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\documentclass[reqno]{amsart} 

\AtBeginDocument{{\noindent\small 
{\em Electronic Journal of Differential Equations},
Vol. 2001(2001), No. 16, pp. 1--20.\newline
ISSN: 1072-6691. URL: http://ejde.math.swt.edu or http://ejde.math.unt.edu
\newline ftp ejde.math.swt.edu \quad ejde.math.unt.edu (login: ftp)}
\thanks{\copyright 2001 Southwest Texas State University.} 
\vspace{1cm}}

\begin{document} 

\title[\hfilneg EJDE--2001/16\hfil A deformation theorem]
{ A deformation theorem in the noncompact  nonsmooth setting 
  and its applications } 

\author[Gianni Arioli \hfil EJDE--2001/16\hfilneg]
{ Gianni Arioli }

\address{ Gianni Arioli \hfill\break 
Dipartimento di Scienze e T.A., C.so Borsalino 54, 15100 Alessandria Italy}
\email{ gianni@unipmn.it \quad http://www.mfn.unipmn.it/$\sim$gianni }


\date{}
\thanks{Submitted September 15, 2000. Published February 23, 2001.}
\thanks{Supported by MURST project ``Metodi variazionali ed
Equazioni Differenziali Non Lineari''}
\subjclass[2000]{35D05, 35J20, 35J60}
\keywords{Quasilinear elliptic differential systems, Equivariant category,
\hfill\break\indent Nonsmooth critical point theory }


\begin{abstract}
 We build a deformation for a continuous functional defined on a Banach space
 and invariant with respect to an isometric action of a noncompact group. 
 Under these assumptions the Palais-Smale condition does not hold. When the
 functional is also invariant with respect to the action of a compact Lie
 group, we prove that the deformation can be chosen to be equivariant with
 respect to the same action. In the second part of the paper a system of
 periodic quasilinear partial differential equations invariant under the 
 action of some compact Lie group is considered. Using the deformation 
 technique developed in the first part, we prove the existence of infinitely 
 many solutions.
\end{abstract}

\maketitle

\newtheorem{theorem}{Theorem}[section]
\newtheorem{lemma}[theorem]{Lemma}
\newtheorem{proposition}[theorem]{Proposition}
\newtheorem{remark}{Remark}[section]
\newtheorem{definition}{Definition}[section]
\numberwithin{equation}{section}


\section{Introduction}
%
In the last decade the nonsmooth critical point theory introduced by
Degiovanni and his collaborators \cite{cd,cdm,dm} has been applied to a large
variety of nonlinear differential problems whose Lagrangian functional lacks
differentiability. The core of the theory is the concept of \emph{weak slope},
which plays the part of the norm of the derivative, although it is defined
also for lower semi-continuous functionals. One defines the critical points as
points with zero weak slope and proves that they provide weak solutions for
the differential equation. The beauty of the theory consists in the fact that
most topological tools used in critical point theory, such as the mountain
pass theorem, the saddle point theorem, Morse theory and index theory, can be
applied up to small adaptations. All these tools rely on the existence of a
deformation in the regions where there are no critical points. Accordingly to
\cite{cdm,dm}, in order to build a deformation a certain compactness is
required on the functional, more precisely the nonsmooth version of the
well-known Palais-Smale condition. On the other hand, if a functional is
invariant under the action of a noncompact topological group, a typical
example being $\mathbb{Z}^{n}$, then the Palais-Smale condition cannot hold.
Functionals which are invariant under the action of $\mathbb{Z}^{n}$ are very
common: as an example consider the problem of looking for homoclinic solutions
in the theory of dynamical system. It is in that setting that a weaker version
of the Palais Smale condition has been introduced, namely the $\overline{PS}$
condition (see \cite{s} and also \cite{as} where the symmetry with respect to
a compact Lie group has been considered). It turns out that the $\overline
{PS}$ condition is sufficient to build a deformation under suitable
circumstances. The proof relies in following the negative gradient flow and
proving that it does not come to an end, unless it comes close to a critical
point. Due to the nonexistence of the gradient, it is not possible to extend
directly the proof to the nonsmooth setting. In the first part of this paper
we take a different approach and we build a deformation for continuous
functionals on a Banach space, invariant with respect to some isometric action
of a noncompact group. In order to achieve this goal, we build a deformation
by using the tools of the abstract nonsmooth critical point theory, without
recurring to the (possible) integral form of the functional. This approach is
very natural in this context, and provides a flexible abstract result.

In the second part of the paper we apply this technique to a system of $m$
quasilinear differential equations defined and periodic in $\mathbb{R}^{n}$
(and therefore invariant with respect to the natural isometric action of
$\mathbb{Z}^{n}$ on $\mathbb{R}^{n}$). We prove the existence of a (weak)
solution in the general case, and the existence of infinitely many (weak)
solutions when the system is also invariant under the action of some compact
Lie group $G$ on $\mathbb{R}^{m}$. A similar system has been considered in
\cite{ag} in the case where the nonlinear part of the functional is compact;
that case excludes the periodic setting.

We remark that the simpler case of a single equation, symmetric with respect
to the action of $\mathbb{Z}_2$, comes as a corollary, and indeed it
requires weaker assumptions.

\section{Equivariant nonsmooth critical point theory}
%
The following definitions provide the basic tools of the nonsmooth critical
point theory in the equivariant setting introduced in \cite{cdm,dm} and are
given for the convenience of the reader. For a detailed introduction we refer
to the above cited papers and to \cite{cd}. We remark that the non equivariant
definitions correspond exactly to the following ones, if one just forgets
about the invariant-equivariant requirements. The main definitions concerning
representation theory are summarized in Section 8.

\begin{definition}
Let $(X,d)$ be a metric $G-$space, let $I\in C(X,\mathbb{R})$ be an invariant
function and let $x\in X$. We denote by $|d_{G}I|(x)$ the supremum of the
$\sigma\in\lbrack0,+\infty)$ such that there exist an invariant neighborhood
$U$ of $x$, $\delta>0$ and a continuous map $\mathcal{H}:U\times
\lbrack0,\delta]\longrightarrow X$ such that for all $y\in U$ and for all
$t\in\lbrack0,\delta]$
\[
d(\mathcal{H}(y,t),y)\leq t\text{,\hspace{1cm}}I(\mathcal{H}(y,t))\leq
I(y)-\sigma t
\]
and $\mathcal{H}(\cdot,t)$ is equivariant for all $t\in\lbrack0,\delta]$; the
extended real number $|d_{G}I|(x)$ is called the \emph{equivariant weak slope}
of $I$ at $x$.
\end{definition}

\begin{definition}
Let $(X,d)$ be a metric $G-$space, let $I\in C(X,\mathbb{R})$ be an invariant
function; a point $x\in X$ is said to be $G-$critical for $I$ if
$|d_{G}I|(x)=0$. A real number $c$ is said to be a $G-$critical value for $I$
if there exists $x\in X$ such that $I(x)=c$ and $|d_{G}I|(x)=0$.
\end{definition}

The equivariant version of the compactness condition of Palais-Smale has been
defined in this context (see \cite{dm}):

\begin{definition}
Let $(X,d)$ be a metric $G-$space, let $I\in C(X,\mathbb{R})$ be an invariant
function. A sequence $\{x_m\}\subset X$ is called a $G-$Palais-Smale
sequence ($G-$(PS)) if there exists $K>0$ such that $|I(x_m)|\leq K$ and
$|d_{G}I|(x_m)\to 0$.

$I$ satisfies the $G-$(PS) condition if all its $G-$(PS) sequences are precompact.
\end{definition}

The following definition provides a tool to relate the weak slope to the norm
of the directional derivatives, when $X$ is a Banach space.

\begin{definition}
Let $X$ be a Banach space, let $I\in C(X,\mathbb{R})$ and let $Y$ be a dense
subspace of $X$. If the directional derivative of $I$ exists for all $x$ in
$X$ in all the directions $y\in Y$ we say that $I$ is weakly Y-differentiable
and we call weak Y-slope in $x$ the extended real number
\[
\Vert I_{Y}'(x)\Vert_{\ast}:=\sup\{I'(x)[\phi]:\ \phi\in
Y,\ \Vert\phi\Vert_{X}=1\}.
\]
\end{definition}

Consider now the Hilbert space $H=H^{1}(\mathbb{R}^{n},\mathbb{R}^{m})$, i.e.
the closure of $C_{c}^{\infty}(\mathbb{R}^{n},\mathbb{R}^{m})$ (the space of
smooth vector functions with compact support in $\mathbb{R}^{n}$) with respect
to the norm induced by the scalar product $(\psi,\phi)=$ $\int_{{\mathbb{R}
^{n}}}\langle D_i\psi,D_i\phi\rangle+\langle\psi,\phi\rangle$. The
following theorem provides the connection between the equivariant nonsmooth
critical point theory and applications to an invariant continuous functional
defined on $H$. In order to do so, we extend to the equivariant setting
Theorem 1.5 in \cite{c}, which states that the weak slope provides an upper
limit for the $C_{c}^{\infty}(\mathbb{R}^{n})$-slope.

