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\AtBeginDocument{{\noindent\small
{\em Electronic Journal of Differential Equations},
Vol. 2004(2004), No. 37, pp. 1--17.\newline
ISSN: 1072-6691. URL: http://ejde.math.txstate.edu or http://ejde.math.unt.edu
\newline ftp ejde.math.txstate.edu  (login: ftp)}
\thanks{\copyright 2004 Texas State University - San Marcos.}
\vspace{9mm}}

\begin{document}

\title[\hfilneg EJDE-2004/37\hfil hemivariational inequalities]
{An existence result for hemivariational inequalities}

\author[Zsuzs\'{a}nna D\'{a}lyay \& Csaba Varga  \hfil EJDE-2004/37\hfilneg]
{Zsuzs\'{a}nna D\'{a}lyay \& Csaba Varga} % in alphabetical order

\address{Zsuzs\'{a}nna D\'{a}lyay \hfill\break
University of Szeged, Juh\'{a}sz Gyula Teacher Training Faculty,
Department of Mathematics 6721,
Szeged, Boldogasszony Sgt. 6. Hungary}
\email{dalyay@jgytf.u-szeged.hu}

\address{Csaba Varga \hfill\break
``Babe\c{s}- Bolyai" University,
Department of  Mathematics, Str. M Kog\u{A}lniceanu, Nr.1,
Cluj-Napoca, Romania}
\email{csvarga@cs.ubbcluj.ro}

\date{}
\thanks{Submitted November 17, 2003. Published March 16, 2004.}
\thanks{Z. D\'{a}lyai was partially supported by contract HPRN-CT-1999-00118 from the EU}
\thanks{C. Varga was partially supported by contract HPRN-CT-1999-00118 from the EU,
and by  \hfill\break\indent  the Research Center of Sapientia Foundation}
\subjclass{35A15, 35J60, 35H30}
\keywords{Variational methods, discontinuous nonlinearities, \hfill\break\indent
principle of symmetric criticality}

\begin{abstract}
 We present a general method for obtaining solutions  for an
 abstract class of hemivariational inequalities. This result
 extends many results to the nonsmooth case.
 Our proof is based on a nonsmooth version of the Mountain Pass
 Theorem with Palais-Smale or with Cerami compactness condition.
 We also use the Principle of Symmetric Criticality for locally
 Lipschitz functions.
\end{abstract}

\maketitle
\numberwithin{equation}{section}
\newtheorem{theorem}{Theorem}[section]
\newtheorem{corollary}[theorem]{Corollary}
\newtheorem{application}[theorem]{Application}
\newtheorem{proposition}[theorem]{Proposition}
\newtheorem{remark}[theorem]{Remark}

\section{Introduction}

Let ($X, \|\cdot\|$) be a real, separable, reflexive Banach space,
and let $(X^{\star}, \|\cdot\|_{\star})$ be its dual. Also assume
that the inclusion $X \hookrightarrow L^{l}(\mathbb{R}^N)$ is
continuous with the embedding constants $C(l)$, where $l \in [p,
p^{\star}]$ ($p \geq 2, p^{\star}=\frac{ Np}{ N-p}$). Let us
denote by $\|\cdot\|_{l}$ the norm of $L^{l}(\mathbb{R}^N)$. Let
$A: X \to X^{\star}$ be a potential operator with the potential
$a: X \to \mathbb{R}$, i.e. $a$ is G\^{a}teaux differentiable and
$$ \lim _{t \to 0} \frac{ a(u +tv)-a(u)}{ t} = \langle A(u), v
\rangle , $$ for every $u, v \in X$.   Here $\langle \cdot, \cdot
\rangle$ denotes the duality pairing between $X^{\star}$ and $X$.
For a potential we always assume that $a(0)=0$. We suppose that $A
: X \to X^{\star}$ satisfies the following properties:

\begin{itemize}
  \item  $A$ is hemicontinuous, i.e. $A$ is continuous on line
  segments in $X$ and $X^{\star}$ equipped with the weak topology.
  \item  $A$ is homogeneous of degree $p-1$, i.e.
  for every $u \in X$ and $t > 0$ we have $A(tu)= t^{p-1} A(u)$.
  Consequently, for a homogeneous hemicontinuous operator of degree
  $p-1$, we have $a(u)= \frac{ 1}{ p} \langle A(u), u  \rangle$.
  \item  $A : X \to X^{\star}$ is a strongly monotone
  operator, i.e. there exists a function $\kappa : [0, \infty) \to
  [0, \infty)$ which is positive on $(0, \infty)$ and
  $ \lim _{t \to \infty} \kappa (t)= \infty$ and such that
  for all $u, v \in X$,
$$
\langle A(u) - A(v), u -v \rangle \geq \kappa(\|u -v \|)\|u-v\|\,.
$$
\end{itemize}
In this paper we suppose that the operator $A : X \to
X^{\star}$ is a potential, hemicontinuous, strongly monotone
operator, homogeneous of degree $p-1$.

Let $f : \mathbb{R}^{n} \times \mathbb{R} \to \mathbb{R}$
be a measurable function which satisfies the following growth
condition:
\begin{itemize}
\item[(F1)] $|f(x, s)| \leq c(|s|^{p-1} + |s|^{r-1})$,
for a.e. $x \in \mathbb{R}^N$, for all $s \in \mathbb{R}$

\item[(F1')] The embedings  $X \hookrightarrow L^{r}(\mathbb{R}^{n})$
are compact ($p < r < p^{\star}$).
\end{itemize}
Let $F : \mathbb{R}^N \times \mathbb{R} \to \mathbb{R}$
be the function defined by
\begin{equation}\label{monoton1}
F(x, u) = \int _{0}^{u} f(x, s)ds, \quad \mbox{for  a.e. }
x \in \mathbb{R}^N, \; \forall  s \in \mathbb{R}.
\end{equation}
For a.e. $x \in \mathbb{R}^N$ and for every $u, v \in
\mathbb{R}$, we have:
\begin{equation}\label{monoton2}
|F(x, u) - F(x, v)| \leq c_{1}|u - v|\left(|u|^{p-1}+ |v|^{p-1}+
|u|^{r-1}+ |v|^{r-1} \right),
\end{equation}
where $c_{1}$ is a constant which depends only of $u$ and $v$.
Therefore, the function $F(x, \cdot)$ is locally Lipschitz and we
can define the partial Clarke derivative, i.e.
\begin{equation}\label{monoton3}
F^{0}_{2}(x,u; w) =  \limsup _{y \to u , \; t
\to 0^{+}} \frac{ F(x, y + t w) - F(x, y)}{ t},
\end{equation}
for every $u, w \in \mathbb{R}$ and for a.e. $x \in \mathbb{R}$.

Now, we formulate the hemivariational inequality problem that
will be studied in this paper:

 \emph{Find $u \in X$ such that}
\begin{equation}\label{P}
 \langle Au, v \rangle  +  \int _{\mathbb{R}^N}
F^{0}_{2}(x, u(x); - v(x))dx \geq 0, \quad \forall\,  v \in X.
\end{equation}

When the function $f : \mathbb{R}^{n} \times \mathbb{R} \to
\mathbb{R}$ is continuous, the problem \eqref{P} is reduced to
the problem:

\emph{Find $u \in X$ such that}
\begin{equation} \label{P'}
\langle Au, v \rangle  =  \int _{\mathbb{R}^N} f(x, u(x))v(x)dx, \quad  \forall\,
 v \in X.
\end{equation}
Such problems have been studied by many authors, see
\cite{Bart1, BartWa, BarWill1, BarWill2, FanZhao, GazRad, Megrez, MontRad}.

To study the existence of solutions of the problem \eqref{P} we
introduce the functional $\Psi : X \to \mathbb{R}$ defined
by $\Psi(u)=a(u) - \Phi(u)$, where  $a(u)= \frac{
1}{ p} \langle A(u), u \rangle$ and $\Phi(u) =
 \int _{\mathbb{R}^N} F(x, u(x))dx$. From
Proposition \ref{kritpont} we will see that the critical points of
the functional $\Psi$ are the solutions of the problem \eqref{P}.
Therefore it is enough to study the existence of critical points
of the functional $\Psi$. Considering such a problem is motivated
by the works of Clarke \cite{Clarke}, D. Motreanu and P.D.
Panagiotopoulos \cite{MotPan} and by the recent book of D.
Motreanu and V. R\u{a}dulescu \cite{MotRad} , where several
applications are given.

To study the existence of the critical point of
the function $\Psi$ is necessary to impose some condition on
function $f$:
\begin{itemize}
\item[(F2)] There exists $\alpha > p$, $\lambda  \in
[0, \frac{ \kappa(1)(\alpha -p)}{C^p (p)}[$ and a continuous function
$g: \mathbb{R} \to \mathbb{R}_{+}$, such that for a.e. $x \in
\mathbb{R}^N$ and for all $ u \in \mathbb{R}$ we have
\begin{equation}\label{monoton4}
\alpha F(x,u) + F^{0}_{2}(x,u;-u) \leq g(u),
\end{equation}
where $ \lim_{|u| \to \infty} g(u)/ |u|^p = \lambda$.

