\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}_0(\Omega)}}
\def\xseg{\bar{x}}
\def\useg{\bar{u}}
\def\d{\mathop{\rm dist}}
\def\d{{\rm diam(\Omega)}}
%\date{}
\title[Mountain Pass Solutions]{Mountain Pass Solutions for Nonlocal
Equations}
%
\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{$p$-Laplacian equations; Kirchhoff-type problems; multiple
solutions; mountain pass theorem.}
%
%\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}
This work is devoted to study the existence of solutions to nonlocal
equations involving the $p$-Laplacian.
More precisely, we prove the existence of at least one nontrivial weak
solution, and under additional assumptions, the existence
of infinitely many weak solutions. In order to apply mountain pass
results, we require rather general assumptions on
on the local operator. Finally, a concrete application is presented.
\end{abstract}
\maketitle
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\section{Introduction}
The purpose of this paper is to investigate existence and multiplicity
properties for the following class of nonlocal Dirichlet problems:
\begin{equation} \tag{$D_{M,f}^p$} \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 \ \Omega
\\ {\,u|_{\partial \Omega}=0.}\\
\end{array}
\right.
\end{equation}

Here and in the sequel, $\Omega$ is a bounded domain in
$({\R}^{N},|\cdot|)$ with smooth
boundary $\partial \Omega$, $p>1$, and $\Delta_p u:=\div(|\nabla
u|^{p-2}\nabla u)$ denotes the usual $p$--Laplace operator. Further,
$f:\bar\Omega\times \erre\rightarrow \erre$ and $M:[0,+\infty)\to
[0,+\infty)$ are suitable continuous maps.\par
Nonlocal boundary value problems like problem \eqref{Nostro} model
several physical
and biological systems where $u$ describes a process which depend on
the average of itself, as for
example, the population density. We refer the reader, for instance, to Alves and Corr\^{e}a \cite{8}, Alves and Figueiredo \cite{alves}, Andrade and Ma \cite{9}, Chipot and Lovat \cite{10}, Chipot and Rodrigues \cite{11}, and
Vasconcellos \cite{12}.

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.

 Moreover, 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}. This equation 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 Arosio and Panizzi \cite{AP}, Autuori, Colasuonno and Pucci \cite{aupp2}, Autuori, Pucci and Salvatori \cite{aupp1},  Counsin, Frota, Lar'kin and
Medeiros \cite{CFL}, and D'Ancona and Spagnolo \cite{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 of \eqref{Nostro} (for $p=2$) 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).
We also refer to Colasuonno and Pucci \cite{copp}, which covers for the first time in the literature the degenerate case of
the Kirchhoff equation for the existence of solutions in the elliptic
case. In other words, paper \cite{copp} includes the degenerate setting that corresponds to $M(0)=0$.
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.

In our context, problem \eqref{Nostro} is studied exploiting classical
variational methods. More precisely we apply the celebrated mountain
pass theorem (briefly MPT) to this kind of equations motivated by the
current literature where the MPT has been intensively applied to find
solutions to
quasilinear elliptic equations of $p$-Laplacian type.\par
\indent Technically, this approach is realizable checking that the
associated energy functional verifies the usual compactness
Palais-Smale condition. Motivated by this fact, we
suppose some restrictions on the behavior of the continuous function
$M$.\par
 More precisely, we assume that
there exists a constant $m_0$ such that:
\begin{itemize}
\item[$(C_M^1)$] $0< m_0\leq M(t),\,\,\,\forall\, t\in [0,+\infty)$.
\end{itemize}
\indent In addition to the above hypotheses, we require that:
 \begin{itemize}
 \item[$(C_M^2)$] \textit{There exists} $t_0\geq 0$ \textit{such that}
 $$
\displaystyle \displaystyle \widehat{M}(t)\geq t [M(t)]^{p-1},
$$
\textit{for every} $t\in [t_0,+\infty)$, \textit{where} $\displaystyle
\widehat{M}(t):=\int_0^t [M(s)]^{p-1}ds.
$
\end{itemize}

\indent As proved in Lemma \ref{lemmaSer}, in our setting, the
potential $\widehat{M}$
has a sublinear growth. This fact permit to overcome some technical
difficulties treating problem \eqref{Nostro} due to the presence of
the nonlocal term $$M\Big(\displaystyle\int_\Omega|\nabla
u(x)|^pdx\Big).$$