\begin{theorem}
Let $\mathbb{R}^{m}$ be a representation space for $G$, let $J:H\to 
\mathbb{R}$ be defined by
\[
J(u)=\int_{\mathbb{R}^{n}}L(x,u,\nabla u)dx,
\]
where $L:\mathbb{R}^{n}\times\mathbb{R}^{m}\times\mathbb{R}^{mn}
\to \mathbb{R}$ satisfies the following assumptions:
\[
\begin{array}
[c]{l}
L(x,s,\xi)\text{ is measurable with respect to }x\text{ for all }(s,\xi
)\in\mathbb{R}^{m}\times\mathbb{R}^{mn}\\
L(x,s,\xi)\text{ is of class }C^{1}\text{ with respect to }(s,\xi)\text{ for
a.e. }x\in\mathbb{R}^{n}
\end{array}
\]
and there exist $h_1\in L^{1}(\mathbb{R}^{n},\mathbb{R})$, $h_2\in
L_{loc}^{1}(\mathbb{R}^{n},\mathbb{R})$, $h_{3}\in L_{loc}^{\infty}
(\mathbb{R}^{n},\mathbb{R})$ and $c>0$ such that for all $(s,\xi)\in
\mathbb{R}^{m}\times\mathbb{R}^{mn}$ and a.e. $x\in\mathbb{R}^{n}$ the
following inequalities hold:
\[
\begin{array}
[c]{c}
|L(x,s,\xi)|\leq h_1(x)+c(|s|^{\frac{2n}{n-2}}+|\xi|^{2})\\
\left|  {\frac{\partial L}{\partial s}}(x,s,\xi)\right|  \leq h_2
(x)+h_{3}(x)(|s|^{\frac{2n}{n-2}}+|\xi|^{2})\\
\left|  {\frac{\partial L}{\partial\xi}}(x,s,\xi)\right|  \leq h_2
(x)+h_{3}(x)(|s|^{\frac{2n}{n-2}}+|\xi|^{2}).
\end{array}
\]
Furthermore $L$ is invariant with respect of the action of $G$ on
$\mathbb{R}^{n}\times\mathbb{R}^{m}\times\mathbb{R}^{mn}$ defined by
\[
(x,s,(\xi_1,\dots,\xi_m))\mapsto(x,gs,(g\xi_1,\dots,g\xi_m)).
\]
Then $J$ is continuous, invariant with respect to the action of $G$ on $H$
defined by $g(u)(x):=g(u(x))$, weakly $C_{c}^{\infty}(\Omega)-$differentiable
and for all $u\in H$ we have
\begin{equation}
|d_{G}J|(u)\geq\Vert J_{C_{c}^{\infty}}'(u)\Vert_{\ast}\text{.}
\label{eq:greater}
\end{equation}
In particular, if $u$ is a $G-$critical point of $J$, then the Euler equation
\[
div\left(  \frac{\partial L}{\partial\xi}(x,u,\nabla u)\right)  -\frac
{\partial L}{\partial s}(x,u,\nabla u)=0
\]
is satisfied in distributional sense.
\end{theorem}

\begin{proof}
By the Sobolev embedding theorem $J$ is continuous,
\[
J'(u)[v]=\int_{\mathbb{R}^{n}}\left[  \frac{\partial L}{\partial\xi
}(x,u,\nabla u)\nabla v+\frac{\partial L}{\partial s}(x,u,\nabla u)v\right]
dx
\]
for all $u\in H$ and all $v\in C_{c}^{\infty}(\mathbb{R}^{n})$ and the
functional $\{u\mapsto J'(u)[v]\}$ is continuous for all $v\in
C_{c}^{\infty}(\mathbb{R}^{n})$. The invariance of $J$ is straightforward.
Choose $u_0\in H$; if $\Vert J_{C_{c}^{\infty}}'(u_0)\Vert_{\ast
}=0$ there is nothing to prove, otherwise let $0<\sigma<\Vert J_{C_{c}
^{\infty}}'(u_0)\Vert_{\ast}$. Then there exists $v_0\in
C_{c}^{\infty}(\mathbb{R}^{n})$, $||v_0||=1$, such that $J'
(u_0)[v_0]>\sigma$, and there exists $\delta>0$ such that $J^{\prime
}(u)[v_0]>\sigma$ for all $u\in B_{2\delta}(u_0)$. Let $\eta
:H\to \lbrack0,1]$ be a smooth function satisfying $\eta(u)=1$ if $u\in
B_{\delta}(u_0)$ and $\eta(u)=0$ if $u\notin B_{2\delta}(u_0)$. Let
$U=\mathcal{O}_{G}(B_{\delta}(u_0))$ and define the function $v:U\to 
H$ by
\[
v(u)=\alpha(u)\int_{G}gv_0\eta(g^{-1}u)dg,
\]
where $dg$ is the Haar measure and $\alpha(u)=\left[  \int_{G}\eta
(g^{-1}u)dg\right]  ^{-1}$; by Fubini Theorem $||v(u)||=||v||=1$. Define a
continuous map $\mathcal{H}:U\times\lbrack0,\delta]\longrightarrow H$ by
\[
\mathcal{H}(u,t)=u-tv(u).
\]
The map $\mathcal{H}(\cdot,t)$ is equivariant for all $t\in\lbrack0,\delta]$,
because
\[
v(g_0u)=\int_{G}gv_0\eta(g^{-1}g_0u)dg=\int_{G}g_0\tilde{g}v_0
\eta(\tilde{g}^{-1}u)d\tilde{g}=g_0v(u)
\]
so $\mathcal{H}(gu,t)=gu-tv(gu)=g(u-tv(u)).$ Since $||v(u)||=1$, then
$||\mathcal{H}(u,t)-u||\leq t$. By the invariance of $L$ and Fubini Theorem it
follows that
\begin{align*}
\int_{\mathbb{R}^{n}}\frac{\partial L}{\partial s}(x,u,\nabla u)v(u)dx  &
=\alpha(u)\int_{\mathbb{R}^{n}}\frac{\partial L}{\partial s}(x,u,\nabla
u)\int_{G}gv_0\eta(g^{-1}u)dgdx\\
&  =\alpha(u)\int_{G}\eta(g^{-1}u)\int_{\mathbb{R}^{n}}\frac{\partial
L}{\partial s}(x,u,\nabla u)gv_0dxdg\\
&  =\alpha(u)\int_{G}\eta(g^{-1}u)\int_{\mathbb{R}^{n}}\frac{\partial
L}{\partial s}(x,g^{-1}u,g^{-1}\nabla u)v_0dxdg
\end{align*}
and analogously
\[
\int_{\mathbb{R}^{n}}\frac{\partial L}{\partial\xi}(x,u,\nabla u)\nabla
(v(u))dx=\alpha(u)\int_{G}\eta(g^{-1}u)\int_{\mathbb{R}^{n}}\frac{\partial
L}{\partial s}(x,g^{-1}u,g^{-1}\nabla u)\nabla v_0dxdg.
\]
For all $u\in B_{2\delta}(u_0)$ we have
\[
\int_{\mathbb{R}^{n}}\frac{\partial L}{\partial s}(x,u,\nabla u)v_0
+\frac{\partial L}{\partial s}(x,u,\nabla u)v_0dx=J'(u)[v_0
]>\sigma
\]
and for all $u\notin B_{2\delta}(u_0)$ and $\eta(u)=0$, therefore
\begin{align*}
&J'(u)[v(u)]\\
 &  = \int_{\mathbb{R}^{n}}\frac{\partial L}{\partial\xi
}(x,u,\nabla u)\nabla(v(u))+\frac{\partial L}{\partial s}(x,u,\nabla
u)v(u)dx\\
&=  \alpha(u)\int_{G}\eta(g^{-1}u)\int_{\mathbb{R}^{n}}\frac{\partial
L}{\partial s}(x,g^{-1}u,g^{-1}\nabla u)\nabla v_0+\frac{\partial
L}{\partial s}(x,g^{-1}u,g^{-1}\nabla u)v_0dxdg\\
&\geq  \sigma\alpha(u)\int_{G}\eta(g^{-1}u)dg=\sigma,
\end{align*}
$J(\mathcal{H}(u,t))-J(u)=tJ'(u-sv_0)[v_0]$ for some $s\in
\lbrack0,t]$ and $J(\mathcal{H}(u,t))\leq J(u)-\sigma t$. By the definition of
equivariant weak slope we have $|d_{G}J|(u)\geq\sigma$ and by the
arbitrariness of $\sigma$ we obtain (\ref{eq:greater}).
\end{proof}

\section{The deformation theorem}
%
In this section we build a deformation in the abstract setting of a Banach
space with an isometric action of a discrete topological group. Let $B$ be a
Banach space, let $\mathbb{D}$ be a discrete topological group acting on $B$
with a linear isometry $\ast:\mathbb{D}\times B\to  B$, $(k,u)\mapsto
k\ast u$.

\begin{definition}
For $l\in\mathbb{N}$, ${\bar{k}}=(k^{1},\dots,k^{l})\mathbf{\in}\mathbb{D}
^{l}$ and $\bar{u}=(u^{1},\dots,u^{l})\in B^{l}$, let
\[
\bar{k}\ast\bar{u}:=\sum_ik^{i}\ast u^{i}.
\]
Given a sequence $\{{\bar{k}}_m\}\subset\mathbb{D}^{l},$ we say that
$\{{\bar{k}}_m\}$ diverges or ${\bar{k}}_m\to \infty$ if for all
$i\neq j$ the sequence $\{k_m^{i}(k_m^{j})^{-1}\}$ does not admit any
convergent subsequence (i.e. $\{k_m^{i}(k_m^{j})^{-1}\}$ does not admit
any constant subsequence).
\end{definition}