\item[(F2')] There exists $\alpha \in ( \max \{p, p^{\star}
\frac{ r - p}{p^{\star} - p} \}, p^{\star})$ and a constant $C >0$
such that for a.e. $x \in \mathbb{R}^N$ and for all $u \in
\mathbb{R}$ we have
\begin{equation}\label{monoton5}
-C|u|^{\alpha} \geq F(x,u) + \frac{ 1}{p} F^{0}_{2}(x,u; -u ).
\end{equation}
\end{itemize}
Next, we impose further assumptions on $f$. First we define
two functions by
\begin{gather*}
\underline{f}(x,s)=  \lim _{\delta\to 0^{+}} \mathop{\rm essinf}
\{ f(x,t) : |t-s| < \delta \},\\
\overline{f}(x,s)=  \lim _{\delta \to 0^{+}} \mathop{\rm esssup}
\{ f(x,t) : |t-s| < \delta \},
\end{gather*}
for every $s \in \mathbb{R}$ and for a.e. $x \in \mathbb{R}^N$. It
is clear that the function $\underline{f}(x, \cdot)$ is lower
semicontinuous and $\overline{f}(x, \cdot)$ is upper
semicontinuous. The following hypothesis on $f$ was introduced by
Chang \cite{Chang}.
\begin{itemize}
 \item[(F3)] The functions $\underline{f}, \overline{f}$
 are $N$-measurable, i.e. for every measurable function
 $u : \mathbb{R}^N \to \mathbb{R}$ the functions
 $x \mapsto \underline{f}(x,u(x)), x \mapsto \overline{f}(x,u(x)) $
 are measurable.

\item[(F4)] For every $\varepsilon >0$, there exists
 $c(\varepsilon) >0$ such that for a.e. $x \in \mathbb{R}^N$ and
 for every $s \in \mathbb{R}$ we have
 $$|f(x,s)| \leq \varepsilon |s|^{p-1} + c(\varepsilon) |s|^{r-1}.$$

\item[(F5)] For the $\alpha \in (p , p^{\star})$ from condition
(F2), there exists a $c^{\star} > 0$ such that for a.e. $x
\in \mathbb{R}^N$ and for all $s \in \mathbb{R}$ we have
$$
F(x,u) \geq c^{\star}(|u|^{\alpha} - |u|^p ).
$$
\end{itemize}

\begin{remark}\label{monremark1} \rm
We observe that if we impose the following condition on $f$,
\begin{itemize}
\item[(F4')] $ \lim_{\varepsilon \to 0^{+}} \mathop{\rm esssup}
\{ \frac{ |f(x, s)|}{ |s|^p } : (x,s) \in \mathbb{R}^N \times
(- \varepsilon, \varepsilon)\}=0$,
\end{itemize}
then this condition with (F1) imply (F4).
\end{remark}

The main result of this paper can be formulated in the following
manner.

\begin{theorem} \label{thm2}
\begin{enumerate}
\item If conditions (F1), (F1'), and (F2)--(F5) hold, then
problem \eqref{P} has a nontrivial solution.

\item If conditions (F1), (F1'), (F2'), (F3), and (F4) hold,
then problem \eqref{P} has a nontrivial solution.
\end{enumerate}
\end{theorem}

Let $G$ be the compact topological group $O(N)$ or a
subgroup of $O(N)$. We suppose that $G$ acts continuously and
linear isometric on the Banach space $X$. We denote by
$$
X^{G}=\{ u \in H  :  gx=x \mbox{ for all } g \in G \}
$$
the fixed point set of the action $G$ on $X$.
It is well known that $X^{G}$ is a closed subspace of $X$.
 We suppose that the potential $a: X \to \mathbb{R}$
 of the operator $A : X \to X^{\star}$ is $G$-invariant and the
 next condition for the function
 $f: \mathbb{R}^N\times \mathbb{R} \to \mathbb{R}$
 holds:
\begin{itemize}
\item[(F6)] For a.e. $x \in \mathbb{R}^N$ and
for every $g \in G, s \in \mathbb{R}$ we have $f(gx, s)=f(x, s)$.
\end{itemize}
In several applications the condition (F1') is replaced by the
condition
\begin{itemize}
\item[(F1'')] The embeddings $X^{G}\hookrightarrow L^{r}(\mathbb{R}^N)$
 are compact ($p < r < p^{\star}$).
\end{itemize}
Now, using the Principle of Symmetric Criticality for locally
Lipschitz functions, proved by  Krawciewicz and Marzantovicz
\cite{KrawMar}, from the above theorem we obtain the following
corollary, which is very useful in the applications.

\begin{corollary} \label{coro3}
We suppose that the potential
$a:X \to \mathbb{R}$ is $G$-invariant and (F6) is satisfied.
 Then the following assertions hold.
\begin{itemize}
\item[(a)]  If  (F1), (F1''), and (F2)--(F5) are fulfilled, then
problem \eqref{P} has a nontrivial solution.

\item[(b)] If (F1), (F1'), (F2'), F3), and (F4)
are fulfilled, then problem \eqref{P} has a nontrivial solution.
\end{itemize}
\end{corollary}

Next, we give an example of a discontinuous  function $f$ for
which the problem \eqref{P} has a nontrivial solution. \smallskip

\noindent {\bf Example.}  Let $(a_{n}) \subset \mathbb{R}$ be a
sequence with $a_{0}=0, a_{n} >0, n \in \mathbb{N}^{\star}$ such
that the series $ \sum _{n=0} ^{\infty} a_{n}$ is
convergent and $ \sum _{n=0} ^{\infty} a_{n} > 1$. We
introduce the following notation $$A_{n}:=  \sum
_{k=0}^{n} a_{k}, A:=  \sum_{k=0}^{\infty} a_{k}.$$
With these notations we have $A
>1 $ and $A_{n}=A_{n-1}+a_{n}$ for every $n \in
\mathbb{N}^{\star}$. Let $f: \mathbb{R} \to \mathbb{R}$
defined by $f(s)=s|s|^{p-2}\left(|s|^{r-p}+A_{n} \right)$, for all
$s \in (-n-1, -n] \cup [n, n+1), n \in \mathbb{N}$ and $r,s \in
\mathbb{R}$ with $r > p>2$. The function $f$ defined above
satisfies the properties (F1), (F2'), (F3), and (F4).
The discontinuity set of $f$ is
$\mathcal{D}_{f}=\mathbb{Z}^{\star}=\mathbb{Z} \setminus \{0\}$.
It is easy to see that the function $f$ satisfies the conditions
(F1) and (F4'), therefore (F4).
Let $F: \mathbb{R}\to \mathbb{R}$ be the function defined by
$F(u)= \int_{0}^{u} f(s)ds$ with $u \in [n, n+1)$,
when $n \geq 1$. Because $F(u)=F(-u)$, it is sufficient to
consider the case $u> 0$. We have
$F(u)=  \sum_{k=0}^{n-1} \int_{k}^{k+1}f(s)ds +
 \int_{n}^{u} f(s)ds$.
Therefore, for
 $F(u)= \frac{ 1}{ r} u^{r} +\frac{ 1}{ p} A_{n} u^p  -\frac{ 1}{ p}
\sum_{k=0}^{n}a_{k}k^p $, for every $u \in [n, n+1]$. It is easy
to see that $F^{0}(u; -u)=-u f(u)$ for every $u \in (n, n+1]$.
i.e. $F^{0}(u,-u)=-u^{r}-A_{n}u^p $.
Thus,
$$
F(u) + \frac{ 1}{ p}F^{0}(u,-u)
=-\big( \frac{ 1}{ p} -\frac{ 1}{ r}\big)u^{r}
-\frac{ 1}{ p}\sum_{k=0}^{n}a_{k}k^p \leq
-\big( \frac{1}{ p} - \frac{ 1}{r}\big)u^{r}.
$$
If we choose $C=\frac{ 1}{ p}-\frac{ 1}{ r}, \alpha=r>2$, the
condition (F2') is fulfilled.


This paper is organized as follows: In Section 2, some facts about
locally Lipschitz functions are given; In Section 3 a key inequality is
proved; in Section 4 the Palais-Smale and Cerami condition is
verified for the function $\Psi$; in Section 5 we prove Theorem 2
and in the last section we give some concrete applications.

\section{Preliminaries and preparatory results}

Let $(X, \|\cdot\|)$ be a real Banach space and $(X^{\star},
\|\cdot\|_{\star}$) its dual. Let $U \subset X$ be an open set. A
function $\Psi : U \to \mathbb{R}$ is called locally
Lipschitz function if each point $u \in U$ possesses a
neighborhood $N_{u}$ of  $u$ and a constant $K>0$ which depends on
$N_{u}$ such that
 $$
|f(u_{1})-f(u_{2})|\leq K\| u_{1}-u_{2}\|, \quad  \forall\, u_{1},\ u_{2} \in N_{u}.
$$
The generalized directional derivative of a locally Lipschitz
function $\Psi : X \to \mathbb{R}$ in $u\in U$  in the
direction $ v\in X$ is defined by
$$
\Psi^{0}(u;v)=\limsup_{\scriptstyle {\it w\to u}\frac
 \scriptstyle \it t\searrow 0}\frac{1}{t}(\Psi(w+tv)-\Psi(w)).
$$
It is easy to verify that $\Psi^{0}(u;-v)=(-\Psi)^{0}(u;v)$ for
every $u \in U$ and $v \in X$.

