}
  \{\hat{x}_n\}_{n\geq 1}\subseteq W^{1,p}(Z)\text{ is bounded}.
\end{equation}
Because of \eqref{pr:eq15} and by passing to a suitable subsequence
if necessary, we may assume that
\begin{equation}
\label{pr:eq16}
  |\hat{x}_n(z)|\leq k(z)
\end{equation}
for a.a. $z\in Z$, all $n\geq 1$, with $k\in L^r(Z)_+$.

Suppose that $\{x_n\}_{n\geq 1}\subseteq W^{1,p}(Z)$ is not bounded.
We may assume that $\|x_n\|\to\infty$ and so, because
of \eqref{pr:eq15}, we must have $|\bar{x}_n|\to\infty$. Then
 $$
|x_n(z)|\geq |\bar{x}_n|-|\hat{x}(z)|\geq |\bar{x}_n|-k(z)
 $$
for a.a. $z\in Z$ (see \eqref{pr:eq16}), hence
 $$
|x_n(z)|\to+\infty\quad\text{ as }\quad n\to\infty
$$
for a.a. $z\in Z$.
 From \eqref{pr:eq14}, we see that
\[
  M_1\geq \varphi_1(x_n)=\eta(\hat{x}_n)-\int_Z j(z,x_n(z))\,dz
    \geq\eta(\hat{x}_0)-\int_Zj(z,x_n(z))\,dz,
\]
(see the proof of Proposition \ref{prop2}).  Passing to the limit as
$n\to+\infty$ and using \eqref{pr:eq11}, we obtain
 \begin{align*}
  M_1&\geq c\geq \eta(\hat{x}_0)-\limsup_{n\to\infty}\int_Z j(z,x_n(z))\,dz\\
     &\geq \eta(\hat{x}_0)-\int_Z \limsup_{n\to\infty}j(z,x_n(z))\,dz
\quad \text{ (by Fatou's lemma, see (H1)(iv))}\\
     &= \eta(\hat{x}_0)-\beta,
 \end{align*}
which contradicts the choice of $c$ (see \eqref{pr:eq11}).
This proves that $\{x_n\}_{n\geq 1}\subseteq W^{1,p}(Z)$ is bounded.
 Hence, we may assume that
$$
x_n\stackrel{w}{\to} w\text{ in } W^{1,p}(Z)\quad\text{and}\quad
x_n\to x\text{ in }L^r(Z).
$$
 From \eqref{pr:eq12} and \eqref{pr:eq13}, we have, with
$\epsilon_n\downarrow 0$,
\begin{equation}
\label{pr:eq17}
   \big|{{\langle {A(x_n),x_n-x}\rangle}}-\int_Z u_n(x_n-x)\,dz
-\int_Z h(x_n-x)\,dz\big|\leq \epsilon_n\|x_n-x\|.
\end{equation}
Clearly
 $$
\int_Z u_n(x_n-x)\,dz\to 0, \quad
\int_Z h(x_n-x)\,dz\to 0.
$$
So, if in \eqref{pr:eq17} we pass to the limit as $n\to\infty$,
then we obtain
 $$
\lim_{n\to\infty}{{\langle {A(x_n),x_n-x}\rangle}}=0.
$$
thus by virtue of Proposition \ref{prop3}, we have that
$x_n\to x$ in $ W^{1,p}(Z)$. This proves that $\varphi$ satisfies
the $PS_c$-condition for all $c<\eta(\hat{x}_0)-\beta$.
\end{proof}

Now we are ready for the existence result concerning
Problem~\eqref{pr:prob2}.