In the following, $J:B\to \mathbb{R}$ will denote a continuous
functional, invariant with respect to the action of $\mathbb{D}$. The set
$K:=\{u\in B:|dJ|(u)=0\}$ is the (nonsmooth) critical set. We also assume that
\begin{equation}
\begin{array}
[c]{c}
K\setminus\{0\}=\mathcal{O}_{\mathbb{D}}(\mathcal{K}),\text{ where
}\mathcal{K}\text{ is a compact set,}\\
\mathcal{K}\cap(k\ast\mathcal{K})=\emptyset\text{ whenever }k\neq
1_{\mathbb{D}}
\end{array}
\label{eq:noncompact}
\end{equation}
and
\begin{equation}
\lim_{m\to \infty}||\bar{k}_m\ast\bar{u}||^{2}=\sum_i||u^{i}||^{2}
\label{eq:split}
\end{equation}
for all $l\in\mathbb{N}$, all diverging sequences $\{{\bar{k}}_m
\}\subset\mathbb{D}^{l}$ and all $\bar{u}=(u^{1},\dots,u^{l})\in B^{l}$.

\begin{lemma}
\label{lem:distance}There exists $r_0>0$ such that
\[
d(k\ast\mathcal{K},\mathcal{K}):=\min_{(v,w)\in\mathcal{K}^{2}}\Vert k\ast
v-w\Vert\geq3r_0
\]
for all $k\in\mathbb{D}\setminus\{0\}$ and $||u||>2r_0$ for all $u\in
K\setminus\{0\}$. Also note that $\mathcal{O}_{\mathbb{D}}(u)$ is closed for
all $u\in\mathcal{K}$.
\end{lemma}

\begin{proof}
First note that $\alpha:=\min\limits_{u\in K\setminus\{0\}}||u||^{2}>0$
because $\mathcal{K}$ is compact and $0\notin\mathcal{K}$. Take $(v,w)\in
\mathcal{K}^{2}$; by (\ref{eq:split}) $\Vert k\ast v-w\Vert^{2}\leq
\frac{\alpha}{2}$ only for a finite number of $k\in\mathbb{D}$. By the
compactness of $\mathcal{K}$ we also have $\min\limits_{(v,w)\in
\mathcal{K}^{2}}\Vert k\ast v-w\Vert\leq\frac{\alpha}{2}$ only for a finite
number of $k\in\mathbb{D}$. Since $\mathcal{K}$ is disjoint to $k\ast
\mathcal{K}$ for all $k\neq1_{\mathbb{D}}$ and it is compact we easily infer
the result.
\end{proof}

\begin{remark}
\label{rem:nothing}If the conditions in (\ref{eq:noncompact}) are \textbf{not}
satisfied, then the critical set consists of infinitely many points, up to the
$\mathbb{D}$ symmetry and up to the symmetry with respect to any compact Lie
group, therefore there is no need to prove a multiplicity theorem. The
condition in (\ref{eq:split}) ensures that the action of $\mathbb{D}$
separates in some sense the points of $B$; it could be replaced with a weaker
(but less natural) condition.
\end{remark}

Whenever a functional is periodic, hence invariant with respect to the action
of $\mathbb{D}=\mathbb{Z}^{n}$, the Palais-Smale condition cannot hold. In the
smooth setting it is often the case that a Palais-Smale sequence consists, up
to a subsequence, to the sum of translated sequences each of which converges,
up to translations. We call this type of Palais-Smale sequences
\emph{representable} (see the Definition below for a precise statement) and in
Section 5 we prove that Palais-Smale sequences are representable even in the
nonsmooth setting, when proper assumptions are taken.

\begin{definition}
A Palais-Smale sequence $\{u_m\}\subset B$ at level $c>0$ is said to be
\emph{representable} if there exist an integer $l$, $\bar{u}=(u^{1}
,\dots,u^{l})\in\mathcal{K}^{l}$ and a diverging sequence $\{\bar{k}
_m\}\subset\mathbb{D}^{l}$ such that, up to a subsequence,
\[
\left\|  u_m-\bar{k}_m\ast\bar{u}\right\|  \to 0
\]
and
\[
\sum_{i=1}^{l}J(u^{i})=c.
\]
\end{definition}

For all $k\in\mathbb{D}$ and $b>a>0$ let
\begin{align*}
\mathcal{U}_{a}  &  =\bigcup_{k\in\mathbb{D}}\{w\in B:d(w,k\ast\mathcal{K}
)<a\}\\
\mathcal{T}_{a,b}(k)  &  =\{w\in B:d(w,k\ast\mathcal{K})\in(a,b)\},
\end{align*}
where $d(v,A)=\inf\limits_{w\in A}\Vert v-w\Vert$. In order to prove a
deformation lemma, we give a lower bound for $|dJ|$ in a suitable set; for
this purpose we adapt the proof of Lemma 5.1 in \cite{as} to the present context.

\begin{lemma}
\label{annulus}Assume that all PS sequences at some level $c>0$ are
representable. The following statements hold:

\begin{enumerate}
\item [(a)]$\mu_{{\rho}}:=\inf\{|dJ|(u):u\in\mathcal{U}_{r_0}\setminus
\bar{\mathcal{U}}_{{\rho}}\}>0$ for all ${\rho}\in(0,r_0)$.

\item[(b)] If $\{u_m\}$ is a Palais-Smale sequence such that $\Vert
u_{m+1}-u_m\Vert<\frac{r_0}{2}$ and $J(u_m)$ is bounded away from $0$
for all $m$, then there exists $k\in\mathbb{D}$ such that $d(u_m
,k\ast\mathcal{K})\to 0$.
\end{enumerate}
\end{lemma}

\begin{proof}
(a) It follows from Lemma \ref{lem:distance} that $\mathcal{U}_{r_0
}\setminus\bar{\mathcal{U}}_{{\rho}}=\bigcup_{k\in\mathbb{D}}\mathcal{T}
_{\rho,r_0}(k)$ and the sets $\mathcal{T}_{\rho,r_0}(k)$ are disjoint. If
$\mu_{{\rho}}=0$ for some ${\rho},$ then by the $\mathbb{D}-$invariance of the
functional there exists a sequence $\{u_m\}\subset B$ such that $u_m
\in\mathcal{T}_{{\rho},r_0}(0)$ and $|dJ|(u_m)\to 0$. The sequence
$\{u_m\}$ is bounded, therefore it is Palais-Smale and representable, so
there exist $l$ points $\bar{u}=(u^{1},\dots,u^{l})\in\mathcal{K}^{l}$ and a
sequence $\{{\bar{k}}_m\}\subset\mathbb{D}^{l}$, ${\bar{k}}_m
\to \infty$, such that, up to a subsequence
\[
\Vert u_m-{\bar{k}}_m\ast\bar{u}\Vert\to 0,
\]
therefore, for large $m$ we have ${\bar{k}}_m\ast\bar{u}\in\mathcal{T}
_{\frac{{\rho}}{2},2r_0}(0),$ i.e.
\[
\frac{{\rho}}{2}<d({\bar{k}}_m\ast\bar{u},\mathcal{K})<2r_0.
\]
We show that these inequalities lead to a contradiction: indeed, as ${\bar{k}
}_m\to \infty$, up to a subsequence we have $k_m^{i}\to 
k^{i}\in\mathbb{D}\cup\{\infty\}$ for $m\to \infty$ and $k^{i}
\in\mathbb{D}$ for at most one value of $i$. If $l=1$ and either $k_m
^{1}\to  k^{1}\neq1_{\mathbb{D}}$ or $k_m^{1}\to \infty$, then
$d({\bar{k}}_m\ast\bar{u},\mathcal{K})=d({k}_m^{1}\ast u^{1}
,\mathcal{K})\geq3r_0$ for almost all $m$; if $l=1$ and $k_m
^{1}\to 1_{\mathbb{D}}$, then $d({\bar{k}}_m\ast\bar{u}
,\mathcal{K})=d({k}_m^{1}\ast u^{1},\mathcal{K})\to 0.$ If $l>1$,
then choose $i$ such that $k_m^{i}\to \infty$ and note that by
(\ref{eq:split}) and Lemma \ref{lem:distance} we have $\lim
\limits_{m\to \infty}d({\bar{k}}_m\ast\bar{u},\mathcal{K})\geq\Vert
u^{i}\Vert>2r_0$.

(b) Since $\{u_m\}$ is representable, then there exist $l$ points
$(u^{1},\dots,u^{l})\in\mathcal{K}^{l}$, a sequence $\{\bar{h}_{k}
\}\subset\mathbb{D}^{l}$, $\bar{h}_{k}\to \infty$, and a subsequence
$\{u_{m_{k}}\}$ such that $\Vert u_{m_{k}}-\bar{h}_{k}\ast\bar{u}
\Vert\to 0$. If $l=1$, then (b) holds$:$ indeed, since $\Vert
u_{m+1}-u_m\Vert<\frac{r_0}{2}$, it follows from Lemma \ref{lem:distance}
and (a) that $d(u_m,k^{1}\ast\mathcal{K})<r_0$ for some $k^{1}$ and almost
all $m$. Hence $\bar{h}_{k}=k^{1}$ for $k$ large and $u_{m_{k}}\to 
k^{1}\ast u^{1}$.