 \indent Specifically, studying $p$-superlinear problems as in the
classical literature for the $p$-Laplace operator, we
are conduced to assume $M$ nonincreasing.\par
\indent In the first part of this manuscript, under the previous
assumptions, requiring conditions on the nonlinear part $f$ (among
others the Ambrosetti-Rabinowitz relation) we prove the existence of
at
least one nontrivial weak solution to problem \eqref{Nostro}; see
Theorem \ref{Esistenza}. This result is related to \cite[Theorem
3]{ACM} where the authors studied problem $(D_{M,f}^2)$.\par
 We just observe that in our context, on the contrary of the cited
result, we do not require
$$
\lim_{t\rightarrow 0}\frac{f(x,t)}{t}=0,
$$
uniformly with respect to $x\in\bar\Omega$; see condition
\eqref{atzero} in Theorem \ref{Esistenza}. Further, we assume that
condition $(C_M^2)$ holds true for every $t\geq t_0,$ where $t_0$ can
be different from zero.\par
 The second part of this work is devoted to a multiplicity result. In
presence of symmetries ($f$ is assumed to be odd), the existence of
infinitely many weak solutions to problem \eqref{Nostro} is achieved
exploiting the ${\mathbb{Z}}_2$-symmetric version of the MPT; see
Theorem \ref{ZMPT}.\par
\indent The plan of the paper is as follows. Section 2 and 3 are
devoted to our abstract framework and preliminaries. Successively, in
Sections 4 and 5 we give the main results; see Theorems
\ref{Esistenza} and \ref{Theorem2}. Finally, some concrete examples of
applications are presented in the last section (see Example
\ref{esempio0} and \ref{esempio}).

\section{Abstract Framework}\label{section2}
 Let $W^{1,p}_0(\Omega)$ $(p>1)$ be the
usual Sobolev space, equipped with the norm
$$
\|u\|:=\Big(\int_{\Omega}|\nabla
u(x)|^{p}dx\Big)^{1/p}.
$$

\indent Further, let $W^{-1,p'}(\Omega)$, with $1/p+1/p'=1$, be its
topological
dual and denote the duality brackets for the pair
$(W^{-1,p'}(\Omega),W^{1,p}_0(\Omega))$ by
$\langle\cdot,\cdot\rangle$.
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\|\,,
\end{equation}
\noindent for every $u\in \H$. Moreover, when $p\geq N$, this
inequality holds for any $q\in [1,+\infty[$, since $p^*=+\infty$.\par
For the sake of completeness, we recall that a $C^1$-functional
$\phi:X\to\R$, where $X$ is a real Banach
space with topological dual $X^*$, satisfies the \textit{Palais-Smale
condition at level} $\mu\in\R$, (briefly $\textrm{(PS)}_{\mu}$) when
\begin{itemize}
\item[$\textrm{(PS)}_{\mu}$] {\it Every sequence $\{u_n\}$ in $X$ such
that
$$
\phi(u_n)\to \mu, \quad{\rm and}\quad \|\phi'(u_n)\|_{X^*}\to0,
$$
as $n\rightarrow \infty$, possesses a convergent subsequence.}
\end{itemize}
We say that $\phi$ satisfies the \textit{Palais-Smale condition} (in
short $\textrm{(PS)}$) if $\textrm{(PS)}_{\mu}$ holds for every
$\mu\in \erre$.\par
With the above notations, our main tools are the classical MPT and its
${\mathbb{Z}}_2$-symmetric version recalled respectively in the next
Theorems \ref{MPT} and \ref{ZMPT}. As pointed out by Brezis and
Browder \cite{brebro}, the mountain pass theorem ``{\it extends ideas
already present in Poincar\'e and Birkhoff}".

\begin{theorem}\label{MPT} $($\cite{AR} and Theorem 2.2 in \cite[p.
7]{Ra}$)$
Let $(X,\|\cdot\|)$ be a real Banach space and let $\phi\in
C^1(X;\erre)$ such that $\phi(0_X)=0$ and satisfying the $(\rm PS)$
condition. Suppose that$:$
 \begin{itemize}
 \item[$(I_1)$] There exist constant $\rho,\alpha>0$ such that
$\phi(u)\geq \alpha$ if $\|u\|=\rho$.
  \item[$(I_2)$] There exists $e\in X$ with $\|e\|> \rho$ such that
$\phi(e)\leq 0$.
 \end{itemize}
 Then $\phi$ possesses a critical value $c\geq \alpha$, which can be
characterized as
 $$
 c:=\inf_{\gamma\in \Gamma}\max_{u\in \gamma([0,1])}\phi(u),
 $$
 where
 $$
 \Gamma:=\{\gamma\in C([0,1];X): \gamma(0)=0\,\,\wedge\,\,
\gamma(1)=e\}.
 $$
\end{theorem}

\begin{theorem}\label{ZMPT} $($\cite[p. 5]{Ra}$)$
Let $X$ be an infinite dimensional real Banach space and let $\phi\in
C^1(X;\erre)$ be even, satisfying the $(\rm PS)$ condition, and
$\phi(0_X)=0$. Suppose that condition $(I_1)$ holds in addition to the
following$:$
 \begin{itemize}
  \item[$(I_2')$] For each finite dimensional subspace $X_1\subset X$,
the set
  $$
  \{u\in X_1:\phi(u)\geq 0\}
  $$
  is bounded in $X$.
 \end{itemize}
 Then $\phi$ has an unbounded sequence of critical values.
\end{theorem}

\smallskip
 \noindent We cite the monograph by Krist\'{a}ly, R\u{a}dulescu and Varga \cite{k2} as general reference on
variational setting adopted in this paper.

