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%French
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\begin{document}

\begin{frontmatter}


\selectlanguage{english}
\title{Multiple symmetric solutions for a Neumann problem with lack of compactness}

\vspace{-2.6cm}

\selectlanguage{french}
\title{Solutions sym\'etriques multiples pour un probl\`eme de Neumann sans compacit\'e}

\selectlanguage{english}
\author[GM]{Giovanni Molica Bisci}
\ead{gmolica@unirc.it}
\author[VR1,VR2]{Vicen\c tiu R\u{a}dulescu}
\thanks[label3]{Correspondence address: Vicen\c tiu R\u{a}dulescu, University of Craiova,
Department of Mathematics,
200585 Craiova, Romania. E-mail: {\tt
vicentiu.radulescu@imar.ro}}
\ead{vicentiu.radulescu@imar.ro}
\ead[url]{http://www.inf.ucv.ro/\~{}radulescu}
\address[GM]{Dipartimento MECMAT, University of Reggio Calabria,
 Via Graziella, Feo di Vito, 89124 Reggio Calabria, Italy}
\address[VR1]{Institute of Mathematics ``Simion Stoilow" of the Romanian Academy,  014700
Bucharest, Romania}
\address[VR2]{Department of Mathematics, University of Craiova, Street A.I. Cuza No. 13, 200585 Craiova, Romania}

\begin{abstract}
% Text of abstract in English
The existence of multiple cylindrically symmetric solutions for a class of non-autonomous elliptic Neumann problems in a strip-like domain of the Euclidean space is investigated. The proof combines a recent compactness result and the Palais symmetric critically principle. A concrete application illustrates the main abstract result of this Note.
\vskip 0.5\baselineskip

\selectlanguage{francais}
% Text of abstract in French
\noindent{\bf R\'esum\'e} \vskip 0.5\baselineskip \noindent
On \'etudie l'existence de solutions multiples \`a sym\'etrie cylindrique pour une classe de probl\`emes elliptiques non autonomes de Neumann sans compacit\'e. La preuve combine un r\'esultat r\'ecent de compacit\'e et le principe de Palais de sym\'etrique critique. Une application met en \'evidence le r\'esultat principal de cette Note.
\end{abstract}
\end{frontmatter}

% now the Version fran\c{c}aise abr\'eg\'ee, if it exists
\selectlanguage{francais}
\section*{Version fran\c{c}aise abr\'eg\'ee}
Soit ${\mathcal O}\subset \RR^m$ un domaine born\'e et r\'egulier et soit le cylindre $\Omega:={\mathcal O}\times\RR^n$.
On consid\`ere l'espace des fonctions \`a sym\'etrie cylindrique  d\'efini par
 $$
 W^{1,p}_c(\Omega):=\{u\in W^{1,p}(\Omega):u(x,\cdot)\, \mbox{a\ sym\'etrie\ radiale,\ pour\ tout} \ x\in {\mathcal O}\}.
 $$

Dans cette Note on \'etudie
le probl\`eme non lin\'eaire
$$
\left\{
\begin{array}{ll}
-\Delta_p u+|u|^{p-2}u= \lambda \alpha(x,y)f(u) &\rm dans \, \Omega
\\ \partial u/\partial\nu=0&\rm sur \,
\partial \Omega,
\end{array}
\right. \eqno(P)
$$
o\`u $\Delta_pu=\mbox{div}\, (|\nabla u|^{p-2}\nabla u)$, $f:\RR\rightarrow\RR$ est une fonction continue, $\nu$ est la normale ext\'erieure \`a $\partial\Omega$ et $p>m+n$. On suppose que $\alpha\in L^1(\Omega)$ est une fonction non n\'egative \`a sym\'etrie
cylindrique telle que, pour un certain $\tau>0$,
$$\displaystyle\mbox{essinf}_{(x,y)\in {\mathcal O}\times B_n(0,\tau/2)} \alpha(x,y)>0\,.$$