\begin{theorem}\label{theo5}
If hypotheses {\rm (H1)} hold and $\beta<\int_Z j(z,\hat{x}_0(z))\,dz$,
then \eqref{pr:prob2} admits a nontrivial solution $y_0\in C^1(\bar{Z})$.
\end{theorem}

\begin{proof}
Recall that
$$
\varphi_1(x)=\eta(\hat{x})-\int_Z j(z,x(z))\,dz
$$
for all $x\in W^{1,p}(Z)$ ($\hat{x}=p_V(x)$).
 From the proof of Proposition \ref{prop2}, we know that $\hat{x}_0\in V$
 is a minimizer of the functional $\eta$.
Therefore,
\[
  \varphi_1(x)\geq\eta(\hat{x}_0)-\int_Z j(z,x(z))\,dz
   \geq\eta(\hat{x}_0)-\|\xi\|_1
\]
for all $x\in W^{1,p}(Z)$ (see hypothesis (H1)(iv)).
Hence $\varphi_1$ is bounded below and so
$-\infty<\hat{m}_1=\inf\left\{\varphi_1\in W^{1,p}(Z)\right\}$. Also
\[
   -\infty<\hat{m}_1\leq\varphi_1(\hat{x}_0)
=\eta(\hat{x}_0)-\int_Z j(z,x_0(z))\,dz,
<\eta(x_0)-\beta
\]
by hypothesis.
Then, by virtue of Proposition \ref{prop4}, $\varphi_1$ satisfies the
$PS_{\hat{m}_1}$-condition. So, from   \cite[p.144]{R11},
we infer that there exists $y_0\in W^{1,p}(Z)$ such that
 $$
\varphi_1(y_0)=\hat{m}_1
=\inf\{\varphi_1(x) : x\in W^{1,p}(Z)\}
\leq\varphi_1(c_0)=-\int_Zj(z,c_0)\,dz<0=\varphi_1(0)
$$
(see hypothesis (H1)(v)). It follows that $y_0\neq0$. Also
$\varphi'(y_0)=0$, which implies
\begin{equation}
\label{pr:eq18}
  A(y_0)=u_0+h\quad\text{with }u_0\in N(y_0).
\end{equation}
 From \eqref{pr:eq18} as before, using the nonlinear
Green's identity and nonlinear regularity theory, we infer
that $y_0\in C^1(\bar{Z})$ and solves \eqref{pr:prob2}.
\end{proof}


We remark that in Example \ref{exa2}, we have $j(x)>0$ for $x\neq0$,
so the hypotheses of Theorem~\ref{theo5} are satisfied
for $h\equiv0$.

\section{Multiplicity Theorem}

In this section, we return to \eqref{pr:prob1}, where $h\equiv 0$,
hence $\hat{x}_0=0$ and $\eta(x_0)=0$.
To prove a multiplicity theorem for \eqref{pr:prob1},
we need to strengthen the hypotheses on the nonsmooth potential $j$
as follows:

\begin{itemize}
\item[(H2)]  $j:Z\times\mathbb{R}^N\to\mathbb{R}$ is a function
such that $j(z,0)\to0$ a.e. on $Z$, and satisfies hypotheses
(H1)(i)--(v) and
\begin{itemize}
   \item[(vi)] $\beta=\int_Z \limsup_{|x|\to\infty}j(z,x)\,dz<0$
 and there exists $\eta\in L^\infty(Z)_+$, $\eta\neq 0$ such that
 $$ \eta(z)\leq \liminf_{x\to 0}\frac{j(z,x)}{|x|^p} $$
 uniformly for a.a. $z\in Z$;
\item[(vii)] $j(z,x)\leq \frac{\lambda_1}{p}|x|^p$ for a.a. $z\in Z$
and all $x\in\mathbb{R}$ and with $\lambda_1>0$ being the first
nonzero eigenvalue of $(-\Delta_{p}\,, W^{1,p}(Z))$ (i.e.,
the second eigenvalue).
\end{itemize}
\end{itemize}
The Euler functional $\varphi: W^{1,p}(Z)\to\mathbb{R}$ for
\eqref{pr:prob1} is defined by
 $$
\varphi(x)=\frac{1}{p}\|Dx\|_p^p-\int_Z j(z,x(z))\,dz
$$
for all $x\in W^{1,p}(Z)$. We know that $\varphi$ is Lipschitz
continuous on bounded sets, hence it is locally Lipschitz
(see \cite{R7}, p.83).