\section{Notations and Preliminaries}
Set $\Phi: W^{1,p}_0(\Omega)\rightarrow \erre$ 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}
for all $u\in W^{1,p}_0(\Omega)$, where we recall that
\begin{gather*}
\widehat{M}(t):=\int_0^t [M(s)]^{p-1}ds,
\end{gather*}
for every $t\in [0,+\infty)$.\par
 Then the
Fr\'echet
derivative  $\Phi':W^{1,p}_0(\Omega)\rightarrow W^{-1,p'}(\Omega)$ is
given 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 every $u,v\in W^{1,p}_0(\Omega)$.\par
\indent Moreover, it is well-known that the operator $A:W^{1,p}_0(\Omega)\rightarrow W^{-1,p'}(\Omega)$ 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*}
for every $u,v\in W^{1,p}_0(\Omega)$, satisfies the $(S_+)$ property.\par
 This means that for every sequence $\{u_n\}\subset W^{1,p}_0(\Omega)$ such that $u_n\rightharpoonup u$ (weakly) in $W^{1,p}_0(\Omega)$ and
\begin{equation}\label{S}
\limsup_{n\to\infty} \langle A(u_n),u_n-u\rangle \leq0,
\end{equation}
 then $u_n\rightarrow u$ (strongly) in $W^{1,p}_0(\Omega)$.\par

\section{Existence Result}
Put
$$
\lambda_{1,p}(\Omega):=\inf\left\{\frac{\displaystyle\int_{\Omega}|\nabla
u(x)|^{p}dx}{\displaystyle\int_{\Omega}|
u(x)|^{p}dx}:u\in W^{1,p}_0(\Omega)\setminus\{0\}\right\}.
$$
Our first result extends a pioneering property established by
Ambrosetti and Rabinowitz \cite{AR} under similar assumptions,
provided that $M\equiv 1$ and $p=2$.
\begin{theorem}\label{Esistenza}
Let us assume that $M:[0,+\infty)\to [0,+\infty)$ is a continuous map
such that condition $(C_M^1)$ and $(C_M^2)$ hold. Further, require
that $f:\bar\Omega\times\erre\rightarrow \erre$ is a continuous
function that verifies$:$
\begin{itemize}
\item[$\rm h_1)$] The subcritical growth condition$:$
$$
 |f(x,t)|\leq c(1+|t|^{q-1}),\qquad\forall\, x\in
\bar\Omega,\,\forall\,t\in \erre
$$
where $c>0$ and $p<q<p^{*}$$;$
\item[$\rm h_2)$] The Ambrosetti-Rabinowitz $($briefly $(\rm AR)$$)$
condition: the functional $F(x,\xi):=\displaystyle\int_0^\xi f(x,t)dt$
is $\theta$-superhomogeneous at infinity, that is, there exists
$t^{\star}>0$ such that
$$
0<\theta F(x,\xi)\leq f(x,\xi)\xi\qquad\forall\,x\in
\bar\Omega,\,\forall\, |\xi|\geq t_{\star}
$$
    where $\theta>p$.
\end{itemize}
We assume that
\begin{equation}\label{atzero}
 \displaystyle\limsup_{t\rightarrow 0}\frac{f(x,t)}{|t|^{p-2}t}\leq
\lambda,
 \end{equation}
 uniformly for $x\in\bar \Omega$, where
 $$
 \lambda< m_0^{p-1}\lambda_{1,p}(\Omega).
 $$
Then the Dirichlet problem
\begin{equation} \tag{$D_{M,f}^p$} \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
\\ {\,u|_{\partial \Omega}=0,}\\
\end{array}
\right.
\end{equation}
\noindent has at least one nontrivial weak solution in
$W^{1,p}_0(\Omega)$.
\end{theorem}

For the sake of completeness we recall that a \textit{weak solution}
of problem \eqref{Nostro} is a function $u\in
W^{1,p}_0(\Omega)\setminus\{0\}$ such that
\begin{eqnarray*}
\langle\Phi'(u),v\rangle =\int_\Omega f(x,u(x))v(x)dx,
\end{eqnarray*}
\noindent for every $v\in W^{1,p}_0(\Omega)$.\par
 Further we note that, if $f:\bar\Omega\times \erre\rightarrow \erre$
is locally Lipschitz, then every weak solution of problem
\eqref{Nostro} is  classical.