The generalized gradient of $\Psi$ in $u\in X$ is defined as being
the subset of $X^{\star}$ such that
$$
\partial \Psi(u)=\{z\in X^{\ast}:\langle z,\ v\rangle \leq \Psi^{0}(u; v),\;
\forall\,  v\in X \},
$$
where $\langle \cdot, \cdot \rangle $ is
the duality pairing between $X^{\star}$ and $X$. The subset
$\partial \Psi(u) \subset X^{\star}$ is nonempty, convex and
$w^{\star}$-compact and we have
$$
\Psi^{0}(u; v)=\max\{\langle z,v\rangle :z\in \partial \Psi(u)\},\; \forall\, v\in X.
$$
If $\Psi_{1}, \Psi_{2} : U \to \mathbb{R}$ are two locally
Lipschitz functions, then
$$
(\Psi_{1} + \Psi_{2})^{0}(u;\ v)\leq  \Psi_{1}^{0}(u;\ v)+\Psi_{2}^{0}(u;\ v)
$$
for every $u\in U$ and $v \in X$. We define the function
$\lambda_{\Psi}(u)= \inf \{\|x^{\star}\|_{\star} : x^{\star} \in \Psi(u) \}$.
This function is lower semicontinuous and this infimum is attained,
because $\partial \Psi(u)$ is $w^{\star}$-compact. A point $u \in
X$ is a critical point of $\Psi$, if $\lambda_{\Psi}(u)=0$, which
is equivalent with $\Psi^{0}(u; v) \geq 0$ for every $v \in X$.
For a real number $c\in \mathbb{R}$ we denote by
$$
K_c=\{u \in X:\lambda_{\Psi}(u)=0, \ \Psi(u)=c\}.
$$

\begin{remark}\label{psmejegyzes} \rm
If $\Psi : X \to \mathbb{R}$ is locally Lipschitz and we take
$u \in X$ and $\mu > 0$,
the next two assertions are equivalent:
\begin{itemize}
\item[(a)]  $\Psi^{0}(u, v) + \mu \|v\| \geq 0 $, for all $v\in X$;
\item[(b)]  $\lambda_{\Psi}(u) \leq \mu$.
\end{itemize}
\end{remark}

Now, we define the following terms.
\begin{itemize}
\item[(i)] $\Psi$  satisfies the $(PS)$-condition at level $c$
(in short, $(PS)_{c}$) if every sequence $\{x_n\} \subset X$ such that
$\Psi(x_n)\to c$ and $\lambda_{\Psi}(x_n)\to 0$ has a convergent
subsequence.

\item[(ii)] $\Psi$  satisfies the $(CPS)$-condition at level $c$
 (in short, $(CPS)_{c}$) if  every
sequence $\{x_n\} \subset X$ such that $\Psi(x_n)\to c$
and $(1+\|x_n\|)\lambda_{\Psi}(x_n)\to 0$ has a convergent
subsequence.
\end{itemize}
It is clear that $(PS)_c$ implies  $(CPS)_c$.

Now, we consider a globally Lipschitz function $\varphi:X
\to \mathbb{R}$ such that $\varphi(x)\geq 1$, for all
$x\in X$ (or, generally, $\varphi(x)\geq \alpha$, $\alpha> 0$).
We say that
\begin{itemize}
\item[(iii)] $\Psi$ satisfies the $(\varphi-PS)$-condition at level
$c$ (in short, $(\varphi-PS)_{c}$) if  every sequence
$\{x_n\}\subset X$ such that $\Psi(x_n)\to c$ and
$\varphi(x_n)\lambda_{\Psi}(x_n)\to 0$ has a convergent
subsequence.
\end{itemize}
The compactness $(\varphi-PS)_{c}$-condition in (iii)
contains the assertions (i) and (ii) in the sense that  if
$\varphi\equiv 1$ we get the $(PS)_{c}$-condition and if
$\varphi(x)=1+\|x\|$ we have the $(C)_{c}$-condition.

In the next we use the following version of the Mountain Pass
Theorem, see Krist\'{a}ly-Motreanu-Varga \cite{KrMoVa}, which
contains the classical result of Chang \cite{Chang} and
Kourogenis-Papageorgiu \cite{{Papa}}.


\begin{proposition}[Mountain Pass Theorem] \label{mountaincer}
 Let X be a Banach space, $\Psi: X \to R$ a locally Lipschitz
function with $\Psi(0)\leq 0$ and $\varphi :X \to R$ a globally
Lipschitz function such that $\varphi(x)\geq 1$, $\forall x \in
X.$ Suppose that there exists a point  $x_1 \in X$ and constants
$\rho, \alpha >0$ such that
\begin{itemize}
\item[(i)] $\Psi(x)\geq \alpha$, $\forall x\in X$ with $\|x\|=\rho$
\item[(ii)] $\| x_1 \|>\rho$  and $\Psi(x_1)<\alpha$
\item[(iii)] The function $\Psi$ satisfies the
$(\varphi-PS)_c$-condition, where
$$
c=\inf_{\gamma \in\Gamma}\max_{t \in [0,1]}\Psi(\gamma(t)),
$$
with
$\Gamma =\{\gamma \in C([0,1],X): \gamma(0)=0, \gamma(1)=x_1 \}$.
\end{itemize}
Then the minimax value $c$ in (iii) is a critical value of
$\Psi$, i.e. $K_c$ is nonempty, and, in addition, $c\geq \alpha$.
\end{proposition}


Let $G$ be a compact topological group which acts linear
isometrically on the real Banach space $X$, i.e. the action
$G\times X\to X$ is continuous and for every $g\in~G,\ g:X\to X$
is a linear isometry. The action on $X$ induces an action of the
same type on the dual space $X^*$ defined by $(gx^*)(x)=x^*(gx)$,
for all $g\in G,\ x\in X$ and $x^*\in X^*$. Since
\[
\|gx^*\|_{\star}=\sup_{\|x\|=1}|(gx^*)(x)|=\sup_{\|x\|=1}|x^*(gx)|,
\]
the isometry assumption for the action of $G$ implies
\[
\|gx^*\|_{\star}= \sup_{\|x\|=1}|x^*(x)|=\|x^*\|_{\star},\
\forall \ x^*\in X^*,\ g\in G.
\]

We suppose that $\Psi:X\to \mathbb{R}$ is a locally Lipschitz and
$G$-invariant function, i.e., $\Psi(gx)=\Psi(x)$ for every $g\in
G$ and $x\in X$. From Krawcewicz-Marzantowicz [10] we have the
relation
\[
g\partial \Psi(x)=\partial \Psi(gx)=\partial \Psi(x),
\mbox{ for every }g\in G\mbox{ and }x\in X.
\]
Therefore,  the subset $\partial \Psi(x)\subset X^*$ is $G$-invariant,
so the function \\ $\lambda_{\Psi}(x)=\inf_{w\in \partial
\Psi(x)}\|w\|_{\star},\ x\in X$, is $G$-invariant. The fixed
points set of the action $G$, i.e.
$X^{G}=\{x \in X \ | gx=x \;\forall \, g \in G \}$ is a closed
linear subspace of $X$.

We conclude this section with the Principle of Symmetric Criticality,
first proved by Palais \cite{Palais} for differentiable functions
and for locally Lipschitz proved by Krawciewicz and Marzantovicz
\cite{KrawMar}.

\begin{theorem}\label{princisim}
Let $\Psi: X \to \mathbb{R}$ be a $G$-invariant locally
Lipschitz function and $u \in X^{G}$ a fixed point. Then  $u \in
X^{G}$ is a critical point of $\Psi$ if and only if $u$ is a
critical point of $\Psi^{G}= \psi|_{X^{G}}: X^{G} \to
\mathbb{R}$.
\end{theorem}

\section{Some basic lemmas}

Define the function $\Phi : X \to \mathbb{R}$ by
\begin{equation}\label{function}
\Phi(u) =  \int _{\mathbb{R}^N} F(x, u(x))dx, \quad \forall \, u \in X,
\end{equation}
where the function $F$ is defined in (\ref{monoton1}).