Soit
$
B_{(1/2,1)}(n,p+1):=\displaystyle\int_{1/2}^{1}t^{n-1}(1-t)^{p}dt.
$
On d\'efinit $$
\sigma(n,p):=2^{p-n}(2^n-1);\qquad
g(p,N):=\frac{1+2^{n+p}NB_{(1/2,1)}(n,p+1)}{2^n};
\qquad
 \omega^{(n)}_\tau:=\tau^n\frac{\pi^{n/2}}{\Gamma \Big(\displaystyle 1+\frac{n}{2}\Big)},$$
et
$$
\kappa:=\left(\frac{p\tau^p}{\displaystyle\omega_\tau^{(n)}|{\mathcal O}|_m\left({\sigma(n,p)}+ \tau^p g(p,n)\right)}\right)^{1/p}\,.
$$

Soit
$
    \|\alpha\|_{({\mathcal O},\tau/2)}:=\displaystyle\int_{{\mathcal O}\times B_{n}(0,\tau/2)}\alpha(x,y)dxdy.
    $

Le r\'esultat principal de cette Note est le suivant.


\smallskip
\begin{theoreme} \label{th-1fr}
Supposons qu'il existe $\delta$ et $\gamma$ avec
$
\delta>\kappa \gamma>0
$ et
tels que les propri\'et\'es suivantes soient satisfaites$:$
\begin{itemize}
\item [$(\rm{h}_0)$] $F(\xi):=\displaystyle\int_0^\xi f(s)ds\geq 0$, pour tout $\xi\in [0,\delta]$$;$
\item [$(\rm{h}_1)$] si $$
\widetilde{\kappa}:=\frac{\kappa \|\alpha\|_{({\mathcal O},\tau/2)}}{pC_p^{p}\|\alpha\|_{L^1}}\,,$$ alors
$$
\displaystyle\frac{\displaystyle\max_{|\xi|\leq \gamma}F(\xi)}{\gamma^{p}}<\widetilde{\kappa}\frac{F(\delta)}{\delta^p}\,;
$$
\item [$(\rm{h}_2)$]
il existe $c_0>0$ et $0<s<p-1$ tels que pour tout $t\in\RR$, $$|f(t)|\leq
c_0(1+|t|^{s})\,.$$
\end{itemize}

Alors, pour tout $\lambda$ dans l'intervalle
$$
\Lambda_{(\gamma,\delta)}:=\left]\frac{\delta^p}{\displaystyle \kappa  \|\alpha\|_{({\mathcal O},\tau/2)} F(\delta)},\frac{\gamma^{p}}{\displaystyle pC_p^{p}\|\alpha\|_{L^1} \max_{|\xi|\leq \gamma} F(\xi)}\right[,
$$
\noindent le probl\`eme
$($P$)$ admet au moins trois solutions faibles \`a sym\'etrie cylindrique.
\end{theoreme}



\vspace{0.3cm}
$$\line(1,0){300}$$
\selectlanguage{english}

\vspace{1cm}
Let ${\mathcal O}\subset \RR^m$ be a bounded domain with smooth boundary and set $\Omega:={\mathcal O}\times \RR^n$.
Define the space of cylindrically symmetric
functions by
 $$
 W^{1,p}_c(\Omega):=\{u\in W^{1,p}(\Omega):u(x,\cdot)\, {\rm is\,radially\,symmetric\, for\ all} \ x\in {\mathcal O}\}.
 $$

 The aim of this paper is to establish the existence of multiple cylindrically symmetric weak solutions
 for the following non-autonomous elliptic Neumann problem.
$$
\left\{
\begin{array}{ll}
-\Delta_p u+|u|^{p-2}u= \lambda \alpha(x,y)f(u) &\rm in \, \Omega
\\ \partial u/\partial\nu=0&\rm on \,
\partial \Omega.
\end{array}
\right. \eqno(P)
$$
Here  $\nu$ denotes the outward
unit normal to $\partial\Omega$, $p>m+n$ is a real number, and $\Delta_pu=\mbox{div}\, (|\nabla u|^{p-2}\nabla u)$.
We assume that $f:\RR \rightarrow \RR$ is a continuous function, $\alpha$ is a nonnegative cylindrically symmetric function, and $\lambda$ is a positive real parameter.