We show that $l\geq2$ cannot occur. By (\ref{eq:split}) and Lemma
\ref{lem:distance} we have $d(\bar{h}_m\ast\bar{z},\bar{h}_{k}
\ast\mathcal{K})\geq2r_0$ for a fixed arbitrary $k$ and large $m,$ say
$m\geq m_{k}$. So for all $k$ there exists $h\geq m_{k}$ such that
$d(u_{h},\bar{h}_{k}\ast\mathcal{K})>r_0$. Since $\Vert u_{m+1}-u_m
\Vert<\frac{r_0}{2}$ and $\Vert u_{m_{k}}-\bar{h}_{k}\ast\bar{u}\Vert
<\frac{r_0}{2}$ whenever $k$ is large, for such $k$ there exists $h_{k}\geq
m_{k}$ with $d(u_{h_{k}},\bar{h}_{k}\ast\mathcal{K})\in(\frac{r_0}{2}
,r_0)$. But this is impossible because $\{u_{h_{k}}\}$ is a PS-sequence and
for all sequences $\{{\bar{k}}_m\}\subset\mathbb{D}^{l}$ ($l\geq2$) such
that ${\bar{k}}_m\to \infty$ as $m\to \infty$ we have
\[
\liminf_{m\to \infty}\left\{  \inf\left\{  |dJ|(u):d(u,\bar{k}_m
\ast\mathcal{K})\in\left(  \frac{r_0}{2},r_0\right)  \right\}  \right\}
>0\,.
\]
To prove this claim by contradiction, assume that there is a sequence
$\{u_m\}$ with $d(u_m,\bar{k}_m\ast\mathcal{K})\in\left(  \frac{r_0
}{2},r_0\right)  $, $|dJ|(u_m)\to 0$ and ${\bar{k}}_m
\to \infty$. Then there exist $l'$ points $(u^{1}
,\dots,u^{l'})=\bar{u}\in\mathcal{K}^{l'}$ and a sequence
$\{\bar{h}_m\}$ such that $\bar{h}_m\to \infty$ and, up to a
subsequence, $\Vert u_m-\bar{h}_m\ast\bar{u}\Vert\to 0$. Hence for
$m$ large enough $d(\bar{h}_m\ast\bar{u},\bar{k}_m\ast\mathcal{K}
)\in\left(  \frac{r_0}{4},2r_0\right)  $. If $l\neq l'$ or
$|h_m^{i}(k_m^{j})^{-1}|\to \infty$ for some $i$ and all $j$, then
by (\ref{eq:split}) and Lemma \ref{lem:distance} we have $\lim
\limits_{m\to \infty}d(\bar{h}_m\ast\bar{u},\bar{k}_m
\ast\mathcal{K})\geq\Vert u^{i}\Vert>2r_0$. So $l=l'$ and up to a
subsequence $h_m^{i}(k_m^{i})^{-1}\to  a^{i}\in\mathbb{D}$ for each
$i$ (possibly after relabelling the $i$'s in $\bar{k}_m$). Then again by
(\ref{eq:split})
\[
\lim_{m\to \infty}d(\bar{h}_m\ast\bar{u},\bar{k}_m\ast
\mathcal{K})^{2}=\sum_{i=1}^{l}d(a^{i}\ast u^{i},\mathcal{K})^{2}.
\]
Since the distances on the right-hand side are either $0$ or exceed $2r_0$,
the above limit is not in $\left(  \frac{r_0}{2},2r_0\right)  $.

So far we have proved that $l$ is necessarily equal to $1$ and consequently
$\{u_m\}$ has a subsequence $u_{m_{k}}\to  k^{1}\ast u^{1}$. The same
argument shows that any subsequence of $\{u_m\}$ has a subsequence
converging to an element of $k^{1}\ast\mathcal{K}$ (with the same $k^{1}$).
Hence the conclusion.
\end{proof}

Choose $0<{\delta}<r_0/5$ and for all $u\in B$ let
\[
\sigma(u)=\inf\{|dJ|(v):v\in B_{\delta}(u)\}.
\]

\begin{lemma}
If $u\in B$ satisfies $d(u,K)>\delta$ and $\inf\limits_{v\in B_{\delta}
(u)}J(v)>0$, then $\sigma(u)>0$. If $u\in B$ satisfies $d(u,K)\leq\delta$,
then $\sigma(u)=0$.
\end{lemma}

\begin{proof}
If $\sigma(u)=0$ and $\inf\limits_{v\in B_{\delta}(u)}J(v)>0$, then there
exists a Palais-Smale sequence $\{v_{n}\}$ in $B_{\delta}(u)$ at positive
level. By Lemma \ref{annulus}(b) $v_{n}$ converges to a critical point (up to
a subsequence), therefore $d(u,K)\leq\delta$. On the other hand, if
$d(u,K)\leq\delta$ then there exists a critical point in $B_{\delta}(u)$,
hence $\sigma(u)=0$.
\end{proof}

\begin{lemma}
\label{lem:hu}For all $u\in B$ such that $d(u,K)\geq2\delta$ there exists a
continuous map $\mathcal{H}_{u}:B_{\delta}(u)\times\lbrack0,\delta
/2]\to  B$ such that
\begin{gather*}
||\mathcal{H}_{u}(v,t)-v||\leq t\\
J(\mathcal{H}_{u}(v,t))\leq J(v)\\
J(\mathcal{H}_{u}(v,t))\leq J(v)-\sigma(u)t\text{ for all }v\in B_{\delta
/2}(u)\\
\mathcal{H}_{u}(v,t)=v\text{ for all }v\notin B_{\delta}(u)\text{.}
\end{gather*}

\begin{proof}
It follows from Theorem 2.11 in \cite{cdm} by replacing $\delta$ with
$\delta/2$, $\sigma$ with $\sigma(u)$, setting $C=B_{\delta/2}(u)$ and
recalling the definition of $\sigma(u)$.
\end{proof}

\begin{remark}
\label{rem:Hexists}By the definition of weak slope, for all $u\in B$ and all
$\sigma<|dJ|(u)$ there exists $\delta_{u}$ and a continuous map $\mathcal{H}
_{u}:B_{\delta_{u}}(u)\times\lbrack0,\delta_{u}]\to  B$ such that for
all $v\in B_{\delta_{u}}(u)$ and all $t\in\lbrack0,\delta_{u}]$
\begin{gather*}
||\mathcal{H}_{u}(v,t)-v||\leq t\\
J(\mathcal{H}_{u}(v,t))\leq J(v)\\
J(\mathcal{H}_{u}(v,t))\leq J(v)-\sigma t.
\end{gather*}
We point out that in the above statement we can choose $\sigma=\sigma(u)$,
indeed either $\sigma(u)<|dJ|(u)$ and there is nothing to prove, or
$\sigma(u)=|dJ|(u)$, but in this case $|dJ|(v)\geq\sigma(u)$ for all $v\in
B_{\delta}(u)$ and the statement follows again by Theorem 2.11 in \cite{cdm}.
\end{remark}
\end{lemma}

\begin{theorem}
There exist a continuous map $\eta:X\times\lbrack0,+\infty)\to  X$ such
that for all $u\in X$ and $t\in\lbrack0,+\infty)$ we have
\begin{gather*}
||\eta(u,t)-u||\leq t\\
J(\eta(u,t))\leq J(u)
\end{gather*}
Furthermore, if $u\notin\mathcal{U}_{2\delta}$, then $J(\eta(u,t))\leq
J(u)-\sigma(u)$ for all $t\in\lbrack0,\delta/5]$.
\end{theorem}

\begin{proof}
For all $u\in B$ let $\mathcal{H}_{u}$ be the map as in Lemma \ref{lem:hu} if
$d(u,K)\geq2\delta$ or as in Remark \ref{rem:Hexists} if $d(u,K)<2\delta$. The
proof follows exactly as the proof of Theorem 2.8 in \cite{cdm}. The function
$\tau(u)$ must satisfy $\tau(u)=\delta/5$ for all $u\notin\mathcal{U}
_{2\delta}$.
\end{proof}

For all $u\in B$ let $\bar{\sigma}(u)=\inf\limits_{v\in B}\{\sigma
(v)+||v-u||\}$ and
\[
\eta_1(u,t)=\left\{
\begin{array}
[c]{ccc}
\eta\left(  u,\frac{t}{\bar{\sigma}(u)}\right)  & \text{if} & d(u,k)\geq
2\delta\\
\eta\left(  u,\left(  \frac{2d(u,k)}{\delta}-3\right)  \frac{t}{\bar{\sigma
}(u)}+\left(  4-\frac{2d(u,k)}{\delta}\right)  t\right)  & \text{if} &
3\delta/2\leq d(u,k)\leq2\delta\\
\eta\left(  u,t\right)  & \text{if} & d(u,k)\leq3\delta/2
\end{array}
\right.
\]
and let $\tau_1(u)=\frac{\delta}{5}\bar{\sigma}(u)$; note that $\bar{\sigma
}$ is Lipschitz continuous, and so is $\tau_1$. We have
\begin{equation}
\begin{array}
[c]{l}
||\eta_1(u,t)-u||\leq\frac{t}{\bar{\sigma}(u)}\\
t\leq\tau_1(u)\Rightarrow J(\eta_1(u,t))\leq J(u)-t.
\end{array}
\label{eta1}
\end{equation}
Define recursively for $m\geq2$
\begin{equation}
\begin{array}
[c]{l}
\eta_m(u,t)=\left\{
\begin{array}
[c]{lll}
\eta_1(\eta_{m-1}(u,\tau_{m-1}(u)),t-\tau_{m-1}(u)) & \text{if} & t\geq
\tau_{m-1}(u)\\
\eta_{m-1}(u,t) & \text{if} & 0\leq t\leq\tau_{m-1}(u)
\end{array}
\right. \\
\tau_m(u)=\tau_{m-1}(u)+\tau_1(\eta_{m-1}(u,\tau_{m-1}(u))).
\end{array}
\label{taun}
\end{equation}