\begin{remark}\label{fontos1} \rm
The following two results are true for the general growth condition
($f_{1}$), but it is sufficient to prove them in the case when the
function $f$ satisfies the growth condition $|f(x, s)| \leq
c|u|^{p-1}$ for a.e. $x \in \mathbb{R}^N, \forall \ s \in
\mathbb{R}$. For simplicity we denote $h(u)=c|u|^{p-1}$ and in
the next two results we use only that the function $h$ is monotone
increasing, convex and $h(0)=0$.
\end{remark}

\begin{proposition}\label{lipprop}
The function $\Phi : X \to \mathbb{R}$, defined by
$\Phi(u) =  \int _{\mathbb{R}^N} F(x, u(x))dx$ is
locally Lipschitz on bounded sets of $X$.
\end{proposition}

\begin{proof} For every $u, v \in X$, with $\|u\|, \|v\| < r$, we have
\begin{align*}
&\|\Phi(u)-\Phi(v)\|\\
&\leq  \int _{\mathbb{R}^N}|F(x, u(x)) - F(x, v(x))| dx \\
&\leq c_{1}\int _{\mathbb{R}^N} |u(x) - v(x)|[h(|u(x)|)+ h(|v(x)|)]\\
&\leq  c_{2} \big( \int_{\mathbb{R}^N}|u(x)-v(x)|^p \big)^{1/p}
\big[\big(\int_{\mathbb{R}^N}(h(|u(x)|)^{p'}dx\big)^{1/p'}
+\big( \int_{\mathbb{R}^N}(h(|v(x)|)^{p'}dx\big)^{1/p'}\big]\\
&\leq c_{2}\|u-v\|_{p}[\|h(|u|)\|_{p'}+\|h(|v|)\|_{p'})\\
&\leq C(u, v) \|u -v\|,
\end{align*}
where $\frac{ 1}{ p} + \frac{1}{ p'}=1$ and we used the H\"{o}lder
inequality, the subadditivity of the norm
$\| \cdot\|_{p'}$ and the fact that the inclusion
$X\hookrightarrow L^p (\mathbb{R}^N)$ is continuous. We observe
that C(u, v) is a constant which depends only of $u$ and $v$.
\end{proof}

\begin{proposition} \label{clarkeder}
If condition (F1) holds, then for every $u, v \in X$, then
\begin{equation}\label{clarkegyen}
\Phi^{0}(u;v) \leq  \int _{\mathbb{R}^N}F^{0}_{2}(x,u(x); v(x))dx.
\end{equation}
\end{proposition}

\begin{proof}
It is sufficient to prove the proposition
for the function $f$, which satisfies only the growth condition
$|f(x, s)| \leq c|u|^{p-1}$ from Remark \ref{fontos1}. Let us fix
the elements $u, v \in X$. The function $F(x, \cdot)$ is locally
Lipschitz and therefore continuous. Thus $F^{0}_{2}(x,u(x);v(x))$
can be expressed as the upper limit of
$\big(F(x, y+ tv(x)) - F(x, y)\big)/t$, where $t \to 0^{+}$
takes rational values and $y \to u(x)$ takes values in a
countable subset of $\mathbb{R}$. Therefore, the map $x
\to F^{0}_{2}(x,u(x); v(x))$ is measurable as the
``countable limsup" of measurable functions in $x$. From condition
(F1) we get that the function
$x \to F_{2}^{0}(x,u(x);v(x))$ is from $L^{1}(\mathbb{R}^N)$.

Using the fact that the Banach space $X$ is separable, there
exists a sequence $w_{n} \in X$ with $\|w_{n} - u\| \to 0$
and a real number sequence $t_{n} \to 0^{+}$, such that
\begin{equation}\label{iranym}
\Phi^{0}(u, v)=  \lim _{n \to \infty}
\frac{ \Phi(w_{n} + t_{n}v)-\Phi(w_{n})}{ t_{n}}.
\end{equation}
Since the inclusion $X \hookrightarrow L^p (\mathbb{R}^N)$ is
continuous, we get $\|w_{n}-u\|_{p} \to 0$. Using
\cite[Theorem IV.9]{Brez}, there exists a subsequence of
$(w_{n})$ denoted in the same way, such that $w_{n}(x) \to
u(x)$ a.e. $x \in \mathbb{R}^N$. Now, let $\varphi_{n} :
\mathbb{R}^N \to \mathbb{R} \cup \{ + \infty\}$ be the
function defined by
\begin{align*}
\varphi_{n}(x) &= -\frac{ F(x,w_{n}(x) + t_{n}v(x)) - F(x, w_{n}(x))}{ t_{n}} \\
&\quad +c_{1} |v(x)|[h(|w_{n}(x)+ t_{n}v(x)|) + h(|w_{n}(x)|)].
\end{align*}
We see that the the functions $\varphi _{n}$ are measurable and
non-negative. If we apply Fatou's lemma, we get
$$
\int _{\mathbb{R}^N} \liminf _{n \to \infty }
\varphi_{n}(x)dx \leq  \liminf _{n \to
\infty} \int _{\mathbb{R}^N} \varphi_{n}(x)dx.
$$
This inequality is equivalent to
\begin{equation}\label{fatou1}
 \int _{\mathbb{R}^N}  \limsup _{n\to \infty}[- \varphi_{n}(x)]dx
 \geq  \limsup_{n \to \infty} \int _{\mathbb{R}^N} [-\varphi_{n}(x)]dx.
\end{equation}

For simplicity in the calculus we introduce the following notation:
\begin{itemize}
\item [(i)]  $\varphi_{n}^{1}(x) = \frac{ F(x, w_{n}(x)
+ t_{n}v(x)) - F(x, w_{n}(x))}{ t_{n}}$;
\item [(ii)]  $\varphi_{n}^{2}(x)=  c_{1}
|v(x)|[h(|w_{n}(x)+ t_{n}v(x)|) + h(|w_{n}(x)|)]$.
\end{itemize}
With these notation, we have $\varphi_{n}(x)= -
\varphi_{n}^{1}(x) + \varphi_{n}^{2}(x)$.

Now we prove the existence of limit $b =  \lim _{n
\to \infty} \int_{\mathbb{R}^N} \varphi_{n}^{2}(x)dx$.
Using the facts that the inclusion $X \hookrightarrow
L^p (\mathbb{R}^N)$ is continuous and $\|w_{n} - u\|
\to 0$, we get $\|w_{n} - u\|_{p} \to 0$.
Using \cite[Theorem IV.9]{Brez},  there exist a
positive function $g \in L^p (\mathbb{R}^N)$, such that
$|w_{n}(x)| \leq g(x)$ a.e. $x \in \mathbb{R}^N$. Considering
that  the function $h$ is monotone increasing, we get
$$
|\varphi_{n}^{2}(x)| \leq c_{1}|v(x)|[h(g(x)+ |v(x)|) + h(g(x))], \ \ {\rm a.e. } \
x\in \mathbb{R}^N.
$$
Moreover, $\varphi_{n}^{2}(x) \to 2c_{1}|v(x)|h(|u(x)|)$
for a.e. $x \in \mathbb{R}^N$.  Thus, using the Lebesque
dominated convergence theorem, we have
\begin{equation}\label{fontos}
b=  \lim _{n \to \infty} \int_{\mathbb{R}^N} \varphi_{n}^{2}(x)dx
 =  \int_{\mathbb{R}^N} 2c_{1}|v(x)|h(|u(x)|) dx.
\end{equation}
If we denote by
$I_{1}=  \limsup _{n \to\infty} \int _{\mathbb{R}^N}
[- \varphi_{n}(x)]dx$, then using
(\ref{iranym}) and (\ref{fontos}), we have
\begin{equation}\label{jobbold}
I_{1}=  \limsup _{n \to \infty} \int
_{\mathbb{R}^N} [- \varphi_{n}(x)]dx = \Phi^{0}(u;v) - b.
\end{equation}
Next we estimate the expression $I_{2}=  \int
_{\mathbb{R}^N}  \limsup _{n \to \infty}[-
\varphi_{n}(x)]dx$. We have the inequality
\begin{equation}\label{fatou2}
  \int_{\mathbb{R}^N}  \limsup _{n \to\infty}[\varphi^{1}_{n}(x)]dx
-  \int_{\mathbb{R}^N}  \lim _{n \to \infty}\varphi^{2}_{n}(x)dx \geq I_{2}.
\end{equation}
Using the fact that $w_{n}(x) \to u(x)$ a.e. $x \in \mathbb{R}^N$ and
$t_{n} \to 0^{+}$, we get
$$
\int _{\mathbb{R}^N}  \lim _{n
\to \infty} \varphi^{2}_{n}(x)dx = 2c_{1}
\int_{\mathbb{R}^N}|v(x)| h(|u(x)|)dx.
$$
On the other hand,
\begin{align*}
\int _{\mathbb{R}^N}  \limsup _{n \to \infty} \varphi_{n}^{1}(x)dx
&\leq  \int _{\mathbb{R}^N}  \limsup_{y\to u(x),\, t \to 0^{+}}
\frac{ F(x,y + t v(x)) - F(x, y)}{ t}dx \\
&= \int_{\mathbb{R}^N} F_{2}^{0}(x,u(x); v(x))dx.
\end{align*}
Using relations (\ref{fatou1}), (\ref{jobbold}),
(\ref{fatou2}) and the above estimates, we obtain the desired
result.
\end{proof}


\section{The Palais-Smale and Cerami compactness condition}

In this section we study the situation when the function $\Psi$
satisfies the $(PS)_{c}$ and $(CPS)_{c}$ conditions. We have the
following result.

\begin{proposition}\label{pskor}
Let $(u_{n}) \subset X$ be a
$(PS)_{c}$ sequence for the function $\Psi : X \to\mathbb{R}$.
If the conditions (F1) and (F2) are
fulfilled, then the sequence $(u_{n})$ is bounded in $X$.
\end{proposition}

\begin{proof}
Because $(u_{n}) \subset X$ is a $(PS)_{c}$
sequence for the function $\Psi$, we have $\Psi(u_{n}) \to c$ and
$\lambda_{\Psi}(u_{n}) \to 0$. From the condition
$\Psi(u_{n}) \to c$ we get $c + 1 \geq \Psi(u_{n})$ for sufficiently
large $n \in \mathbb{N}$.