In the present paper, just requiring a suitable oscillating behaviour of the potential $ F(\xi):=\displaystyle\int_0^{\xi}f(t)dt$, we are able to find a precise interval of values of the parameter $\lambda$ for which  problem $(P)$ admits at least three cylindrically symmetric weak solutions.

Assume $\alpha\in L^{1}(\Omega)$ is a nonnegative cylindrically symmetric function  such that for some $\tau>0$,
\begin{equation}\label{alpha+}
\mbox{essinf}_{(x,y)\in {{\mathcal O}}\times B_n(0,\tau/2)} \alpha(x,y)>0,\end{equation}
where $B_n(0,\tau/2)$ denotes open ball in $\RR^n$ centred in zero and radius $\tau/2$.\par
 We say that $u\in W^{1,p}(\Omega)$ is a weak
solution of problem $(P)$ if for all $v\in W^{1,p}(\Omega)$,
{\small
$$
\int_{\Omega} |\nabla u(x,y)|^{p-2}\nabla u(x,y)\cdot\nabla v(x,y) dxdy+\int_{\Omega} |u(x,y)|^{p-2}u(x,y)v(x,y)dxdy
=\lambda\int_{\Omega} \alpha(x,y)f(u(x,y))v(x,y)dxdy.
$$  }
Then
the weak solutions of problem $(P)$ are the critical points of the $C^1$-functional
$$
J_\lambda(u):={\frac{\|u\|^p_{W^{1,p}}}{p}}-\lambda\int_{\Omega}\alpha(x,y) F(u(x,y))dxdy,
\quad
u\in W^{1,p}(\Omega)$$
\noindent where
$$
\|u\|_{W^{1,p}}:=\Big(\int_{\Omega} |\nabla u(x,y)|^pdxdy+\int_{\Omega}|u(x,y)|^pdxdy\Big)^{1/p}.
 $$
Let $B_{(1/2,1)}(n,p+1)$ denote the generalized incomplete beta function, namely
$
B_{(1/2,1)}(n,p+1):=\displaystyle\int_{1/2}^{1}t^{n-1}(1-t)^{p}dt.
$
Set $$
\sigma(n,p):=2^{p-n}(2^n-1);\qquad
g(p,N):=\frac{1+2^{n+p}NB_{(1/2,1)}(n,p+1)}{2^n};
\qquad
 \omega^{(n)}_\tau:=\tau^n\frac{\pi^{n/2}}{\Gamma \Big(\displaystyle 1+\frac{n}{2}\Big)}.$$
Define
$$
\kappa:=\left(\frac{p\tau^p}{\displaystyle\omega_\tau^{(n)}|{\mathcal O}|_m\left({\sigma(n,p)}+ \tau^p g(p,n)\right)}\right)^{1/p},
$$
in which the symbol $|{\mathcal O}|_m$ stands for the Lebesgue measure of the domain ${\mathcal O}$ in $\RR^m$.\par
Set
$
    \|\alpha\|_{({\mathcal O},\tau/2)}:=\displaystyle\int_{{\mathcal O}\times B_{n}(0,\tau/2)}\alpha(x,y)dxdy.
    $