\subsection{Some 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 $(C_M^2)$ hold. Then there are
constants $m_1>0$ and $m_2\geq 0$ 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 $t_1>t_0$, where $t_0$ appears in hypothesis $(C_M^2)$. By our
assumptions we easily have
 $$
 \frac{[M(t)]^{p-1}}{\widehat{M}(t)}\leq \frac{1}{t},
 $$
 for every $t\in ]t_1,+\infty)$. Integrating the above relation, 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 $t\in]t_1,+\infty)$. Thus
 $$
 \widehat{M}(t)\leq \frac{\widehat{M}(t_1)}{t_1}t,
 $$
 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}
one has the following inequalities:
 \begin{itemize}
 \item[$(\widehat{C}_M)$]
$\displaystyle\frac{m_0^{p-1}}{p}\|u\|^{p}\leq \Phi(u)\leq
{\frac{m_1}{p}\|u\|^{p} +\frac{m_2}{p}},$
 \end{itemize}
 for every $u\in W^{1,p}_0(\Omega)$.
  \begin{remark}\rm{
 We observe that the $(\rm AR)$ condition implies that
 $$
 F(x,\tau \xi)\geq F(x,\xi)\tau^\theta,
 $$
 for every $x\in \bar\Omega$, $|\xi|\geq t_{\star}$ and $\tau\geq 1$.
Indeed, for $\tau=1$, clearly the equality holds. Otherwise, fix
$|\xi|\geq t_{\star}$ and define $g(x,\tau):=F(x,\tau\xi)$, for every
$x\in\Bar\Omega$ and $\tau\in ]1,+\infty)$. By $(\rm AR)$ condition
it follows that
 $$
 \frac{g'(x,\tau)}{g(x,\tau)}\geq \frac{\theta}{\tau},
 $$
 for every $x\in \bar\Omega$ and $\tau> 1$. Integrating in $]1,\tau]$
it follows that
 $$
\int_{1}^{\tau}\frac{g'(x,s)}{g(x,s)}ds= \log
\frac{g(x,\tau)}{g(x,1)}\geq \log \tau^{\theta}.
 $$
In conclusion, since for every $x\in \bar\Omega$, $|\xi|\geq
t_{\star}$ and $\tau> 1$ one has
 \begin{eqnarray*}%\label{A2}
F(x,\tau\xi) &=:& g(x,\tau)\\\nonumber
&\geq& g(x,1)\tau^{\theta}\\\nonumber
&=& F(x,\xi)\tau^\theta.\nonumber
\end{eqnarray*}
 The claim is verified.
 }
 \end{remark}
\subsection{Proof of Theorem \ref{Esistenza}}

Under the assumptions of Theorem \ref{Esistenza}, we define the
$C^1$-functional
 $$
 \phi(u):=\Phi(u)-\int_\Omega F(x,u(x))dx,\qquad u\in W^{1,p}_0(\Omega),
 $$
whose critical points are the weak solutions of problem
\eqref{Nostro}.\par
 In order to prove our result, we apply Theorem \ref{MPT} to this
functional. In the next three lemmas we shall verify the mountain
pass theorem conditions.
\begin{lemma}\label{lemma1}
 Every Palais-Smale sequence for the functional $\phi$
 is bounded in $W^{1,p}_0(\Omega)$.
\end{lemma}

\begin{proof}
Let $\{u_n\}\subset W^{1,p}_0(\Omega)$ be a Palais-Smale sequence,
that is,
\begin{equation}\label{E1}
\phi(u_n)\rightarrow \mu,
\end{equation}
for $\mu\in \erre$ and
\begin{equation}\label{E2}
\quad \|\phi'(u_n)\|_{W^{-1,p'}}\to0.
\end{equation}
\indent We argue by contradiction. So, suppose that the conclusion is
not true. Passing to a subsequence if necessary, we may assume that
$$
\|u_n\|\rightarrow +\infty.
$$
\indent  By conditions $(C_M^1)$ and $(C_M^2)$, it follows that there
exists $n_0\in \enne$ such that
  \begin{eqnarray}\label{A2}
\phi(u_n) &-& \frac{\langle \phi'(u_n),u_n\rangle }{\theta}\\\nonumber
&=&
\left[M(\|u_n\|^p)\right]^{p-1}\left[\frac{\widehat{M}(\|u_n\|^p)}{p
\left[M(\|u_n\|^p)\right]^{p-1}}-\frac{\|u_n\|^p}{\theta}\right]\\\nonumber
               &+&
\int_{\Omega}\left[\frac{f(x,u_n(x))u_n(x)}{\theta}-F(x,u_n(x))\right]dx,\\\nonumber
               &\geq&
m_0^{p-1}\left(\frac{1}{p}-\frac{1}{\theta}\right)\|u_n\|^p\\\nonumber
               &+&
\int_{\Omega}\left[\frac{f(x,u_n(x))u_n(x)}{\theta}-F(x,u_n(x))\right]dx,\nonumber
\end{eqnarray}
for every $n\geq n_0$.\par
 Thus
\begin{eqnarray*}
m_0^{p-1}\left(\frac{\theta-p}{\theta p}\right)\|u_n\|^p&\leq&
\phi(u_n)-\frac{\langle \phi'(u_n),u_n\rangle }{\theta}\\\nonumber
               &-&
\int_{|u_n(x)|>t_{\star}}\left[\frac{f(x,u_n(x))u_n(x)}{\theta}-F(x,u_n(x))\right]dx,\\\nonumber
               &+& M \meas\,(\Omega),\quad\, \forall\, n\geq n_0,
\end{eqnarray*}
where $``\meas\,(\Omega)"$ denotes the standard Lebesgue measure of
$\Omega$ and
$$
M:=\sup\left\{\left|\frac{f(x,t)t}{\theta}-F(x,t)\right|:x\in\bar\Omega,
|t|\leq t_{\star}\right\}.
$$