 Because $\lambda_{\Psi}(u_{n}) \to 0$,
 $\|u_{n}\| \geq  \|u_{n}\|\lambda_{\Psi}(u_{n}) $
for every sufficiently large $n \in \mathbb{N}$. From the
definition of $\lambda_{\Psi}(u_{n})$ results the existence of an
element $z^{\star}_{u_{n}} \in
\partial \Psi(u_{n})$, such that
$\lambda_{\Psi}(u_{n})=\|z^{\star}_{u_{n}}\|_{\star}$. For every
$v \in X$, we have $|z^{\star}_{u_{n}}(v)| \leq
\|z^{\star}_{u_{n}}\|_{\star} \|v\|$, therefore
$\|z^{\star}_{u_{n}}\|_{\star} \|v\| \geq - z^{\star}_{u_{n}}(v)$.
If we take $v=u_{n}$, then $\|z^{\star}_{u_{n}}\|_{\star}
\|u_{n}\| \geq - z^{\star}_{u_{n}}(u_{n}).$

 Using the properties
$\Psi^{0}(u, v)= \max \{ z^{\star}(v) :z^{\star} \in
\partial \Psi(u) \ \}$ for every $v \in X$,
we have $- z^{\star}(v) \geq - \Psi^{0}(u,
v) $ for all $z^{\star} \in
\partial \Psi(u)$ and $v \in X$. If we take $u=v=u_{n}$ and
$z^{\star}=z^{\star}_{u_{n}}$, we get $-
z_{u_{n}}^{\star}(u_{n}) \geq - \Psi^{0}(u_{n}, u_{n})$.
Therefore,  for every $\alpha > 0$, we have
$$
\frac{1}{ \alpha} \|u_{n}\| \geq \frac{1}{ \alpha}
 \|z^{\star}_{u_{n}}\|_{\star} \|u_{n}\| \geq -
 \frac{ 1}{ \alpha} \Psi^{0}(u_{n}, u_{n}).
$$
When we add the above inequality with $c + 1 \geq \Psi(u_{n})$, we
obtain
$$
c + 1 + \frac{ 1}{ \alpha}\|u_{n}\| \geq \Psi(u_{n})
- \frac{ 1}{\alpha} \Psi^{0}(u_{n};u_{n}).
$$
Using the above inequality,  $\Psi^{0}(u, v)
\leq \langle A(u), v \rangle + \Phi^{0}(u, - v)$, and Proposition
\ref{clarkeder} we get
\begin{align*}
&c + 1 + \frac{1}{\alpha} \|u_{n}\| \\
&\geq \Psi(u_{n}) -\frac{1}{\alpha} \Psi^{0}(u_{n};u_{n})\\
&= \frac{1}{p} \langle A(u_{n}), u_{n}\rangle -\Phi(u_{n})
- \frac{ 1}{\alpha}\left( \langle A(u_{n}), u_{n} \rangle
+ \Phi^{0}(u_{n};-u_{n}) \right)\\
&\geq  (\frac{1}{p} - \frac{1}{\alpha})\langle A(u_{n}), u_{n} \rangle
-\int_{\mathbb{R}^N} \big[ F(x, u_{n}(x))
+ \frac{1}{\alpha} F_{2}^{0}(x,u_{n}(x); - u_{n}(x) )\big]dx \\
&\geq (\frac{1}{p} -\frac{1}{\alpha})\langle A(u_{n}), u_{n} \rangle
- \frac{1}{\alpha} \int_{\mathbb{R}^N}g(u_{n}(x))dx .
\end{align*}
The relation $ \lim_{|u|\to \infty} \frac{ g(u)}{ |u|^p }=
\lambda$ assures the existence of a constant $M$, such that
$\int_{\mathbb{R}^N}g(u_{n}(x))dx \leq M + \lambda
\int_{\mathbb{R}^N} |u_{n}(x)|^p dx$. We use again that the
inclusion $X \hookrightarrow L^p (\mathbb{R}^N)$ is continuous,
that $a(u)= \frac{ 1}{p}\langle A(u), u \rangle$ and that $$
a(u)=\|u\|^p  \langle A(\frac{ u}{ \|u\|}), \frac{ u}{ \|u\|}
\rangle \geq \kappa(1) \|u\|^p, $$ to obtain
\begin{align*}
c + 1 + \|u_{n}\|
&\geq (\frac{ 1}{ p} - \frac{ 1}{\alpha}) \langle A(u_{n}), u_{n} \rangle
 -\frac{ \lambda C^p (p)}{ \alpha}\|u_{n}\|^p  - \frac{ M}{ \alpha}\\
&\geq \frac{ \kappa(1)(\alpha - p) - \lambda C^p (p)}{ \alpha}\|u_{n}\|^p
- \frac{M}{ \alpha}.
\end{align*}
 From the above inequality, it results that the sequence $(u_{n})$ is
bounded.
\end{proof}

\begin{proposition}\label{cpskor}
If conditions (F1), (F2') and (F4) hold,
then every $(CPS)_{c} (c>0)$ sequence $(u_{n}) \subset X$ for the
function $\Psi:X \to \mathbb{R}$ is bounded in $X$.
\end{proposition}

\begin{proof} Let $(u_{n}) \subset X$ be a $(CPS)_{c}$
$(c > 0)$ sequence for the function $\Psi$, i.e. $\Psi(u_{n}) \to
c$ and $(1+\|u_{n}\|)\lambda_{\Psi}(u_{n}) \to 0$. From
$(1+\|u_{n}\|)\lambda_{\Psi}(u_{n}) \to 0$, we get
$\|u_{n}\|\lambda_{\Psi}(u_{n}) \to 0$ and $\lambda_{\Psi}(u_{n})
\to 0$. As in Proposition \ref{pskor}, there exists
$z_{u_{n}}^{\star} \in
\partial \Psi(u_{n})$ such that
$$
\frac{ 1}{p}\|z_{u_{n}}^{\star}\|_{\star}\|u_{n}\| \geq-\Psi^{0}(u_{n};
\frac{ 1}{ p} u_{n}).
$$
 From this inequality, Proposition \ref{clarkeder}, condition
(F2') and the property $\Psi^{0}(u;v) \leq \langle
Au, v \rangle + \Phi^{0}(u; -v)$  we get
\begin{align*}
c+1 &\geq \Psi(u_{n}) - \frac{ 1}{ p}\Psi^{0}(u_{n}; u_{n}) \\
&\geq a(u_{n}) - \Phi(u_{n}) - \frac{ 1}{ p} \left[ \langle Au_{n},
u_{n}\rangle + \Phi^{0}(u_{n}; - u_{n})\right] \\
&\geq -  \int_{\mathbb{R}^N} \big[ F(x, u_{n}(x)) +
\frac{ 1}{ p} F_{2}^{0}(x, u_{n}(x); - u_{n}(x))\big]dx \\
&\geq C \|u_{n}\|_{\alpha}^{\alpha}.
\end{align*}
Therefore, the sequence $(u_{n})$ is bounded in $L^{\alpha}(\mathbb{R}^N)$.
 From the condition (F4) follows that, for every
 $\varepsilon > 0$, there exists $c(\varepsilon) > 0$, such that for
  a.e. $x \in \mathbb{R}^N$,
$$ F(x, u(x)) \leq \frac{\varepsilon}{ p}|u(x)|^p +
\frac{c(\varepsilon)}{ r}|u(x)|^{r}. $$ After integration, we
obtain $$ \Phi(u) \leq \frac{ \varepsilon}{p}\|u\|_{p}^p +
\frac{c(\varepsilon)}{r} \|u\|_{r}^{r}. $$ Using the above
inequality, the expression of $\Psi$, and
 $\|u\|_{p}\leq C(p)\|u\|$, we obtain
$$ \frac{\kappa(1)- \varepsilon C^p (p)}{ p}\|u\|^p  \leq \Psi(u)+
\frac{ c(\varepsilon)}{ r}\|u\|_{r}^{r} \leq c + 1 +
\|u\|^{r}_{r}. $$ Now, we study the behaviour of the sequence
$(\|u_{n}\|_{r})$. We have the following two cases:
\begin{itemize}
\item[(i)] If $r=\alpha$, then it is easy to see that the sequence $(\|u_{n}\|_{r})$
is bounded in $\mathbb{R}$.