\noindent The main result in this Note is the following multiplicity property.
\begin{theorem}\label{Main}
Assume there are constants $\delta$ and $\gamma$ such that
$$
\delta>\kappa \gamma>0
$$
and the following properties are fulfilled$:$
\begin{itemize}
\item [$(\rm{h}_0)$] $F(\xi)\geq 0$, for every $\xi\in [0,\delta]$$;$
\item [$(\rm{h}_1)$] if
$$
\widetilde{\kappa}:=\frac{\kappa \|\alpha\|_{(\mathcal{O},\tau/2)}}{pC_p^{p}\|\alpha\|_{L^1}}
$$
then
$$
\displaystyle\frac{\displaystyle\max_{|\xi|\leq \gamma}\,F(\xi)}{\gamma^{p}}<\widetilde{\kappa}\frac{F(\delta)}{\delta^p}\,;
$$
\item [$(\textrm{h}_2)$]
there exist $c_0>0$ and $0<s<p-1$ such that for all $t\in\RR$, $$|f(t)|\leq
c_0(1+|t|^{s})\,.$$
\end{itemize}

 Then for all $\lambda$ belonging to the interval
$$
\Lambda_{(\gamma,\delta)}:=\left]\frac{\delta^p}{\displaystyle \kappa  \|\alpha\|_{(\mathcal{O},\tau/2)} F(\delta)},\frac{\gamma^{p}}{\displaystyle pC_p^{p}\|\alpha\|_{L^1} \max_{|\xi|\leq \gamma} F(\xi)}\right[,
$$
\noindent the problem
$($P$)$ has at least three distinct cylindrically symmetric weak solutions.
\end{theorem}

\smallskip
{\it Proof.}
For all $u\in W^{1,p}_c(\Omega)$ we set
$$
\Phi(u):={\frac{\|u\|^p_{W^{1,p}}}{p}},\quad{\rm and}\quad\Psi(u):=\int_{\Omega}\alpha(x,y) F(u(x,y))dxdy.
$$
Then $\Psi$ is continuously G\^{a}teaux differentiable and
$$
\Psi'(u)(v)=\int_{\Omega} \alpha(x,y)f(u(x,y))v(x,y)dxdy,
$$
\noindent for every $v\in W^{1,p}_c(\Omega)$. Further, $\Psi'$ admits a continuous inverse on $W^{1,p}_c(\Omega)^*$ and $\Psi'$ is a compact operator.

The (restriction) energy functional associated to problem $(P)$ is
$$
J_\lambda(u):=\Phi(u)-\lambda \Psi(u),\quad\mbox{for all}\ u\in W^{1,p}_c(\Omega).
$$

  Set $r:=\gamma^p/(pC_p^p)$, where $C_p$ denotes the best constant of the continuous embedding $W^{1,p}_c(\Omega)\hookrightarrow L^\infty(\Omega)$ (see \cite[Theorem 2.2]{FIK}). Since
$$
\left\{u\in W^{1,p}_c(\Omega): \|u\|_{W^{1,p}}\leq
\displaystyle (pr)^{1/p} \right\}\subseteq \left\{u\in W^{1,p}_c(\Omega): \|u\|_\infty\leq \gamma \right\},
$$
in which
 $\|u\|_{L^{\infty}}:=\mbox{esssup}_{(x,y)\in \Omega} |u(x,y)|,$ we obtain
$$
\frac{\displaystyle\sup_{u\in \Phi^{-1}(]-\infty,r])}\Psi(u)}{r}\leq pC_p^{p}\|\alpha\|_{L^1}\frac{\displaystyle\max_{|\xi|\leq \gamma}F(\xi)}{\gamma^{p}}.
$$