\indent Now, we observe that, the $(\rm AR)$ condition yields
$$
\int_{|u_n(x)|>t_{\star}}\left[\frac{f(x,u_n(x))u_n(x)}{\theta}-F(x,u_n(x))\right]dx\geq
0.
$$
So, we deduce that
\begin{eqnarray*}
m_0^{p-1}\left(\frac{\theta-p}{\theta p}\right)\|u_n\|^p\leq
\phi(u_n)-\frac{\langle \phi'(u_n),u_n\rangle }{\theta}+M
\meas\,(\Omega),
\end{eqnarray*}
for every $n\geq n_0$.\par
Then, for every $n\geq n_0$ one has
\begin{eqnarray*}
C\|u_n\|^p\leq
\left\{\phi(u_n)+{\|\phi'(u_n)\|_{W^{-1,p'}}\frac{\|u_n\|}{\theta}}+M
\meas\,(\Omega)\right\},
\end{eqnarray*}
where $C:=\displaystyle m_0^{p-1}\left(\frac{\theta-p}{\theta
p}\right)>0$.\par
In conclusion, dividing by $\|u_n\|$ and letting $n\rightarrow
\infty$, we obtain a contradiction.
\end{proof}

The above Lemma implies that the $C^1$-functional
$\phi$ verifies the Palais-Smale condition as proved in the next result.
\begin{lemma}\label{lemma2}
The functional $\phi$
satisfies the compactness $(\rm PS)$ condition.
\end{lemma}

\begin{proof}
 Take $\{u_{n}\}\subset W^{1,p}_0(\Omega)$ be a Palais-Smale
sequence. Thus, by Lemma \ref{lemma1}, the sequence $\{u_n\}$ is
necessarily bounded in $W^{1,p}_0(\Omega)$.\par Since
$W^{1,p}_0(\Omega)$ is reflexive, we may extract a subsequence
that for simplicity we call again $\{u_{n}\}$, such that
$u_{n}\rightharpoonup u$ in $W^{1,p}_0(\Omega)$.\par
 We will prove that ${u_{n}}$ strongly converges to $u\in
W^{1,p}_0(\Omega)$.
Exploiting the derivative $\phi'(u_n)(u_n-u)$, we obtain
\begin{eqnarray*}
\langle\Phi'(u_n),u_n-u\rangle &=& \langle
\phi'(u_n),u_n-u\rangle\\\nonumber
               &+& \int_{\Omega}f(x,u_n(x))(u_n-u)(x)dx.\nonumber
\end{eqnarray*}

Since $\|\phi'(u_n)\|_{W^{-1,p'}}\to0$ and the sequence $\{u_n-u\}$ is
bounded in
$W^{1,p}_0(\Omega)$, taking into account that $$|\langle
\phi'(u_n),u_n-u\rangle|\leq\|\phi'(u_n)\|_{W^{-1,p'}}\|u_n-u\|,$$ one
has
\[
\langle \phi'(u_n),u_n-u\rangle\to0.
\]
 \indent Further, by $\textrm{h}_1)$ and taking into account that
$u_n\rightarrow u$ in $L^q(\Omega)$ we obtain
\[
\int_{\Omega}|f(x,u_n(x))||u_n(x)-u(x)|dx\to0.
\]
 \indent We can conclude (by $(C_M^1)$) that
\[ %3.3
\langle A(u_n),u_n-u\rangle\rightarrow0,
\]
as $n\rightarrow \infty$.\par
\indent Since the operator $A$ has the $(S_+)$ property, in conclusion, $u_{n}\to u$ strongly in $W^{1,p}_0(\Omega)$.\par
 Hence, as claimed, the functional $\phi$ fulfills condition $($\rm{PS}$)$.
\end{proof}

\begin{lemma}\label{lemma3}
The functional $\phi$ has the geometry of the MPT. More precisely$:$
\begin{itemize}
\item[1.] There exists $r>0$ such that
$$
\inf_{\|u\|=r}\phi(u)>0.
$$
\item[2.] For some $u_0\in W^{1,p}_0(\Omega)$ one has
$$
\phi(\tau u_0)\rightarrow -\infty,
$$
as $\tau\rightarrow +\infty$.
\end{itemize}
\end{lemma}
\begin{proof}
1. We choose $\varepsilon>0$ small enough, verifying
$$
m_0^{p-1}>\frac{\lambda+\varepsilon}{\lambda_{1,p}(\Omega)}.
$$