\begin{theorem}\label{theo6}
If hypotheses {\rm (H2)} hold, then \eqref{pr:prob1} has at least
two nontrivial solutions $y_0, v_0\in C^1(\bar{Z})$.
\end{theorem}

\begin{proof}
As we already mentioned, since $h\equiv0$, we have $\hat{x}_0=0$
and so $j(z,\hat{x}_0(z))=0$ a.e. on $Z$. Then hypothesis (H2)
permits the use of Theorem \ref{theo5}, which gives a nontrivial
solution $y_0\in C^1(\bar{Z})$ for \eqref{pr:prob1}.

Hypothesis (H2)(vi) implies that, for $\epsilon>0$, we
can find $\delta=\delta(\epsilon)>0$ such that
\begin{equation}
\label{pr:eq19}
  j(z,x)\geq \left(\eta(z)-\epsilon\right)|x|^p
\end{equation}
for a.a. $Z$ and all $|x|\leq\delta$.
If $\xi\in\mathbb{R}$ with $0<|\xi|\leq\delta$, then
\begin{equation}
\label{pr:eq20}
  \varphi(\xi)=-\int_Z j(z,\xi)\,dz
\leq\int_Z (\epsilon-\eta(z))dz\,|\xi|^p,
\end{equation}
(see \eqref{pr:eq19}).
If we choose $0<\epsilon<\frac{1}{|Z|_N}\int_Z\eta(z)\,dz$ (by $|\cdot|_N$ we denote the Lebesgue measure on $\mathbb{R}^N$), then from \eqref{pr:eq20}, we infer that $\varphi(\xi)<0$, so
\begin{equation}
\label{pr:eq21}
   \mu_r=\max_{\partial B_r\cap\mathbb{R}}\varphi<0
\end{equation}
for all $0<r\leq\delta$. Let
 $$
C(p)=\big\{x\in W^{1,p}(Z): \int_Z |x(z)|^{p-2}x(z)\,dz=0\big\}.
$$
Then, for every $x\in C(p)$, we have
 \begin{align*}
  \varphi(x)
&= \frac{1}{p}\|Dx\|_p^p-\int_Z j(z,x(z))\,dz\\
&\geq \frac{1}{p}\|Dx\|_p^p-\frac{\lambda_1}{p}
\|Dx\|_p^p-\frac{\lambda_1}{p}\|x\|_p^p \geq 0,
 \end{align*}
since $x\in C(p)$ (see \cite{R14}) and by using (H2)(vii). Hence,
\begin{equation}
\label{pr:eq22}
   \inf_{C(p)}\varphi = 0.
\end{equation}