 Define $w_\delta\in W^{1,p}_c(\Omega)$  by $w_\delta (y):=\delta w(y)$, where
\[
w(y):= \left\{
\begin{array}{ll}
0 & \mbox{ if $|y|\geq \tau$} \\
\displaystyle\frac{2}{\tau} \left(\tau- |x| \right) & \mbox{ if $\tau/2<|y|<\tau$} \\
1 & \mbox{ if $|y|\leq \tau/2\,.$}
\end{array}
\right.
\]
 We have
$$
\int_{\small\RR^n} |\nabla w(y)|^p \;dy +\int_{\small\RR^n} |w(y)|^p \;dy = \omega_\tau^{(n)}\Big[\frac{\sigma(n,p)}{\tau^p}+ g(p,n)\Big].
$$
 \indent Indeed
$$
\int_{\small\RR^n} |\nabla w(y)|^p \;dy
                    = \frac{2^{p-n}\omega_\tau^{(n)}
}{\tau^{p}}(2^n-1),
$$ and
$$
\int_{\small\RR^n} |w(y)|^p \;dy  = \left(\int_{|y|\leq \tau/2}d\;y+\frac{2^p}{\tau^p}\displaystyle\int_{\tau/2<|y|<\tau}(\tau-|y|)^p\;dy\right)
                    = \omega_\tau^{(n)}g(p,n).
$$
 Note that the last equality holds because
$$
I_p:=\int_{\tau/2<|y|<\tau}(\tau-|y|)^p\;dy=n\omega_\tau^{(n)}\tau^pB_{(1/2,1)}(n,p+1).
$$
\indent Hence, since
$$
\Phi(w_\delta)=\frac{\delta^p|{\mathcal O}|_m}{p}\left(\int_{\small\RR^n} |\nabla w(y)|^p \;dy +\int_{\small\RR^n} |w(y)|^p \;dy\right),
$$
we deduce that
$$
\Phi(w_\delta)=\frac{\delta^p\omega_\tau^{(n)}|{\mathcal O}|_m}{p}\Big[\frac{\sigma(n,p)}{\tau^p}+ g(p,n)\Big].
$$
 Thus, by hypotheses (\ref{alpha+}) and $(\rm{h}_0)$, we deduce that
$$
\Psi(w_\delta)\geq F(\delta)\int_{\mathcal{O}\times B_{n}(0,\tau/2)}\alpha(x,y)dxdy=F(\delta)\|\alpha\|_{(\mathcal{O},\tau/2)}.
$$
\indent Therefore
$$
\frac{\Psi({w}_\delta)}{\Phi({w}_\delta)}=\kappa\frac{F(\delta)\|\alpha\|_{(\mathcal{O},\tau/2)}}{\delta^p}.
$$
Now, from $(\rm{h}_1)$, it follows that
$$\sup_{\Phi(u)\le r}\frac{\Psi(u)}{r}< \frac{\Psi({w}_\delta)}{\Phi({w}_\delta)}\,.$$

On the other hand, by $(\rm{h}_2)$, there exists a positive constant $c_1$ such that
$$\int_\Omega
\alpha(x,y)F(u(x,y))\;dxdy\leq
c_1(\|u\|_{\infty}+\|u\|^{s+1}_{\infty}),$$
for all $u\in
W^{1,p}_c(\Omega)$.\par
Moreover, since $W^{1,p}_c(\Omega)$ is compactly embedded into $L^{\infty}(\Omega)$, it follows that there exists $c_2>0$ such that
for all $u\in W^{1,p}_c(\Omega)$,
 $$J_\lambda(u):=\Phi(u)-\lambda
\Psi(u)\geq\|u\|^{p}_{W^{1,p}}-\lambda
c_2(\|u\|_{W^{1,p}}+\|u\|^{s+1}_{W^{1,p}})\,.$$
 Since
$1<s+1<p$, it follows that for all $\lambda>0$,
$\lim_{\|u\|_{W^{1,p}}\rightarrow\infty}J_\lambda(u)=+\infty\,.$
 Thus, $J_\lambda$ is  coercive for every $\lambda>0$, in particular for all $$\lambda\in \Lambda_{(\gamma,\delta)}\subseteq \Big]\displaystyle\frac{\Phi({w}_\delta)}{\Psi({w}_\delta)},\displaystyle\frac{r}{\displaystyle\sup_{\Phi(u)\le r}\Psi(u)}\Big[.$$

 Hence, by \cite[Theorem 3.6]{BONMAR} and Remark 3.3 of the cited paper, we deduce that for all $\lambda \in \Lambda_{(\gamma,\delta)}$, the functional $J_\lambda$ has at least three distinct critical points.