\indent By condition \eqref{atzero} there exists
$\delta_\varepsilon>0$ such that
$$
\frac{f(x,t)}{|t|^{p-2}t}\leq \lambda+\varepsilon,
$$
for every $x\in \bar\Omega$ and $|t|\leq\delta_\varepsilon$.\par
 Hence, one has
$$
F(x,\xi)\leq \frac{\lambda+\varepsilon}{p}|\xi|^p,
$$
for every $|\xi|\leq \delta_\varepsilon$.\par
As a consequence of the above inequality, using hypotheses $\rm h_1)$,
the Sobolev embedding $W^{1,p}_0(\Omega)\hookrightarrow L^q(\Omega)$
and $(\widehat{C}_M)$, we can write:
\begin{eqnarray*}
\phi(u) &\geq&
\displaystyle\frac{m_0^{p-1}}{p}\|u\|^{p}-\int_{|u(x)|\leq
\delta_\varepsilon}\frac{\lambda+\varepsilon}{p}|u(x)|^pdx\\\nonumber
&-& C\int_{|u(x)|> \delta_\varepsilon}|u(x)|^qdx\\\nonumber
               &\geq&
\displaystyle\frac{m_0^{p-1}}{p}\|u\|^{p}-\frac{\lambda+\varepsilon}{p\lambda_{1,p}(\Omega)}\|u\|^{p}-C\|u\|^{q},\nonumber
\end{eqnarray*}
for a suitable positive constant $C$.\par
Now, set $r:=\|u\|^p$ and observe that for $r>0$ small enough, we have
$$
\frac{1}{p}\left(m_0^{p-1}-\frac{\lambda+\varepsilon}{\lambda_{1,p}(\Omega)}\right)r-Cr^{q/p}>0,
$$
bearing in mind that $q>p$.\par
 Hence
$$
\inf_{\|u\|=r}\phi(u)>0.
$$
\noindent 2. Let us choose an element $u_0\in W^{1,p}_0(\Omega)$ such
that
$$
\meas\,\left(\{x\in \Omega:u_0(x)\geq t_{\star}\}\right)>0.
$$

\indent Being $F(x,\xi)$ a $\theta$-superhomogeneous function if
$|\xi|\geq t_{\star}$, for $\tau>1$, we obtain
\begin{eqnarray*}
\phi(\tau u_0) &\leq& \displaystyle{\frac{m_1}{p}\|\tau u_0\|^{p}
+\frac{m_2}{p}}-\int_\Omega F(x,\tau u_0(x))dx\\\nonumber
&\leq& \displaystyle\tau^p\frac{m_1\| u_0\|^{p}}{p}
-\tau^\theta\int_{|u_0(x)|\geq t_\star}F(x,u_0(x))dx\\\nonumber
               &+& \frac{m_2}{p}+M\meas\,(\Omega),\nonumber
\end{eqnarray*}
where
$$
M:=\sup\left\{\left|F(x,\xi)\right|:x\in\bar\Omega, |\xi|\leq
t_{\star}\right\}.
$$

\indent Thus, the $(\rm AR)$ condition implies that
$$
\phi(\tau u_0)\rightarrow -\infty,
$$
as $\tau\rightarrow +\infty$. This concludes the proof.
\end{proof}
\section{Infinitely Many Solutions}
In this section, we prove the existence of infinitely many weak
solutions for problem \eqref{Nostro} exploiting the
${\mathbb{Z}}_2$-symmetric version of the MPT recalled in Theorem
\ref{ZMPT}. Our main result is the following.

\begin{theorem}\label{Theorem2}
Assume that $f:\bar \Omega\times\erre\rightarrow \erre$ odd  with
respect to the second argument and that the conditions of Theorem
\ref{Esistenza} are satisfied. Then problem \eqref{Nostro} has
infinitely many weak solutions.
\end{theorem}
\begin{proof}
Since the function $f$ is odd in the second argument the functional
$\phi$ is even. Moreover $\phi$ verifies that $\phi(0_{W^{1,p}_0})=0$
and by Lemmas \ref{lemma1} and \ref{lemma2}, it satisfies also the
$(\rm PS)$ condition. On the other hand by Lemma \ref{lemma3}, there
exist positive constants $\rho$ and $\alpha$ such that $\phi(u)\geq
\alpha$ if $\|u\|=\rho$. Hence, we need to prove just that for each
finite dimensional subspace $X_1\subset W^{1,p}_0(\Omega)$, the set
  $$
  \{u\in X_1:\phi(u)\geq 0\}
  $$
  is bounded in $W^{1,p}_0(\Omega)$.\par
  Indeed, let $u\in W^{1,p}_0(\Omega)$ arbitrary and denote
  $$
  \Omega_{<}:=\{x\in \Omega:|u(x)|<t_\star\},
  $$
  as well as
  $$
  \Omega_{\geq}:=\{x\in \Omega:|u(x)|\geq t_\star\}.
  $$