We remark that the definitions of $\eta_m$ and $\tau_m$ match on the
boundaries. Furthermore, it is easy to check by induction that the maps
$\eta_m$ and $\tau_m$ are continuous. Indeed, by the induction assumption
$\tau_m$ is continuous, and $\eta_m$ is also continuous in $(t,u)$ if
$t\neq\tau_{m-1}(u).$ The continuity for all $t$ follows by the following easy
lemma, whose proof is omitted:

\begin{lemma}
Let $A_1,A_2\subset X$ be two closed sets. Let $A_{3}$ be a topological
space, let $f_i:A_i\to  A_{3}$, $i=1,2$, be two continuous
functions satisfying $f_1(x)=f_2(x)$ for all $x\in A_1\cap A_2$. Let
$f:A_1\cup A_2\to  C$ be defined by $f(x)=f_i(x)$ if $x\in A_i
$. Then $f$ is continuous.
\end{lemma}

The following lemma is the crucial step in the construction of the
deformation: it ensures that it is possible to carry on with the deformation
$\eta_m$ as long as the trajectory does not come close to a critical point.

\begin{lemma}
\label{taudiv}If $u\in B$ satisfies $\eta_m(u,\tau_m(u))\notin
\,\mathcal{U}_{2\delta}$, $J(\eta_m(u,\tau_m(u)))$ is bounded away from
$0$ and $\inf\limits_{v\in B_{2\delta}(\eta_m(u,\tau_m(u)))}J(v)>0$ for
all integers $m$, then $\lim\limits_{m\to \infty}\tau_m(u)=+\infty$.
\end{lemma}

\begin{proof}
Assume that $u\in B$ and $\lim\tau_m(u)=c<+\infty.$ Let $u_m=\eta
_m(u,\tau_m(u))$. By (\ref{taun}) if follows that $\tau_1(u_m
)\to 0$ and since $u_m\notin\,\mathcal{U}_{2\delta}$ $\varepsilon
_m=\bar{\sigma}(u_m)\to 0$. By the definition of $\bar{\sigma}$
there exists $\{w_m\}$ such that $d(u_m,w_m)\leq\varepsilon_m$ and
$\sigma(w_m)\leq\varepsilon_m$; by the definition of $\sigma$ there exists
$\{v_m\}$ such that $d(v_m,w_m)\leq\delta$ and $|dJ|(v_m
)\leq\varepsilon_m$; in particular $v_m$ is a Palais-Smale sequence at
positive level and $d(u_m,v_m)\leq\delta+\varepsilon_m$. Furthermore
\begin{align*}
||u_m-u_{m-1}||  &  =||\eta_1(u_{m-1},\tau_m(u)-\tau_{m-1}
(u))-u_{m-1}||\\
&  \leq\frac{\tau_m(u)-\tau_{m-1}(u)}{\bar{\sigma}(u_{m-1})}=\frac{\tau
_1(u_{m-1})}{\bar{\sigma}(u_{m-1})}=\frac{\delta}{5},
\end{align*}
and, if $m$ is large, $||v_m-v_{m-1}||\leq2\delta+2\varepsilon_m
+\delta/5<r_0/2$. Hence by Lemma \ref{annulus}(b) $v_m$ converges, up to a
subsequence, to a critical point of $J$, contradicting $u_m\notin
\,\mathcal{U}_{2\delta}$.
\end{proof}

Let $\mu:=\inf\left\{  \bar{\sigma}(u):u\in\mathcal{U}_{3\delta}
\setminus\mathcal{U}_{2{\delta}}\right\}  .$

\begin{lemma}
$\mu >0$
\end{lemma}

\begin{proof}
If $\mu=0$, then there exists $\{u_{n}\}\subset\mathcal{U}_{3\delta}
\setminus\mathcal{U}_{2{\delta}}$ such that $\bar{\sigma}(u_{n})\to 0$,
by the definition of $\bar{\sigma}$ and $\sigma$, and using the same steps as
in the proof of Lemma \ref{taudiv} there exists a sequence $\{v_{n}
\}\subset\mathcal{U}_{5\delta}\setminus\mathcal{U}_{{\delta/2}}$ such that
$|dJ|(v_{n})\to 0$. This contradicts Lemma \ref{annulus}(a).
\end{proof}

Choose $0<\varepsilon<\min\frac{{\delta}\mu}{4}$ and define $\bar{\eta
}:B\times\lbrack0,2\varepsilon]\to  B$ by
\[
\bar{\eta}(u,t)=\lim\limits_{m\to \infty}\eta_m\left(  u,\min\left\{
t,\frac{\mu}{2}(d(u,K)-2\delta)\right\}  \right)  .
\]

\begin{lemma}
\label{lem:eta}The function $\bar{\eta}$ is well-defined and continuous.
\end{lemma}

\begin{proof}
\emph{Definition.} If $u\in\mathcal{U}_{2\delta}$ there is nothing to prove
because $\eta_m(u,t)\equiv u$ for all $m$. If $u\in B$ satisfies $\eta
_m(u,\tau_m(u))\notin\,\mathcal{U}_{2\delta}$ for all $m$, then by Lemma
\ref{taudiv} there exists $\bar{m}$ such that $\tau_{\bar{m}}(u)\geq
2\varepsilon$ for all $m\geq\bar{m}$, hence $\bar{\eta}(u,t)=\eta_{\bar{m}
}(u,t)$ and the definition is well posed. Assume instead that $u\in B$
satisfies $\eta_m(u,\tau_m(u))\notin\,\mathcal{U}_{2\delta}$ for all $m$
smaller than some integer $\bar{m}$, but $\eta_{\bar{m}}(u,\tau_{\bar{m}
}(u))\in\,\mathcal{U}_{2\delta}$. Then there exists a minimal $t_2\leq
\tau_{\bar{m}}(u)$ such that $\eta_{\bar{m}}(u,t_2)\in\partial
\,\mathcal{U}_{2\delta}$; furthermore there exists $t_1$ (possibly $0$) such
that $\eta_{\bar{m}}(u,t)\in\,\mathcal{U}_{3\delta}$ for all $t_1\leq t\leq
t_2,$ and $t_2-t_1\leq\min\left\{  2\varepsilon,\frac{\mu}
{2}(d(u,K)-2\delta)\right\}  $. By (\ref{eta1}) we have
\begin{align*}
||\eta_{\bar{m}}(t_1,u)-\eta_{\bar{m}}(t_2,u)||  &  \leq\min\left\{
2\varepsilon,\frac{\mu}{2}(d(u,K)-2\delta)\right\}  \left(  \min
\limits_{s\in\lbrack s_1,s_2]}\bar{\sigma}(\eta(s,u))\right)  ^{-1}\\
&  \leq\min\left\{  \frac{\delta}{2},\frac{d(u,K)-2\delta}{2}\right\}
\end{align*}

which is a contradiction because either $u\notin\,\mathcal{U}_{3\delta}$ and
then $||\eta_{\bar{m}}(t_1,u)-\eta_{\bar{m}}(t_2,u)||\geq\delta$, or
$u\in\,\mathcal{U}_{3\delta}$, and then $||\eta_{\bar{m}}(t_1,u)-\eta
_{\bar{m}}(t_2,u)||\geq d(u,K)-2\delta$.

\emph{Continuity.} Fix $(u,t)$, assume that $t<\frac{\mu}{2}(d(u,K)-2\delta)$
(the other case is simpler) and let $m(u,t)$ be the lowest integer satisfying
$\bar{\eta}(u,t)=\eta_m(u,t)$. If $\tau_{m(u,t)}(u)>t$, then $\bar{\eta
}=\eta_m$ in a neighborhood of $(u,t)$ and the continuity of $\bar{\eta}$ at
$(u,t)$ follows by the continuity of $\eta_m$. If $\tau_{m(u,t)}(u)=t$, take
two sequences $t^{h}\to  t$ and $u^{h}\to  u$; then (possibly
neglecting the first terms) either $m(u^{h},t^{h})=m(u,t)$ or $m(u^{h}
,t^{h})=m(u,t)-1$. To see this, note that $\tau_1(u)>0$ for all $u$
satisfying $d(u,K)\geq2\delta$, hence by the continuity of the functions
$\tau_m$ and $\eta_m$ we have $\tau_{m-1}(u^{h})\leq t^{h}\leq\tau
_{m+1}(u^{h})$ for large values of $h$. If $\tau_{m-1}(u)\leq t^{h}\leq
\tau_m(u)$, then $m(u^{h},t^{h})=m(u,t)-1$, otherwise $m(u^{h}
,t^{h})=m(u,t).$ The conclusions follows by the continuity of $\eta_m$ and
$\eta_{m-1}$.
\end{proof}

\begin{lemma}
\label{lem:getdown}For all $u\in J^{c+\varepsilon}\setminus\mathcal{U}
_{3{\delta}}$ and all $t\in\lbrack0,2\varepsilon]$ satisfying $\bar{\eta
}(u,t)\notin\,\mathcal{U}_{2\delta}$ we have $J(\bar{\eta}(u,t))\leq J(u)-t$
and $||\bar{\eta}(u,t)-u||\leq t/\mu_{t},$ where $\mu_{t}=\min\limits_{s\in
\lbrack0,t]}\bar{\sigma}(\bar{\eta}(u,s))$.
\end{lemma}

\begin{proof}
It follows from the definition of $\bar{\eta}$, (\ref{eta1}) and (\ref{taun}).
\end{proof}