 Next, we prove that $J_\lambda$ is an invariant functional with respect to the action of the compact group of linear isometries of $\RR^n$.
   Indeed, let $\sigma:\RR^n\rightarrow \RR^n$ be a linear isometry. For any $u\in W^{1,p}(\Omega)$ define $u_\sigma\in W^{1,p}(\Omega)$ by putting, for any $(x,y)\in \Omega$,
  $$
  u_\sigma(x,y):=u(x,\sigma y).
  $$
  Since $\alpha$ is cylindrically symmetric, using the variable change $z=\sigma y$, we deduce that $J_\lambda(u_\sigma)=J_\lambda(u)$, for every $u\in W^{1,p}(\Omega)$.\par
   Thus, by the Palais symmetric criticality principle \cite{pala} applied to
   the smooth and isometric invariant functional $J_\lambda$, we deduce that
  problem
$(P)$ admits at least three distinct cylindrically symmetric weak solutions. The proof is complete.
\qed



 A direct consequence of the previous result is the following multiplicity property.

\begin{corollary} \label{i2}
Let $f:\RR \rightarrow \RR$ be a nonnegative $($not identically zero$)$ continuous function such that
$$
\lim_{t\rightarrow 0^+
}\frac{f(t)}{t^{p-1}}=0. \leqno{(\ell_0)}
$$
Further, assume that condition $(\rm{h}_2)$ holds. Set $$ S:=\left\{\delta>0:\displaystyle F(\delta)>0\right\}\,.
$$

 Then, for all
$$
\lambda>\frac{1}{\displaystyle \kappa  \|\alpha\|_{(\mathcal{O},\tau/2)}}\inf_{\delta\in S}\frac{\delta^p}{F(\delta)},
$$
 problem
$($P$)$ has at least two nontrivial cylindrically symmetric weak solutions.
\end{corollary}

{\it Proof.}
Fix $\displaystyle
\lambda>\frac{1}{\displaystyle \kappa  \|\alpha\|_{(\mathcal{O},\tau/2)}}\inf_{\delta\in S}\frac{\delta^p}{F(\delta)}.
$ Then, there exists $\bar{\delta}$ such that $F(\bar{\delta})>0$ and $\displaystyle\lambda>\frac{\bar \delta^p}{\displaystyle \kappa  \|\alpha\|_{(\mathcal{O},\tau/2)}F(\bar\delta)}$. By using condition $(\ell_0)$ we have
$$
\lim_{\xi\rightarrow 0^+
}\frac{F(\xi)}{\xi^{p}}=0.
$$
Therefore, there exists $\bar{\gamma}>0$ such that
$
\bar{\gamma}<\bar\delta/\kappa
$
and
$$
\frac{\displaystyle F(\bar{\gamma})}{\bar{\gamma}^{p}}<\min\left\{\widetilde{\kappa}\frac{F(\bar\delta)}{\bar\delta^p},\frac{1}{\displaystyle pC_p^{p}\|\alpha\|_{L^1}\lambda}\right\}.
$$
Hence

$$
\lambda\in\Lambda_{(\bar{\gamma},\bar{\delta})}:=\left]\frac{\bar \delta^p}{\displaystyle \kappa  \|\alpha\|_{(\mathcal{O},\tau/2)}F(\bar\delta)},\frac{\bar\gamma^{p}}{\displaystyle pC_p^{p}\|\alpha\|_{L^1} F(\bar\gamma)}\right[.
$$
All the hypotheses of Theorem \ref{Main} are satisfied and problem $(P)$ admits at least three (two nontrivial) distinct cylindrically symmetric weak solutions. The proof is complete.
\qed


\section*{Acknowledgements}
The authors acknowledge the support through  Grant CNCSIS PCE--47/2011.
This paper has been completed while G.~Molica Bisci was visiting the University of Craiova in October 2012.