  \indent We shall prove that $\phi$ satisfies the following estimate
  \begin{equation}\label{15}
\phi(u) \leq \displaystyle{\frac{m_1}{p}\|u\|^{p}
+\frac{m_2}{p}}-\int_\Omega \gamma(x)|u(x)|^{\theta}dx+\kappa,
\end{equation}
  where $\kappa$ is a suitable positive constant and $\gamma\in
L^{\infty}(\Omega)$, with $\gamma>0$ in $\Omega$.\par
\indent To show the above inequality, let us start observing that by
$\rm h_1)$, the function $F$ satisfies
  \begin{equation}\label{13}
 |F(x,\xi)|\leq C_1(1+|\xi|^{q}),\qquad\forall\, x\in
\bar\Omega,\,\forall\,\xi\in \erre.
\end{equation}

\indent We claim that there exists $\gamma\in L^{\infty}(\Omega)$,
$\gamma>0$ in $\Omega$, such that
\begin{equation}\label{14}
F(x,\xi)\geq \gamma(x)|\xi|^\theta,\qquad\forall\, x\in
\bar\Omega,\,\forall\,|\xi|\geq t_{\star}.
\end{equation}
Indeed, since $F$ is $\theta$-superhomogeneous, we have that
$$
F(x,\xi)\geq \gamma_1(x)|\xi|^\theta,\qquad\forall\, x\in
\bar\Omega,\,\forall\,\xi\geq t_{\star},
$$
where $\displaystyle \gamma_1(x):=F(x,t_\star)/t_\star^\theta$. It is
easy to see that $\gamma_1\in L^\infty(\Omega)$ and $\gamma_1>0$ in
$\Omega$.\par
 In a similar way, it follows that
$$
F(x,\xi)\geq \gamma_2(x)|\xi|^\theta,\qquad\forall\, x\in
\bar\Omega,\,\forall\,\xi\leq -t_{\star},
$$
with $\displaystyle \gamma_2(x):=F(x,-t_\star)/t_\star^\theta$. Also
in this case $\gamma_2\in L^\infty(\Omega)$ and $\gamma_2>0$ in
$\Omega$.\par
\indent Then \eqref{14} holds with
$$
\gamma(x):=\max\{\gamma_1(x),\gamma_2(x)\},
$$
for every $x\in \bar\Omega$.\par
Now, by condition \eqref{13} we conclude that
$$
\int_{\Omega_{<}}F(x,u(x))dx\geq -C_1(t_\star^q+1)\meas\,(\Omega).
$$
Further, inequality \eqref{14} yields
$$
\int_{\Omega_{\geq}}F(x,u(x))dx\geq
\int_{\Omega_{\geq}}\gamma(x)|u(x)|^{\theta}dx.
$$
\indent Then
\begin{eqnarray*}
\phi(u) &\leq& \displaystyle{\frac{m_1}{p}\|u\|^{p}
+\frac{m_2}{p}}-\left(\int_{\Omega_{<}}F(x,u(x))dx+\int_{\Omega_{\geq}}F(x,u(x))dx\right)\\\nonumber
&\leq& \displaystyle{\frac{m_1}{p}\|u\|^{p}
+\frac{m_2}{p}}-\int_{\Omega_{\geq}}\gamma(x)|u(x)|^{\theta}dx+C_1(t_\star^q+1)\meas\,(\Omega)\\\nonumber
               &\leq& \displaystyle{\frac{m_1}{p}\|u\|^{p}
+\frac{m_2}{p}}-\int_{\Omega}\gamma(x)|u(x)|^{\theta}dx+\kappa,\nonumber
\end{eqnarray*}
where
$$
  \kappa:=(\|\gamma\|_\infty
t_\star^\theta+C_1(t_\star^q+1))\meas\,(\Omega).
  $$

  \indent Hence, inequality \eqref{15} is proved.\par
  \noindent At this point, observe that the functional
$\|\cdot\|_\gamma:W^{1,p}_0(\Omega)\rightarrow \erre$ defined by
  $$
  \|u\|_\gamma:=\left(\int_\Omega \gamma(x)|u(x)|^\theta
dx\right)^{1/\theta},
  $$
  is a norm in $W^{1,p}_0(\Omega)$.\par
  Since in $X_1$ the norms $\|\cdot\|$ and $\|\cdot\|_\gamma$ are
equivalent ($X_1$ is finite dimensional), there exists a positive
constant $\kappa_{X_1}$ such that
  $$
  \|u\|\leq \kappa_{X_1}\|u\|_\gamma,
  $$
  for every $u\in W^{1,p}_0(\Omega)$.\par