\item[(ii)] If $r \in (\alpha, p^{\star})$ and
$\alpha > p^{\star} \frac{ r -p}{p^{\star} - p}$, then we have
$$
\|u\|_{r}^{r} \leq \|u\|_{\alpha}^{(1-s)\alpha} \cdot
\|u\|_{p^{\star}}^{sp^{\star}},
$$
where $r=(1-s)\alpha +sp^{\star}, s \in (0,1)$.
\end{itemize}
Using the inequality $\|u\|_{p^{\star}}^{sp^{\star}}\leq
C^{sp^{\star}}(p)\|u\|^{sp^{\star}}$, we obtain
\begin{equation}\label{cpsec}
\frac{ \kappa(1)- \varepsilon C^p (p)}{p}\|u\|^p
\leq c+1+\frac{c(\varepsilon)}{r}\|u\|_{\alpha}^{(1-s)\alpha}\|u\|^{sp^{\star}}.
\end{equation}
When in the inequality (\ref{cpsec}) we take $\varepsilon \in
\left(0, \frac{ \kappa(1)}{ C^p (p)}
\right)$ and use b), we obtain that the sequence $(u_{n})$ is
bounded in $X$.
\end{proof}

The main result of this section is as follows.

\begin{theorem}\label{mainps}
\begin{enumerate}
\item If conditions (F1), (F1'), and  (F2)--(F4) hold, then
$\Psi$ satisfies the $(PS)_{c}$ condition for every $c \in \mathbb{R}$.

\item If conditions (F1), (F1'), (F2'), (F3), and (F4) hold,
then $\Psi$ satisfies the $(CPS)_{c}$ condition for every $c>0$.
\end{enumerate}
\end{theorem}

\begin{proof}
Let $(u_{n}) \subset X$ be a $(PS)_{c} (c\in \mathbb{R})$ or a
$(CPS)_{c} (c > 0)$ sequence for the function $\Psi(u_{n})$.
Using  Propositions \ref{pskor}
\ref{cpskor}, it follows that $(u_{n})$ is a bounded sequence in $X$.
As $X$ is reflexive Banach space, the existence of an
element $u \in X$ results, such that $u_{n} \rightharpoonup u$ weakly in
$X$. Because the inclusions
$X \hookrightarrow L^{r}(\mathbb{R}^N)$ is compact, we have that
$u_{n} \to u$ strongly in $L^{r}(\mathbb{R}^N)$.

Next we estimate the expressions
$I_{n}^{1}=\Psi^{0}(u_{n}; u_{n}- u)$ and $I_{n}^{2}=\Psi^{0}(u;
u- u_{n})$. First we estimate the expression
$I_{n}^{2}=\Psi^{0}(u; u- u_{n})$. We know that $\Psi^{0}(u;
v)=\max \{z^{\star}(v):z^{\star} \in
\partial \Psi(u)\},\; \forall\, v\in X$. Therefore, there exists
$z_{u}^{\star} \in \partial \Psi(u)$, such that $\Psi^{0}(u; v)=
z_{u}^{\star}(v)$ for all $v \in X$.  From the above relation and
from the fact that $u_{n} \rightharpoonup u$ weakly in $X$, we get
$\Psi^{0}(u; u- u_{n})=z^{\star}_{u}(u - u_{n})\to 0$.

Now, we estimate the expression $I_{n}^{1}=\Psi^{0}(u_{n}; u_{n}-
u)$. From $\lambda_{\Psi}(u_{n}) \to 0$ follows the
existence of a positive real numbers sequence $\mu_{n} \to
0$, such that $\lambda_{\Psi}(u_{n}) \leq  \mu_{n}$. If we use the
Remark \ref{psmejegyzes}, we get $\Psi^{0}(u_{n},u_{n}-u)+
\mu_{n}\|u_{n}- u\| \geq 0$.

Now, we estimate the expression $I_{n}=\Phi^{0}(u_{n}; u- u_{n}) +
\Phi(u; u -u_{n})$. For the simplicity in calculus we introduce
the notations $h_{1}(s)=|s|^{p-1}$ and $h_{2}(s)=|s|^{r}$. For
this we observe that if we use the continuity of the functions
$h_{1}$ and $h_{2}$, the condition (F4) implies that for every
$\varepsilon > 0$, there exists a $c(\varepsilon) > 0$ such that
\begin{equation}\label{psegyen1}
\max \left\{|\underline{f}(x, s)|, |\overline{f}(x, s)| \right\}
\leq \varepsilon h_{1}(s) + c(\varepsilon)h_{2}(s),
\end{equation}
 for a.e. $x \in \mathbb{R}^N$ and for all
$s \in \mathbb{R}$. Using this relation and Proposition
\ref{clarkeder}, we have
\begin{align*}
I_{n}&=\Phi^{0}(u_{n}; u- u_{n}) +\Phi(u; u -u_{n}) \\
&\leq  \int _{\mathbb{R}^N} \left
[ F^{0}_{2}(x, u_{n}(x); u_{n}(x)-u(x)) +  F^{0}_{2}(x, u(x);
u(x)-u_{n}(x))\right ]dx\\
&\leq  \int_{\mathbb{R}^N} \left[ \underline{f}(x, u_{n}(x))(u_{n}(x)-u(x))
+ \overline{f}(x, u(x)) (u(x)-u_{n}(x)) \right]dx \\
&\leq 2\varepsilon  \int _{\mathbb{R}^N}\left[
h_{1}(u(x))+ h_{1}(u_{n}(x))\right]|u_{n}(x) - u(x)|dx\\
&\quad+ 2c_{\varepsilon}  \int _{\mathbb{R}^N}
 \left[(h_{2}(u(x))+ h_{2}(u_{n}(x))\right]|u_{n}(x) - u(x)|dx.
\end{align*}
Using H\"{o}lder inequality and that the
inclusion $X \hookrightarrow L^p (\mathbb{R}^N)$ is continuous,
we get
\begin{align*}
I_{n}&\leq 2\varepsilon C(p)\|u_{n} -u\|(\|h_{1}(u)\|_{p'}
+ \|h_{1}(u_{n})\|_{p'}) \\
&\quad + 2c(\varepsilon)\|u_{n}-u\|_{r}(\|h_{2}(u)\|_{r'} +
\|h_{2}(u_{n})\|_{r'}),
\end{align*}
where $\frac{1}{ p} + \frac{ 1}{p'}=1$ and $\frac{ 1}{ r} +\frac{ 1}{ r'} =1$.
Using the fact that the inclusion $X \hookrightarrow
L^{r}(\mathbb{R}^N)$ is compact, we get that $\|u_{n}-u\|_{r}
\to 0$ as $n \to \infty$.  For $\varepsilon
\to 0^{+}$ and $n \to \infty$ we obtain that
$I_{n} \to 0$.

Finally, we use the inequality $\Psi^{0}(u; v) \leq  \langle A(u),
v \rangle + \Phi^{0}(u; -v) $. If we replace $v$ with $-v$, we get
$\Psi^{0}(u, -v) \leq -\langle A(u), v \rangle + \Phi^{0}(u; v)$,
therefore $\langle A(u), v \rangle  \leq \Phi^{0}(u; v) -
\Psi^{0}(u, -v)$.

In the above inequality we replace $u$ and $v$  by $u=u_{n}, v=u-u_{n}$
then $u=u, v=u_{n}-u$ and we get
\begin{gather*}
\langle A(u_{n}), u - u_{n} \rangle \leq \Phi^{0}(u_{n}, u-
u_{n}) - \Psi^{0}(u_{n}; u_{n} - u),\\
\langle A(u),u_{n} - u \rangle  \leq \Phi^{0}(u, u_{n} - u) - \Psi^{0}(u, u-u_{n}).
\end{gather*}
Adding these relations, we have the following key inequality:
\begin{align*}
&\|u_{n} - u\|\kappa(u_{n}-u) \\
&\leq \langle A(u_{n}-u), u_{n}- u\rangle \\
&\leq \left[\Phi^{0}(u_{n}; u- u_{n}) + \Phi(u; u-u_{n})\right]
- \Psi^{0}(u_{n}; u_{n}- u)-\Psi^{0}(u; u- u_{n})\\
&=I_{n}-I_{n}^{1}- I_{n}^{2}.
\end{align*}
Using the above relation and the estimations of $I_{n}, I_{n}^{1}$
and $I_{n}^{2}$, we obtain
$$
 \|u_{n}-u\|\kappa(u_{n}-u) \leq I_{n} +
\mu_{n}\|u_{n} - u\| - z_{u}^{\star}(u_{n} -u).
$$
If $n\to \infty$, from the above inequality we obtain the
assertion of the theorem.
\end{proof}

\begin{remark}\label{reszter} \rm
It is important to observe  then the above results remain true if
we replace the Banach space $X$ with every closed subspace $Y$ of
$X$.
\end{remark}

\section{Proof of Theorem \ref{thm2}}

In this section we prove the main result of this paper, whihc is a result
of Mountain Pass type. First we prove
that the critical points of the function $\Psi: X \to
\mathbb{R}$ defined by $\Psi(u)=a(u)-\Phi(u)$ are solutions of
problem \eqref{P}.