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\end{document}

The main tool is a critical point theorem obtained in \cite{BONMAR} that we recall here in a convenient form.
\begin{theorem}\label{abstract}
 Let $X$ be a reflexive real Banach space, $\Phi:X\to\R$ be a coercive, continuously G$\hat{a}$teaux differentiable and sequentially weakly lower semicontinuous functional whose G$\hat{a}$teaux derivative admits a continuous inverse on $X^*$, $\Psi :X\to \R$ be a continuously G$\hat{a}$teaux differentiable functional whose $G\hat{a}$teaux derivative is compact such that $$ \Phi (0_X)=\Psi (0_X)=0.$$
Assume that there exist $r>0$ and $\bar{x}\in X$, with $r<\Phi(\bar{x})$, such that$:$
\begin{itemize}
\item[$(a_1)$] $\dfrac{\displaystyle\sup_{\Phi(x)\le r}\Psi(x)}{r}< \frac{\Psi(\bar{x})}{\Phi(\bar x)}$$;$
\item[$(a_2)$] For each $\lambda\in \Lambda_{r}:=\Big]\displaystyle\frac{\Phi(\bar{x})}{\Psi(\bar{x})},\displaystyle\frac{r}{\sup_{\Phi(x)\le r}\Psi(x)}\Big[$ the functional $J_\lambda:=\Phi -\lambda \Psi$ is coercive.
\end{itemize}
Then, for each $\lambda \in \Lambda_{r}$, the functional $J_\lambda$ has at least three distinct critical points in $X$.
\end{theorem}

We say that
the derivative of $\Phi$ admits a continuous inverse on the topological dual $X^*$ when %we mean that
there exists a continuous operator $T:X^*\rightarrow X$ such that $T(\Phi'(x))=x$ for every $x\in X$.\par

For completeness, we also recall here the principle
of symmetric criticality that plays a central role in many problems
from the differential geometry, physics and in partial differential
equations.\par
 The \textit{action} of a topological group $G$ on the  Banach space $(X,\|\cdot\|_X)$ is a continuous map
$$
\varsigma: G\times X\rightarrow X: (g,x)\mapsto \varsigma(g,u)=:gu,
$$
\noindent such that
$$
1u=u,\,\,\,
(gh)u=g(hu),\,\,\, u\mapsto gu\,\,\,{\rm is\,linear}.
$$
\indent The action is said to be \textit{isometric} if $\|gu\|_X=\|u\|_X$, for every $g\in G$. Moreover,
 the space of $G$-invariant points is defined by
$$
Fix(G):=\{u\in X:gu=u,\forall g\in G\},
$$
and a map $h:X\rightarrow \RR$ is said to be $G$-\textit{invariant} if $h\circ g=h$ for every $g\in G$.\par
\begin{theorem}[Palais (1979), \cite{pala}]\label{Palais}
Assume that the action of the topological group $G$ on the Banach space $X$ is isometric. If $J\in C^1(X;\RR)$ is $G$-invariant and if $u$ is a critical point of $J$ restricted to $Fix(G)$ then $u$ is a critical point of $J$.
\end{theorem}


 \begin{lemma} \label{kristaly}
With the above notations, the embedding
$$
W^{1,p}_c(\Omega)\hookrightarrow L^{\infty}(\Omega),
$$
is compact.
\end{lemma}
\indent See \cite[Theorem 2.2]{FIK} for details.\par
 \noindent Finally, denote by $C_p$ the best embedding constant of $W^{1,p}(\Omega)$ into $L^{\infty}(\Omega)$, that is
$$
C_p:=\sup_{u\in W^{1,p}(\Omega)\setminus\{0\}}\frac{\|u\|_{L^{\infty}}}{\|u\|_{W^{1,p}}}.
$$ 