We introduce the set
 $$
\Gamma_r=\{\gamma\in C(\bar{B}_r\cap\mathbb{R}, W^{1,p}(Z)):
\gamma|_{\partial B_r\cap\mathbb{R}}=id|_{\partial B_r\cap\mathbb{R}}\},
\quad 0<r\leq\delta,
$$
and define the minimax quantity
\begin{equation}
\label{pr:eq23}
  \hat{c}_r=\inf_{\gamma\in\Gamma_r}\max_{x\in\bar{B}_r
\cap\mathbb{R}}\varphi(\gamma(x)).
\end{equation}
Consider the map $\sigma: W^{1,p}(Z)\to\mathbb{R}$ defined by
$$
\sigma(u)=\int_Z |u|^{p-2}u\,dz.
$$
 Evidently, $\sigma$ is continuous and for
$r,-r\in\gamma(\bar{B}_r\cap\mathbb{R})$, $\gamma\in\Gamma_r$, we have
 $$
\sigma(-r)<0<\sigma(r).
$$
So, by Bolzano's theorem, we can find
$u\in\gamma(\bar{B}_r\cap\mathbb{R})$
(recall that the set is connected), such that
$$
\sigma(u)=\int_Z|u|^{p-2}u\,dz=0,
$$
hence $u\in C(p)$.
Therefore, we have $u\in\gamma(\bar{B}_r\cap\mathbb{R})\cap C(p)$,
which implies
\begin{equation}
\label{pr:eq24}
  \hat{c}_r\geq 0
\end{equation}
(see \eqref{pr:eq22} and \eqref{pr:eq23}).
Suppose that $\{0,y_0\}$ are the only critical points of $\varphi$.
 Set
 $$
b=0=\varphi(0)\text{ and }a=\hat{m}=\inf\varphi=\varphi(y_0)
$$
(see the proof of Theorem \ref{theo5}). We know that $a<b$
(also from Theorem \ref{theo5}).
Moreover, by virtue of Proposition \ref{prop4}, $\varphi$ satisfies the
$PS_c$-condition for every $c\in(a,b)$ (recall that
$\eta(\hat{x}_0)=0$ and by hypothesis (H2)(vi), $\beta<0$).
Also $K_a=\{y_0\}$ and $y_0$ is a minimizer of $\varphi$.
Finally, by hypothesis, $\varphi$ has no critical values in $(a,b)$.
Therefore, we can apply Theorem \ref{theo1} and obtain a continuous deformation
$h:{0,1}\times\dot{\varphi}^b\to\dot{\varphi}^b$ such that
$h(t,\cdot)|_{K_a}=id|_{K_a}$ for all $t\in[0,1]$ and
\begin{equation}
\label{pr:eq25}
   h(1,\dot{\varphi}^b)\subseteq\dot{\varphi}^a\cup K_a=\{y_0\}
\end{equation}
(since $\dot{\varphi}^a=\emptyset$) and
\begin{equation}
\label{pr:eq26}
  \varphi(h(t,x))\leq\varphi(x)\quad
\text{ for all }t\in[0,1]\text{ and all }x\in\dot{\varphi}^b.
\end{equation}
We consider the map $\gamma_0:\bar{B}_r\cap\mathbb{R}\to W^{1,p}(Z)$
defined by
\begin{equation}
\label{pr:eq27}
  \gamma_0(x)=\begin{cases}
    y_0 & \text{if }\|x\|\leq\frac{r}{2},\\
    h\big(\frac{2(r-\|x\|)}{r},\frac{rx}{\|x\|}\big)
 & \text{if }\|x\|>\frac{r}{2},
  \end{cases}
\end{equation}
for all $x\in\bar{B}_r\cap\mathbb{R}$. If $\|x\|=\frac{r}{2}$,
then $h\big(\frac{2(r-\|x\|)}{r},\frac{rx}{\|x\|}\big)=h(1,2x)=y_0$
(see \eqref{pr:eq25} and \eqref{pr:eq21}). Hence, it follows
that $\gamma_0$ is continuous.

If $\|x\|=r$, then $\gamma_0(x)=h(0,x)=x$ (since $h$ is a deformation).
Therefore, $\gamma\in\Gamma$. Moreover,
from \eqref{pr:eq27} and \eqref{pr:eq26} and since
$\varphi(y_0)=a\leq\mu_r<0$ (see \eqref{pr:eq21}), we have
 $$
\varphi(\gamma_0(x))\leq\mu_r<0 \quad
\text{ for all }x\in\bar{B}_r\cap\mathbb{R},
$$
which implies
\begin{equation}
\label{pr:eq28}
  \hat{c}_r<0.
\end{equation}
(see \eqref{pr:eq23} and recall $\gamma_0\in\Gamma$).
Comparing \eqref{pr:eq21} and \eqref{pr:eq28}, we reach a contradiction.
This means that there is one more critical point
$v_0\not\in\{0,y_0\}$ of $\varphi$. Then, as before,
we check that $v_0\in W^{1,p}(Z)$ is a solution for
\eqref{pr:prob1} and nonlinear regularity theory
implies that $v_0\in C^1(\bar{Z})$.
\end{proof}


\begin{thebibliography}{00}

\bibitem{R1} G. Anello;
\emph{Existence of infinitely many weak solutions for a Neumann problem},
 Nonlin. Anal. {\bf 57}(2004), 199--209.