\begin{proposition}\label{kritpont}
If $0 \in \partial \Psi(u)$, then $u$ solves the problem \eqref{P}.
\end{proposition}

\begin{proof}
 Because $0 \in \partial \Psi(u)$, we have
$\Psi^{0}(u;v) \geq 0$ for every $v \in X$. Using the Proposition
\ref{clarkeder} and a property of Clarke derivative we obtain
\begin{align*}
0\leq \Psi^{0}(u;v)
&\leq \langle u, v \rangle   + (-\Phi)^{0}(u;v)\\
&= \langle A(u), v \rangle   + \Phi^{0}(u;-v) \\
&\leq \langle A(u), v \rangle  +  \int_{\mathbb{R}^N} F_{2}^{0}(x,u(x),-v(x))dx,
\end{align*}
 for every $v \in X$.
\end{proof}

\begin{proof}[Proof of Theorem \ref{thm2}]
Using (1) in Theorem \ref{mainps}, and conditions (F1)--(F4), it
follows that the functional
$\Psi(u)=\frac{ 1}{ p} \langle A(u), u\rangle -\Phi(u)$ satisfies
the $(PS)_{c}$ condition for every $c \in \mathbb{R}$.
 From Proposition \ref{mountaincer} we verify the
following geometric hypotheses:
\begin{gather}\label{mountain1}
\exists \, \alpha, \rho > 0, \quad \mbox{such  that } \Psi(u)
\geq \beta \mbox{ on } B_{\rho}(0)=\{ u \in X :\|u\| =\rho\},\\
\label{mountain2}
\Psi(0)=0 \quad\mbox{and there exists $v \in H \setminus B_{\rho}(0)$
such  that $\Psi(v) \leq 0$}.
\end{gather}

For the proof of relation (\ref{mountain1}), we use the relation
(F4), i.e. $|f(x,s)| \leq \varepsilon |s|^{p-1}+c(\varepsilon)|s|^{r-1}$.
Integrating this inequality and using
that the inclusions $X \hookrightarrow L^p (\mathbb{R}^N)$,
$X\hookrightarrow L^{r}(\mathbb{R}^N)$ are continuous, we get that
\begin{align*}
\Psi(u) &\geq \frac{ \kappa(1) - \varepsilon
C(p)}{ p} \langle A(u), u \rangle -\frac{ 1}{r}c(\varepsilon)C(r)\|u\|^{r}_{r}\\
&\geq \frac{ \kappa(1) - \varepsilon
C(p)}{ p} \|u\|^p  - \frac{1}{ r}c(\varepsilon)C(r)\|u\|^{r}.
\end{align*}
 The right member of the inequality is a function
 $\chi: \mathbb{R}_{+}\to \mathbb{R}$ of the form $\chi(t)= At^p - Bt^{r}$,
where $A=\frac{ \kappa(1) - \varepsilon C(p)}{ p}$,
$B =\frac{ 1}{r}c(\varepsilon)C(r) $. The function $\chi$ attains its global
maximum in the point $t_{M}= (\frac{pA}{ r B})^{\frac{ 1}{r-p}}$.
 When we take $\rho=t_{M}$ and $\beta \in ]0, \chi(t_{M})]$, it is easy to
see that the condition (\ref{mountain1}) is fulfilled.

 From (F5) we have $\Psi(u) \leq \frac{1}{ p} \langle A(u), u \rangle + c^{\star}
\|u\|_{p}^p - c^{\star}\|u\|_{\alpha}^{\alpha}$. If we fix an
element $v \in H \setminus\{0\}$ and in place of $u$ we put $tv$,
then we have
$$
\Psi(tv) \leq (\frac{ 1}{ p} \langle
A(v), v \rangle + c^{\star} \|v\|_{p}^p )t^p  -
c^{\star}t^{\alpha}\|v\|_{\alpha}^{\alpha}.
$$
 From this we see that if $t$ is large enough, $t v \notin B_{\rho}(0)$ and
$\Psi(tv) < 0.$ So, the condition (\ref{mountain2}) is satisfied and
Proposition \ref{mountaincer} assures the existence of a nontrivial
critical point of $\Psi$.

Now when we use (2) in Theorem \ref{mainps},
from conditions  (F1), (F2'), (F3), and (F4), we get that the function $\Psi$
satisfies the condition $(CPS)_{c}$ for every $c >0$. We use again the
Proposition \ref{mountaincer}, which assures the existence of a
nontrivial critical point for the function $\Psi$. It is sufficient
to prove only the relation $(\ref{mountain2})$, because
$(\ref{mountain1})$ is proved in the same way.

To prove the relation $(\ref{mountain2})$ we fix an element $u \in
X$ and we define the function $h:(0, + \infty) \to \mathbb{R}$ by
$h(t)= \frac{ 1}{ t}F(x,t^{1/p}u) - C \frac{ p}{\alpha- p}
t^{\frac{\alpha}{ p}-1 }|u|^{\alpha}$. The function $h$ is locally
Lipschitz. We fix a number $t >1$, and from the Lebourg's main
value theorem follows the existence of an element $\tau \in (1,t)$
such that $$ h(t)-h(1) \in \partial_{t}h(\tau)(t-1), $$ where
$\partial_{t}$ denotes the generalized gradient of Clarke with
respect to $t \in \mathbb{R}$. From the Chain Rules we have $$
\partial_{t}F(x,t^{1/p}u)\subset \frac{ 1}{ p}
\partial F(x, t^{1/p}u) t^{\frac{1}{p}-1}u.
$$
Also we have
$$
\partial_{t} h(t) \subset - \frac{ 1}{t^{2}}F(x, t^{1/p}u) + \frac{1}{ t}
\partial F(x, t^{1/p}u) t^{\frac{1}{p}-1}u - C
t^{\frac{\alpha}{p}-2}|u|^{\alpha}.
$$
Therefore,
\begin{align*}
h(t)-h(1) &\subset \partial_{t}h(\tau)(t-1)\\
&\subset - \frac{ 1}{ t^{2}} \left[F(x,t^{1/p}u) -
 t^{1/p}u \partial F(x, t^{1/p}u) +C|t^{1/p}u|^{\alpha}\right](t-1).
\end{align*}
Using the relation (F2'), we obtain that $h(t) \geq h(1)$ ;
therefore,
$$
\frac{ 1}{ t} F(x, t^{1/p}u)-C \frac{ p }{ \alpha - p}
t^{\frac{\alpha}{p}-1}|u|^{\alpha} \geq F(x, u)
- C\frac{ p}{ \alpha - p}|u|^{\alpha}.
$$
 From this inequality, we get
\begin{equation}
F(x, t^{1/p}) \geq t F(x,u) + C \frac{
p}{ \alpha - p}[t^{\alpha/p} -t]|u|^{\alpha},
\end{equation}
for every $t >1$ and $u \in \mathbb{R}$.
Let us fix an element $u_{0} \in X \setminus \{0\}$; then for every
$t>1$, we have
\begin{align*}
\Psi(t^{1/p}u_{0}) &= \frac{1}{ p} \langle A(t^{1/p}u_{0}),
t^{1/p}u_{0} \rangle -\int_{\mathbb{R}^N}F(x, t^{1/p}u_{0}(x))dx\\
&\quad \leq \frac{ t}{ p} \langle Au_{0}, u_{0} \rangle - t
\int_{\mathbb{R}^N} F(x, u_{0}(x))dx -C \frac{ p}{ \alpha - p}
[t^{\alpha/p} - t]\|u_{0}\|_{\alpha}^{\alpha}.
\end{align*}

If $t$ is sufficiently large, then for $v_{0}=t^{1/p}u_{0}$
we have $\Psi(v_{0}) \leq 0$. This completes the proof.
\end{proof}

\section{Applications}

In the first two examples we suppose that $X$ is a Hilbert space
with the inner product $\langle \cdot, \cdot \rangle$.


 Let $f: \mathbb{R}^N \times
\mathbb{R} \to \mathbb{R}$ be a measurable function as in
the introduction of this paper.

\begin{application} \label{appl1} \rm
We consider the function $V \in \mathcal{C}(\mathbb{R}^N,
\mathbb{R})$ which satisfies the following conditions:
\begin{itemize}
  \item [(a)] $V(x) >0$ for all $x \in \mathbb{R}^N$
  \item [(b)] $V(x) \to + \infty$ as $|x| \to +  \infty$.
\end{itemize}
Let $X$ be the Hilbert space defined by
$$
X= \{ u \in H^{1}(\mathbb{R}^N) :\int (|\nabla u(x)|^{2} +
V(x)|u(x)|^{2}) dx < \infty\},
$$
with the inner product
$$
\langle u, v \rangle = \int (\nabla u \nabla v + V(x) u v)dx.
$$
It is well known that if the conditions (a) and (b) are fulfilled
then the inclusion $X \hookrightarrow L^{2}(\mathbb{R}^N)$ is
compact \cite{grosters}, therefore the condition
(F1') is satisfied.
\end{application}

Now we formulate the problem.