Let $c\geq\inf\{J(u):u\in K\}$. If $\varepsilon\geq c/2,$ change $\varepsilon$
to $c/2$.

\begin{lemma}
\label{deflemma}There exists a continuous map $f:B\to  B$ such that
$f(J^{c+\varepsilon}\setminus\mathcal{U}_{3{\delta}})\subset J^{c-\varepsilon
}$.
\end{lemma}

\begin{proof}
For all $u\in J^{c+\varepsilon}$ let $f(u)=\bar{\eta}(u,2\varepsilon)$. Let
$u\in J^{c+\varepsilon}\setminus\mathcal{U}_{3{\delta}}$; as in Lemma
\ref{lem:eta} we prove that $\bar{\eta}(u,t)\notin\mathcal{U}_{2{\delta}}$ for
all $t\in\lbrack0,2\varepsilon]$. By Lemma \ref{lem:getdown} we have
$J(f(u))\leq c-\varepsilon$.
\end{proof}

Assume now that $G$ is some compact Lie group, $B$ is a $G-$space and $J$ is
invariant with respect to the action of $G$ (besides being invariant with
respect to the action of $\mathbb{D}$). Then it is possible to use the
equivariant weak slope in all the previous proofs and obtain the same result
of Lemma \ref{deflemma} with an equivariant deformation, that is the following theorem:

\begin{theorem}
\label{th:def}There exists a continuous $G-$map $f:B\to  B$ such that
$f(J^{c+\varepsilon}\setminus\mathcal{U}_{3{\delta}})\subset J^{c-\varepsilon
}$.
\end{theorem}

\section{Applications to quasilinear elliptic systems}
%
In the sequel we prove multiplicity results for systems of quasilinear
elliptic equations defined in $\mathbb{R}^{n}$ and periodic. More precisely we
extend some results proved in \cite{ag} to the case where the whole system is periodic.

We consider the following system of $m$ quasilinear elliptic equations in
$\mathbb{R}^{n}$, $n\geq3$:
\[
-D_{j}(a_{ij}(x,u)D_iu_{k})+{\frac{1}{2}}\frac{\partial a_{ij}}{\partial
u_{k}}(x,u)D_iu_{l}D_{j}u_{l}+b_{jk}(x)u_{j}=\frac{\partial F}{\partial
u_{k}}(x,u)
\]
where $u:\mathbb{R}^{n}\to \mathbb{R}^{m}$, the indices $i,j$ run from
$1$ to $n$, the indices $k,l$ run from $1$ to $m$ and the assumptions on
$a_{ij}$, $b_{jk}$ and $F$ are given below. In the sequel the sum over
repeated indices is understood and we assume a more concise vectorial notation
by setting $\nabla=\left\{  \frac{\partial}{\partial u_1},\dots
,\frac{\partial}{\partial u_m}\right\}  $ and denoting the scalar product in
$\mathbb{R}^{m}$ by $\langle\cdot,\cdot\rangle$; the system takes the form
\begin{equation}
-D_{j}(a_{ij}(x,u)D_iu)+{\frac{1}{2}}\nabla a_{ij}(x,u)\langle D_i
u,D_{j}u\rangle+b(x)u=\nabla F(x,u). \label{equation}
\end{equation}
In order to prove the existence of weak solutions of (\ref{equation}) in a
suitable functional space $E,$ we look for critical points of the functional
$J:E\to \mathbb{R}$ defined by
\begin{equation}
J(u)=\int_{\mathbb{R}^{n}}{\frac{1}{2}}a_{ij}(x,u)\langle D_iu,D_{j}
u\rangle+{\frac{1}{2}}\langle b(x)u,u\rangle-F(x,u). \label{Jlambda}
\end{equation}
This functional is continuous, but not locally Lipschitz continuous if the
coefficients $a_{ij}$ depend on $u$; however, by using the nonsmooth critical
point theory developed in \cite{cdm,dm} it is possible to define critical
points in a generalized sense: according to this theory, a critical point $u$
of $J$ solves (\ref{equation}) in distributional sense and, a posteriori, it
is a weak solution.

In this paper we consider equation (\ref{equation}) in the case where
$a_{ij}(x,u)$, $b_{jk}(x)$ and $F(x,u)$ are periodic, and therefore the system
is invariant under the action of $\mathbb{Z}^{n}$. We prove a general
existence result and a multiplicity result if the equation is also invariant
under a suitable action of a compact Lie group. The problems we have to deal
with, in order to prove such results, concern the lack of compactness of the functional.

The following condition (A1) is standard in this kind of problems: the matrix
$\left[  a_{ij}(x,s)\right]  $ satisfies an ellipticity property and the
matrix $\left[  \langle s,\nabla a_{ij}(x,s)\rangle\right]  $ is semipositive
definite. More precisely:

\begin{enumerate}
\item [\textbf{(A1)}] The matrix $\left[  a_{ij}(x,s)\right]  $ satisfies
\begin{equation}
\begin{array}
[c]{l}
a_{ij}\equiv a_{ji}\\
a_{ij}(x,s)\in L^{\infty}(\mathbb{R}^{n}\times\mathbb{R}^{m},\mathbb{R})\\
\nabla a_{ij}(x,s)\in L^{\infty}(\mathbb{R}^{n}\times\mathbb{R}^{m}
,\mathbb{R}^{m})\\
a_{ij}(x,\cdot)\in C^{1}(\mathbb{R}^{m})\text{ for a.e. }x\in\mathbb{R}^{n}\\
a_{ij}(x,s)=a_{ij}(x+k,s)\text{ for a.e. }x\in\mathbb{R}^{n}\text{ and all
}k\in\mathbb{Z}^{n}
\end{array}
\label{a}
\end{equation}
and there exists $\nu_1>0$ such that for a.e. $x\in\mathbb{R}^{n}$, all
$s\in\mathbb{R}^{m}$ and all ${\xi}\in\mathbb{R}^{n}$
\begin{equation}
a_{ij}(x,s){\xi}_i{\xi}_{j}\geq\nu_1|{\xi}|^{2} \label{elliptic}
\end{equation}
and
\begin{equation}
\langle s,\nabla a_{ij}(x,s)\rangle{\xi}_i{\xi}_{j}\geq0.
\label{semipositivity}
\end{equation}
\end{enumerate}

The assumption (\ref{semipositivity}) is the natural extension of a
semipositivity condition which is standard in this kind of quasilinear
equations. In some sense it means that the ellipticity of the matrix $\left[
a_{ij}(x,s)\right]  $ increases for increasing values of $|s|$.\bigskip

The following assumption (A2) is a control required on the growth of
ellipticity of the matrix $\left[  a_{ij}(x,s)\right]  $ which seems to be
necessary in order to have the Palais-Smale condition when $m\geq2$.

\begin{enumerate}
\item [\textbf{(A2)}]There exist $K>0$ and a function $\psi:[0,+\infty
)\to \lbrack0,+\infty)$ continuously differentiable almost everywhere
and satisfying
\end{enumerate}

\begin{equation}
\begin{array}
[c]{rl}
\text{(i)} & \psi(0)=0\text{ and }\lim\limits_{t\to+\infty}\psi(t)=K\\
\text{(ii)} & \psi'(t)\ge0\text{ for all }t\in[0,+\infty)\\
\text{(iii)} & \psi'\text{ is non-increasing}\\
\text{(iv)} &
\begin{array}
[c]{l}
\sum\limits_{k=1}^{m}\left|  \frac\partial{\partial s_{k}}a_{ij}(x,s)\xi
_i\xi_{j}\right|  \le2e^{-4K}\psi'(|s|)a_{ij}(x,s)\xi_i\xi
_{j}\text{ for all }s\in\mathbb{R}^{m}\\
\text{all }\xi\in\mathbb{R}^{n}\text{ and a.e. }x\in\mathbb{R}^{n}.
\end{array}
\end{array}
 \label{psi}
\end{equation}

In some sense, $\psi$ is a measure of the growth of ellipticity of the
differential operator; we assume that such growth is ``not too large''. We
refer to \cite{a,ag} for more remarks about (A2).

\begin{enumerate}
\item [\textbf{(B)}]The matrix $[b_{jk}(x)]$ is symmetric, periodic
($b_{jk}(x)=b_{jk}(x+k)$ for all $k\in\mathbb{Z}^{k}$ and all $x\in\mathbb{R}
$), $b_{jk}\in L^{\infty}(\mathbb{R}^{n},\mathbb{R})$ for all $j,k$ and there
exists $\nu_2>0$ such that for a.e. $x\in\mathbb{R}^{n}$ and all ${\xi}
\in\mathbb{R}^{n}$
\begin{equation}
b_{jk}(x){\xi}_{j}{\xi}_{k}\geq\nu_2|{\xi}|^{2}. \label{eq:b}
\end{equation}