\bibitem{R2} G. Barletta and N. S. Papageorgiou;
\emph{A multiplicity theorem for the Neumann $p$-Laplacian with an
asymmetric nonsmooth potential}, J. Global Optim.
{\bf 39}(2007), 365--392.

\bibitem{R3} G. Bonanno and P. Candito;
\emph{Three solutions to a Neumann problem for the elliptic equations
involving the $p$-Laplacian}, Archiv der Math {\bf 80}(2003), 424--429.

\bibitem{R4} F. Cammaroto, A. Chinni, and B. DiBella;
\emph{Some multiplicity results for quasilinear Neumann problems},
Archiv der Math {\bf 86}(2006), 154--162.

\bibitem{R5} S. Carl, V. K. Le, and D. Motreanu;
\emph{Nonsmooth Variational Problems and their Inequalities:
Comparison Principles and Applications}, Springer, New York (2007).

\bibitem{R6} K.-C. Chang;
\emph{Infinite Dimensional Morse Theory and Multiple Solution Problems},
Birk\-h\"auser, Boston (1993).

\bibitem{R7} F. H. Clarke;
\emph{Optimization and Nonsmooth Analysis},
Wiley Interscience, New York (1983).

\bibitem{R8} J.-N. Corvellec;
\emph{On the second deformation lemma},
Topol. Meth. Nonlin. Anal. {\bf 17}(2001), 55--66.

\bibitem{R9} F. Faraci;
\emph{Multiplicity results for a Neumann problem involving
the p-Laplacian}, J. Math. Anal. Appl. {\bf 277}(2003), 180--188.

\bibitem{R10} M. Filippakis, L. Gasinski, and N. S. Papageorgiou;
\emph{Multiplicity results for nonlinear Neumann problems},
Canad. J. Math. {\bf 58}(2006), 64--92.

\bibitem{R11} L. Gasinski and N. S. Papageorgiou;
 \emph{Nonsmooth Critical Point Theory and Nonlinear Boundary
Value Problems}, Chapman \& Hall / CRC, Boca Raton (2005).

\bibitem{R12} L. Gasinski and N.S. Papageorgiou;
 \emph{Nonlinear Analysis}, Chapman \& Hall / CRC, Boca Raton (2006).

\bibitem{R13} S. Marano and D. Motreanu;
\emph{Infinitely many critical points of nondifferentiable functions
and applications to a Neumann-type problem involving the p-Laplacian},
J. Diff. Eqns {\bf 182}(2002), 108--120.

\bibitem{R14} D. Motreanu and N. S. Papageorgiou;
\emph{Existence and multiplicity for Neumann problems},
J. Diff. Eqns {\bf 232}(2007), 1--35.

\bibitem{R15} D. Motreanu and V. Radulescu;
 \emph{Variational and Nonvariational Methods in Nonlinear Analysis
and Boundary Value Problems}, Kluwer, Dordrecht (2003).

\bibitem{R16} B. Ricceri;
\emph{On a three critical points theorem},
Archiv der Math {\bf 75}(2000), 220--226.

\bibitem{R17} B. Ricceri;
\emph{Infinitely many solutions of the Neumann problem
for elliptic equations involving the p-Laplacian},
Bull London Math Soc {\bf 33}(2001), 331--340.

\bibitem{R18} X. Wu and K. K. Tan;
\emph{On the existence and multiplicty of solutions of Neumann
boundary problems for quasilinear elliptic equations},
Nonlin. Anal. {\bf 65}(2006), 1334--1347.

\end{thebibliography}

\end{document}