{\it Find a positive  $u \in X$ such that for every $v \in X$ we
have}
\begin{equation}
 \int_{\mathbb{R}^N} (\nabla u\nabla v + V(x) u v)dx +
\int_{\mathbb{R}^N} F^{0}_{2}(x, u(x);-v(x) )dx \geq 0.
\label{P1}
\end{equation}

We have the following result.

\begin{corollary}\label{schrodinger}
If conditions (F1), (F2'), (F3), (F4),
and (a), (b) hold, the problem \ref{P1} has a nontrivial
positive solution.
\end{corollary}

\begin{proof} We replace the function $f$ by $f_{+}:
\mathbb{R}^N \times \mathbb{R} \to \mathbb{R}$ defined by
\begin{equation}\label{poz}
f_{+}(x, u)=\begin{cases}
 f(x, u) &\mbox{if }  u\geq 0;\\
0, & \mbox{if } u < 0
\end{cases}
\end{equation}
and use (2) in Theorem \ref{thm2}.
\end{proof}

\begin{remark}\label{gazolla} \rm
The above result improves a result in Gazolla-R\u{a}dulescu \cite{GazRad}.
\end{remark}

\begin{application} \label{app2} \rm
 Now, we consider $Au:= -\bigtriangleup u + |x|^{2}u$ for $u \in D(A)$,
 where
$$
D(A):= \{ u\in L^{2}(\mathbb{R}^N) :Au \in L^{2}(\mathbb{R}^N)\}.
$$
Here $|\cdot|$ denotes the Euclidian norm of $\mathbb{R}^N$.
In this case the Hilbert space $X$ is defined by
$$
X= \{ \ u \in L^{2}(\mathbb{R}^N) :  \int
_{\mathbb{R}^N} (|\nabla u|^{2} + |x|^{2} u^{2}) dx <\infty\},
$$
with the inner product
$$
\langle u, v \rangle = \int_{\mathbb{R}^N} (\nabla u
\nabla v + |x|^{2} u v)dx.
$$
The inclusion $X \hookrightarrow L^{s}(\mathbb{R}^N)$ is compact
for $s \in [2, \frac{ 2N}{ N-2})$, see
Kavian \cite[Exercise 20, pp. 278]{Kavian}. Therefore, the
condition (F1') is satisfied.
\end{application}

Now, we formulate the next problem.

{\it Find a positive  $u \in X$ such that for every $v \in X$ we
have}
\begin{equation}
 \int_{\mathbb{R}^N} (\nabla u \nabla v + |x|^{2} u v)dx +
\int_{\mathbb{R}^N} F^{0}_{2}(x, u(x);-v(x) )dx \geq 0.
\label{P2}
\end{equation}
\begin{corollary}\label{oscilatie}
If (F1), (F2), (F3), and (F4) hold, then problem \eqref{P2} has a positive
solution.
\end{corollary}
The proof of this corollary is similar to that of Corollary \ref{schrodinger}.

\begin{remark}\label{varga} \rm
This result improves a result from Varga \cite{varga},
where the condition (F5) was used.
\end{remark}

\begin{application} \label{app3} \rm
In this example we suppose that $G$
is a subgroup  of the group $O(N)$.
Let $\Omega$ be an unbounded domain in $\mathbb{R}^N$ with
smooth boundary $\partial \Omega$, and the elements of $G$ leave
$\Omega$ invariant, i.e. $g(\Omega)=\Omega$ for every $g \in G$.
We suppose that $\Omega$ is compatible with $G$, see the book of
Willem \cite{W} Definition 1.22. The action of $G$ on
$X=W_{0}^{1,p}$ is defined by $$gu(x):=u(g^{-1}x).$$ The subspace
of invariant function $X^{G}$ is defined by
$$
X^{G}:= \{u \in X :gu=u, \ \forall g \in G \ \}.
$$
The norm on $X$ is defined by
$$\|u\|= \Big(  \int
_{\Omega} (|\nabla u |^p  + |u|^p )dx \Big)^{1/p}.
$$
If $\Omega$ is compatible with $G$, then the embeddings
$X\hookrightarrow L^{s}(\Omega)$, with $p < s < p^{\star}$ are
compact, see the paper of  Kobayashi and  Otani
\cite{kobotani}. Therefore the condition (F2'') is satisfied.
\end{application}

We consider the potential $a: X \to \mathbb{R}$ defined by
$a(u)= \frac{ 1}{ p} \|u\|^p $. This
function is $G$-invariant because the action of $G$ is isometric
on $X$. The Gateaux differential $A:X \to X^{\star}$  of
the function $a: X \to \mathbb{R}$ is given by
$$
\langle Au, v \rangle =  \int _{\Omega}
\left(|\nabla u|^{p-2} \nabla u \nabla v + |u|^{p-2} u v\right)dx.
$$
The operator $A$ is homogeneous of degree $p-1$ and
strongly monotone, because $p \geq 2$.

Now, we formulate the following problem.

{\it Find   $u \in X \setminus \{0\}$ such that for every $v \in X$ we have}
\begin{equation}
 \int _{\Omega} \left(|\nabla u|^{p-2}
\nabla u \nabla v + |u|^{p-2} u v \right)dx +
\int_{\Omega} F^{0}_{2}(x, u(x);-v(x) )dx \geq 0.
\label{P3}
\end{equation}
We have the following result.


\begin{corollary}\label{plaplace1}
If we suppose that the condition (F6) is true, then the following
assertions hold.
\begin{itemize}
\item[(a)]  If conditions (F1)--(F5) are fulfilled,
then  problem \eqref{P} has a nontrivial solution.

\item[(b)] If conditions (F1), (F2'), (F3), and (F4) are fulfilled,
then problem \eqref{P} has a nontrivial symmetric solution.
\end{itemize}
\end{corollary}

\begin{remark}\label{plapmeg1} \rm
The result (a) from Corollary \ref{plaplace1} is similar to the a
result obtained by Kobayashi,  \^{O}tani \cite{kobotani}, but
the difference is that in the paper \cite{kobotani} the ``Principle
of Symmetric Criticality" was used for Szulkin type functional,
see \cite{szulkin}.
\end{remark}

\begin{application} \label{aap4} \rm
In this case we consider
$\Omega=\tilde{\Omega} \times \mathbb{R}^N, N-m \geq 2, \tilde{\Omega}
\subset \mathbb{R}^{m} (m \geq 1)$ is open bounded and
$2 \leq p \leq N$.  We consider the Banach space $X=W_{0}^{1,p}(\Omega)$
with the norm $\|u\|= ( \int_{\Omega} |\nabla u|^p )^{1/p}$.
Let $G$ be a subgroup of $O(N)$
defined by $G=id^{m} \times O(N-m)$. The action of $G$ on $X$ is
defined by $gu(x_{1}, x_{2})=u(x_{1}, g_{1} x_{2})$ for every
$(x_{1}, x_{2}) \in \tilde{\Omega} \times \mathbb{R}^{N-m}$ and
$g=id^{m} \times g_{1} \in G$. The subspace of invariant function
is defined by
$$
X^{G}=W_{0, G}^{1,p}=\{ u \in X : gu=u, \;\forall \, g \in G\}.
$$
The action of $G$ on $X$ is isometric, that is
$$ \|gu\|=\|u\|, \; \forall \, g \in G.
$$
If $2\leq p \leq N$, from a result of Lions \cite{Lion} follows
that the embeddings $X \hookrightarrow L^{s}(\Omega), p < s <
p^{\star}$ are compact. Therefore the condition
$(f_{2}^{\prime\prime})$ is true.
In this case condition (F6) will be replaced by
\begin{itemize}

\item[(F6')]  $f(x,y_{1}, u)=f(x,y_{2}, u)$ for every
$y_{1}, y_{2} \in \mathbb{R}^{N-m}$ ($N - m \geq 2$), $|y_{1}|=|y_{2}|$; i.e.,
the function $f(x, \cdot, u)$ is spherically symmetric on
$\mathbb{R}^{N-m}$.
\end{itemize}

We consider the potential $a: X \to \mathbb{R}$ defined by
$a(u)= \frac{ 1}{ p} \|u\|^p $. This
functional is $G$-invariant because the action of $G$ is isometric
on $X$. The Gateaux differential $A:X \to X^{\star}$  of
the functional $a: X \to \mathbb{R}$ is given by
$$
\langle Au, v \rangle =  \int _{\Omega} |\nabla
u|^{p-2} \nabla u \nabla v dx.
$$
The operator $A$ is homogeneous of degree $p-1$ and strongly monotone,
because $p \geq 2$.
\end{application}

Now, we formulate the following problem.

{\it Find   $u \in X \setminus \{0\}$ such that for every
$v \in X$ we have}
\begin{equation}
\int _{\Omega} |\nabla u|^{p-2}\nabla u \nabla v dx
+  \int_{\Omega} F^{0}_{2}(x,u(x);-v(x) )dx \geq 0.
\label{P4}
\end{equation}

We have the following result.

\begin{corollary}\label{sferical}
\begin{itemize}
\item[(a)] If conditions (F1)--(F5), and (F6) hold, then
problem \eqref{P4} has a nontrivial solution.
\item[(b)] If conditions (F1), (F2'), (F3), (F4), and
(F6') hold, then problem \eqref{P4} has a
nontrivial solution.
\end{itemize}
\end{corollary}

\subsection*{Acknowledgments}
The authors want to express their gratitude to Dr. Alexandru
Krist\'{a}ly for his helpful discussions.


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\end{document}