\item[\textbf{(F1)}] The function $F:\mathbb{R}^{n}\times\mathbb{R}
^{m}\to \mathbb{R}$ is measurable with respect to the first $n$
variables and $C^{1}$ with respect to the other ones; furthermore we require
\[
\begin{array}
[c]{ll}
F(x,0)=0 & \text{for a.e. }x\in\mathbb{R}^{n}\\
F(x,s)=F(x+k,s) & \text{for a.e. }x\in\mathbb{R}^{n}\text{, all }
k\in\mathbb{Z}^{n}\text{ and all }s\in\mathbb{R}^{m}
\end{array}
\]
\bigskip

\item[\textbf{(F2)}] There exists $p\in(2,2^{\ast})$ and $a>0$ such that
\begin{equation}
0<pF(x,s)\leq\langle s,\nabla F(x,s)\rangle\qquad\text{for all }s\in
\mathbb{R}^{m}\setminus\{0\}\text{ and for a.e. }x\in\mathbb{R}^{n} \label{g}
\end{equation}
and
\begin{equation}
|\nabla F(x,s)|\leq\alpha(1+|s|^{p-1})\quad\text{for all }s\in\mathbb{R}
^{m}\text{ and for a.e. }x\in\mathbb{R}^{n}. \label{subcrit}
\end{equation}
\bigskip
\end{enumerate}

The following assumption is standard for superlinear problems: it relates the
properties of the matrix $[a_{ij}]$ with the function $F$ and it is used to
prove that Palais-Smale sequences are bounded.

\begin{enumerate}
\item [\textbf{(AF)}]There exists $\gamma\in(0,p-2)$ such that ($p$ as in (F2))
\end{enumerate}

\begin{equation}
\langle s,\nabla a_{ij}(x,s)\rangle\xi_i\xi_{j}\le\gamma a_{ij}(x,s)\xi
_i\xi_{j}\quad\text{for a.e. }x\in\mathbb{R}^{n},\text{ for all }
s\in\mathbb{R}^{m}\text{ and }\xi\in\mathbb{R}^{n}\,. \label{AIJ}
\end{equation}

\begin{theorem}
\label{th:1}Assume (A1), (A2), (AF), (B), (F1) and (F2). Then (\ref{equation})
admits a nontrivial weak solution $u\in H$.
\end{theorem}

Due to the periodicity of all coefficients, the system is invariant under the
action of $\mathbb{Z}^{n}$ given by $u(x)\mapsto k\ast u(x)=u(x+k)$. Assume
that the system is also invariant with respect to the action of a compact Lie
group; in that case we prove the following:

\begin{theorem}
\label{th:2}Assume (A1), (A2), (AF), (B), (F1) and (F2). Assume that all
coefficients in (\ref{equation}) are equivariant under the action of an
admissible compact Lie group. Then (\ref{equation}) admits infinitely many
geometrically distinct weak solutions in $H$.
\end{theorem}

The case of a single equation is simpler, although the only symmetry that can
be considered is with respect to the action of $\mathbb{Z}_2$. In this case
we do not need assumption (A2) and we can prove the following:

\begin{theorem}
\label{th:3}Assume (A1), (AF), (B), (F1) and (F2). Assume that $m=1$,
$a_{ij}(x,s)=a_{ij}(x,-s)$ and $F(x,u)=F(x,-u)$. Then (\ref{equation}) admits
infinitely many geometrically distinct weak solutions in $H$.
\end{theorem}

For the definition of admissibility we refer to Section 8. For examples of
nontrivial matrixes $a_{ij}$ satisfying (A1) and (A2) and for examples of
admissible representations we refer to \cite{a,ag}.

\section{Palais-Smale sequences}

By assumption (AF) it is possible to evaluate $J'(u)[u]$ for all $u\in
H$, hence all Palais-Smale sequences satisfy $|J'(u_m)[u_m]|\leq
C\Vert u_m\Vert$ for some $C$. The following result was proved in \cite{cg}
for a single equation:

\begin{lemma}
\label{ps}All Palais-Smale sequences are bounded.
\end{lemma}

\begin{proof}
Taking into account the remark above, we can compute $J(u_m)-{\frac{1}{p}
}J'(u_m)[u_m]$, and taking into account (\ref{g}) and (\ref{AIJ})
we have
\[
\frac{p-2-\gamma}{2p}\int_{\mathbb{R}^{n}}a_{ij}(x,u_m)\langle D_i
u_m,D_{j}u_m\rangle{+}\langle b(x)u_m,u_m\rangle\leq C(\Vert
u_m\Vert+1)
\]
and the result follows by (\ref{elliptic}) and (\ref{eq:b}).
\end{proof}

By $\omega\subset\subset\mathbb{R}^{n}$ we denote an open regular bounded
subset $\omega$ of $\mathbb{R}^{n}$.

\begin{lemma}
\label{exp}Assume (A1) and (A2), let $\{u^{h}\}\subset H$ be a bounded
sequence and set
\[
w^{h}=-D_{j}(a_{ij}(x,u^{h})D_iu^{h})+{\frac{1}{2}}\nabla a_{ij}
(x,u^{h})\langle D_iu^{h},D_{j}u^{h}\rangle.
\]
If $\{w^{h}\}\subset H^{-1}$ and it is strongly convergent to some $w$ in
$H^{-1}(\omega)$ for all $\omega\subset\subset\mathbb{R}^{n}$, then, up to a
subsequence, $\{u_m\}$ converges strongly in $H^{1}(\omega)$ for all
$\omega\subset\subset\mathbb{R}^{n}$.
\end{lemma}

\begin{proof}
Since $\{u^{h}\}$ is bounded, then $u^{h}\rightharpoonup u$ for some $u$ up to
a subsequence. Each component $u_{l}^{h}$ satisfies (2.5) in \cite{bm} and
since Theorem 2.1 in the same paper can be extended to unbounded domains, we
infer that $D_iu_{l}^{h}\to  D_iu_{l}$ a.e. in $\mathbb{R}^{n}$ for
all $l=1,\dots,m$ (see also \cite{dmm}). As in \cite{ag} (see Lemma 6.1) for
all $v\in H\cap L^{\infty}$ we have
\begin{equation}
\int_{{\mathbb{R}^{n}}}a_{ij}(x,u)\langle D_iu,D_{j}v\rangle+{\frac{1}{2}
}\int_{{\mathbb{R}^{n}}}\langle\nabla a_{ij}(x,u),v\rangle\langle D_i
u,D_{j}u\rangle=w[v] \label{tech}
\end{equation}
where $w[v]$ represents the duality product between $w\in H^{-1}$ and $v\in H$.

Now choose $\omega\subset\subset\mathbb{R}^{n}$ and a positive smooth cut-off
function $\chi:\mathbb{R}^{n}\to \lbrack0,1]$ with compact support
$\Omega$ and such that $\chi=1$ on $\omega$. From (\ref{tech}) and by a
density argument, we have
\[
\int_{\mathbb{R}^{n}}a_{ij}(x,u)\langle D_iu,D_{j}(\chi u)\rangle+{\frac
{1}{2}}\langle\nabla a_{ij}(x,u),\chi u\rangle\langle D_iu,D_{j}
u\rangle\ =w[\chi u]
\]
and by Fatou's Lemma, we get
\[
\liminf_{m\to \infty}\int_{\mathbb{R}^{n}}\langle\nabla a_{ij}
(x,u_m),\chi u_m\rangle\langle D_iu_m,D_{j}u_m\rangle\geq
\int_{\mathbb{R}^{n}}\langle\nabla a_{ij}(x,u),\chi u\rangle\langle
D_iu,D_{j}u\rangle;
\]
therefore,
\begin{equation}
\limsup_{m\to \infty}\int_{\mathbb{R}^{n}}a_{ij}(x,u_m)\langle
D_iu_m,D_{j}(\chi u_m)\rangle\leq\int_{\mathbb{R}^{n}}a_{ij}(x,u)\langle
D_iu,D_{j}(\chi u)\rangle. \label{lsup}
\end{equation}
This implies that $\nabla u_m\to \nabla u$ in $L^{2}(\omega)$, indeed
by (\ref{elliptic})
\begin{align*}
&\int_{{\omega}}|\nabla u_m-\nabla u|^{2} \\
 &  \leq\frac{1}{\nu}\int_{{\omega
}}a_{ij}(x,u_m)\langle D_i(u_m-u),D_{j}(u_m-u)\rangle\\
&  \leq\frac{1}{\nu}\int_{{\Omega}}a_{ij}(x,u_m)\langle D_i(u_m
-u),(D_{j}u_m)\chi\rangle-\frac{1}{\nu}a_{ij}(x,u_m)\langle D_i
(u_m-u),(D_{j}u)\chi\rangle;
\end{align*}
\begin{align*}
\frac{1}{\nu}\int_{{\Omega}}a_{ij}(x,u_m)\langle D_i(u_m-u),(D_{j}
u_m)\chi\rangle &  ={\frac{1}{\nu}}\int_{{\Omega}}a_{ij}(x,u_m)(\langle
D_i(\chi(u_m-u)),D_{j}u_m\rangle\\
&  -\langle(D_i\chi)(u_m-u),D_{j}u_m\rangle),
\end{align*}
\[
\int_{{\Omega}}a_{ij}(x,u_m)\langle D_i(u_m-u),(D_{j}u)\chi
\rangle\to 0
\]
because $a_{ij}(x,u_m)D_i(u_m-u)\rightharpoonup0$ in $L^{2}(\Omega)$.
\[
\int_{{\Omega}}a_{ij}(x,u_m)\langle(D_i\chi)(u_m-u),D_{j}u_m
\rangle\to 0
\]
because $u_m\to  u$ in $L^{2}(\Omega)$. Hence
\begin{align*}
\int_{{\omega}}|\nabla u_m-\nabla u|^{2}  &  \leq\frac{1}{\nu}\int_{{\Omega
}}a_{ij}(x,u_m)D_i(\chi(u_m-u))D_{j}u_m+o(1)\\
&  \leq\frac{1}{\nu}\int_{{\Omega}}a_{ij}(x,u)D_i(\chi u)D_{j}u-\frac{1}
{\nu}a_{ij}(x,u_m)D_i(\chi u)D_{j}u_m+o(1)\\
& \leq o(1)
\end{align*}
because of (\ref{lsup}) and
\[
a_{ij}(x,u_m)D_{j}u_m\rightharpoonup a_{ij}(x,u)D_{j}u\text{ in }
L^{2}(\Omega)
\]
for all $i=1,...,n$.
\end{proof}