  \indent Consequently, we have that
  $$
  \phi(u)\leq
\displaystyle{\frac{m_1\kappa_{X_1}^p}{p}\|u\|^{p}_\gamma}-\|u\|_\gamma^{\theta}+\kappa+\frac{m_2}{p},
  $$
  for every $u\in X_1$.\par
  Hence
  $$
  \displaystyle{\frac{m_1\kappa_{X_1}^p}{p}\|u\|^{p}_\gamma}-\|u\|_\gamma^{\theta}+\kappa+\frac{m_2}{p}\geq
0,
  $$
  for every
  $$
  u\in\{u\in X_1:\phi(u)\geq 0\}.
  $$

  \indent Since $\theta>p$ we conclude that the above set is bounded.
The proof is complete.
\end{proof}

\begin{remark}\rm{
Note that condition \eqref{atzero} in Theorem \ref{Theorem2} can be
removed if, instead of Theorem \ref{ZMPT}, we apply the more general
Theorem 9.12 contained in Rabinowitz \cite[p. 55]{Ra}.
}
\end{remark}

\section{Some Examples}
In this section $\Omega$ is an open and bounded subset of $\erre^3$
with smooth boundary $\partial \Omega$. We present two simple
applications of our main results taking $p=2$.\par
\smallskip
The first one is a direct consequence of Theorem \ref{Esistenza}.

\begin{example}\label{esempio0}\rm
 Consider the following nonlocal equation with the Dirichlet boundary
condition:
\begin{equation} \tag{$E_1$} \label{Nostrom}
\left\{
\begin{array}{ll}
-M\Big(\displaystyle\int_\Omega|\nabla
u(x)|^2dx\Big)\Delta u= u^3+u^4 & \rm in \quad \Omega
\\ {\,u|_{\partial \Omega}=0,}
\end{array}
\right.
\end{equation}
where
$$
M\Big(\displaystyle\int_\Omega|\nabla
u(x)|^2dx\Big):=2+\frac{\sin \left(\displaystyle\int_\Omega|\nabla
u(x)|^2dx\right)}{1+\left(\displaystyle\int_\Omega|\nabla
u(x)|^2dx\right)^2}.
$$
\noindent
Owing to Theorem \ref{Esistenza}, problem \eqref{Nostrom} admits one
nontrivial classical solution. Indeed, a direct computation ensures
that the continuous function
$$\displaystyle M(t):=2+\frac{\sin t}{1+t^2},\quad (\forall\,t\geq
0)$$ satisfies conditions $(C_M^1)$ and $(C_M^2)$. Further, the
locally Lipschitz continuous function $f:\erre \rightarrow \erre$
given by $$f(t):=t^{3}+t^4,\quad (\forall\,t\in \erre)$$ verifies all
the hypotheses of Theorems \ref{Esistenza}.
\end{example}

\begin{remark}\rm{
We note that in Example \ref{esempio0} condition $(C_M^2)$ is verified
for every $t\geq t_0$, where $t_0$ is the unique positive solution of
the following real equation
$$
\int_0^t\left(2+\frac{\sin s}{1+s^2}\right)ds-t\left(2+\frac{\sin
t}{1+t^2}\right)=0.
$$}
\end{remark}

In conclusion, the existence of infinitely many solutions for the next
elliptic nonlocal problem (namely \eqref{Nostro2}) can be studied by
using Theorem \ref{Theorem2}.

\begin{example}\label{esempio}\rm
  Consider the following problem in presence of an odd nonlinearity:
\begin{equation} \tag{$E_2$} \label{Nostro2}
\left\{
\begin{array}{ll}
-M\Big(\displaystyle\int_\Omega|\nabla
u(x)|^2dx\Big)\Delta u= u^{3}+u & \rm in \quad \Omega
\\ {\,u|_{\partial \Omega}=0,}
\end{array}
\right.
\end{equation}
where
$$
M\Big(\displaystyle\int_\Omega|\nabla
u(x)|^2dx\Big):=1+\frac{\cos \Big(\displaystyle\int_\Omega|\nabla
u(x)|^2dx\Big)}{1+\Big(\displaystyle\int_\Omega|\nabla
u(x)|^2dx\Big)^2}.
$$
\noindent Owing to Theorem \ref{Theorem2}, problem \eqref{Nostro2}
admits infinitely many classical solutions.
\end{example}

\medskip
 \noindent \textbf{Acknowledgements.} G. Molica Bisci is supported by
the 2012 GNAMPA Project titled  ``{\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}