The case of a single equation is much simpler and has been already treated in
\cite{cg} (see Lemma 3), where the following result was proved.

\begin{lemma}
\label{lem:cg}Assume $m=1$ and (A1), let $\{u^{h}\}\subset H$ be a bounded
sequence and set
\[
w^{h}=-D_{j}(a_{ij}(x,u^{h})D_iu^{h})+{\frac{1}{2}}a_{ij}'
(x,u^{h})D_iu^{h}D_{j}u^{h}.
\]
If $\{w^{h}\}\subset H^{-1}$ and it is strongly convergent to some $w$ in
$H^{-1}(\omega)$ for all $\omega\subset\subset\mathbb{R}^{n}$, then, up to a
subsequence, $\{u_m\}$ converges strongly in $H^{1}(\omega)$ for all
$\omega\subset\subset\mathbb{R}^{n}$.
\end{lemma}

The following proposition collects the previous results:

\begin{proposition}
\label{remark2}Assume (A1), (AF), (B), (F1), (F2) and either $m=1$ or (A2).
Let $\{u_m\}$ be a Palais-Smale sequence for $J$. There exists $\bar{u}\in
${$H$} such that (up to a subsequence)

\begin{enumerate}
\item [(i)]$u_m\rightharpoonup\bar{u}$ in $H$

\item[(ii)] $u_m\to\bar{u}$ in $H^{1}(\omega)$ for every $\omega
\subset\subset\mathbb{R}^{n}$

\item[(iii)] $\bar{u}$ is a weak solution of (\ref{equation}).
\end{enumerate}
\end{proposition}

\begin{proof}
Note first that $\{u_m\}$ is bounded by Lemma \ref{ps} and (i) follows. To
obtain (ii) it suffices to apply Lemma \ref{exp} or \ref{lem:cg} with
$\beta_m=\alpha_m+b(x)u_m+\nabla F(x,u_m)\in H^{-1}$ where $\alpha
_m\to 0$ in $H^{-1}$: indeed, if $u_m\rightharpoonup u$ in $H^{-1}
$, then $\beta_m\to \beta$ in $H^{-1}(\omega)$ for all $\omega
\subset\subset\mathbb{R}^{n}$ with $\beta=b(x)u+\nabla F(x,u)$, (recall the
compact embedding $H^{1}(\omega)\subset L^{2}(\omega)$ and see Theorem 2.2.7
in \cite{cd}). Finally, by (\ref{tech}) $\bar{u}$ is a distributional solution
of (\ref{equation}) and since $D_{j}(a_{ij}(x,u)D_iu)+b(x)u+\nabla F(x,u)\in
H^{-1},$ we also have $\nabla a_{ij}(x,u)\langle D_iu,D_{j}u\rangle\in
H^{-1}$ and the system is solved in a weak sense.
\end{proof}

\begin{remark}
\label{trans}The functional is invariant with respect to the action of
$\mathbb{Z}^{n},$ therefore the same results of Proposition \ref{remark2}
holds for $\{k_m\ast u_m\}$ whenever $\{k_m\}$ is a sequence in
$\mathbb{Z}^{n}$ and $\{u_m\}$ is a PS sequence.
\end{remark}

\section{The representation theorem}
%
To study the behavior of Palais-Smale sequences we need the
concentration compactness lemma \cite{l}, which we state in a form suitable
for our purposes.

\begin{lemma}
\label{cc}Let $\{\rho_{k}\}$ be a sequence of functions $\rho_{k}
:\mathbb{R}^{n}\to \lbrack0,+\infty)$ such that $\int_{\mathbb{R}^{n}
}\rho_{k}(x)dx\to \lambda>0$. Then there exists a subsequence (we still
denote it $\rho_{k})$ such that one of the following occurs:

\begin{itemize}
\item  Concentration. There exists a sequence $\{y_{k}\}\subset\mathbb{R}^{n}$
such that for all $\varepsilon>0$ there exist $R>0$ such that $\liminf
\limits_{k\to \infty}\int_{B_{R}(y_{k})}\rho_{k}(x)dx\geq
\lambda-\varepsilon.$

\item  Dichotomy. There exists $\alpha\in(0,\lambda)$ such that for all
$\varepsilon>0$ there exist a sequence $\{y_{k}\}\subset\mathbb{R}^{n}$ and
$R>0$ such that for all $R'>R$
\[
\limsup_{k\to \infty}\left(  \left|  \int_{B_{R}(y_{k})}\rho
_{k}(x)dx-\alpha\right|  +\left|  \int_{B_{R'}^{c}(y_{k})}\rho
_{k}(x)dx-(\lambda-\alpha)\right|  \right)  \leq\varepsilon.
\]

\item  Vanishing. For all $R>0$, $\sup\limits_{y\in\mathbb{R}^{n}}\int
_{B_{R}(y)}\rho_{k}(x)dx\to 0$ as $k\to \infty$.
\end{itemize}
\end{lemma}

Let $\{u_m\}$ be a Palais-Smale sequence for $J$ at positive level. By Lemma
\ref{ps} $\{u_m\}$ is bounded and we may assume that $||u_m||^{2}
\to \lambda>0$. We apply the concentration compactness technique on
$\rho_m(x)=|\nabla u_m(x)|^{2}+|u_m(x)|^{2}$. In order to exclude the
vanishing case, we make use of the following lemma:

\begin{lemma}
\label{lem:lions}Let $\{u_m\}$ be a bounded sequence in $H$ satisfying
\[
\lim_{m\to \infty}\sup_{x\in\mathbb{R}^{n}}\int_{B_{r}(x_m)}
|u_m|^{2}\,=0
\]
for some $r>0$.
Then $u_m\to 0$ in $L^{s}(\mathbb{R}^{n})$ for each $s\in(2,\infty)$,
$\int_{\mathbb{R}^{n}}F(x,u_m)\to 0$ and $\int_{\mathbb{R}^{n}}\nabla
F(x,u_m)u_m\to 0.$
\end{lemma}

\begin{proof}
The first part is due to P.L. Lions \cite[Lemma I.1]{l}, \cite[Lemma 1.21]{w}.

By (\ref{subcrit}) for all $\varepsilon>0$ there exists $c_{\varepsilon}>0$
satisfying
\[
|\nabla F(x,u)|\leq\varepsilon|u|+c_{\varepsilon}|u|^{p-1}
\]
for all $x\in\mathbb{R}^{n}$ and all $u\in\mathbb{R}$. Hence by H\"{o}lder
inequality,
\[
{\int_{\mathbf{R}^{N}}}|\nabla F(x,u_m)u_m|{\,}\leq\varepsilon
||u_m||_{L^{2}}^{2}+c_{\varepsilon}||u_m||_{L^{p}}^{p}.
\]
Since $\varepsilon$ was chosen arbitrarily, $||u_m||_{L^{2}}^{2}\leq
||u_m||^{2}$ and $u_m\to 0$ in $L^{p}$ by the first part of the
Lemma, it follows that
\[
{\int_{\mathbf{R}^{N}}}\nabla F(x,u_m)u_m{\,}\to 0
\]
and by a similar argument,
\[
{\int_{\mathbf{R}^{N}}}F(x,u_m)\to 0.
\]
\end{proof}

\begin{lemma}
Vanishing cannot occur.
\end{lemma}

\begin{proof}
If by contradiction $\{u_m\}$ is a vanishing sequence, then taking into
account Lemma \ref{lem:lions} we have
\[
2J(u_m)-J'(u_m)[u_m]=-{\frac{1}{2}}\int_{\mathbb{R}^{n}}
\langle\nabla a_{ij}(x,u_m),u_m\rangle\langle D_iu_m,D_{j}u_m
\rangle+o(1).
\]
We conclude by noting that $2J(u_m)-J'(u_m)[u_m]\to  c>0$
and
\[
\limsup-{\frac{1}{2}}\int_{\mathbb{R}^{n}}\langle\nabla a_{ij}(x,u_m
),u_m\rangle\langle D_iu_m,D_{j}u_m\rangle+o(1)\leq0
\]
because of (\ref{semipositivity}).
\end{proof}

\begin{lemma}
\label{lem:conc}If concentration holds, then there exists $\{k_m
\}\subset\mathbb{Z}^{n}$ such that $k_m\ast u_m\to  u\neq0$, up to
a subsequence.
\end{lemma}

\begin{proof}
By the definition of concentration, there exists a sequence $\{y_{k}
\}\subset\mathbb{R}^{n}$ such that for all $\varepsilon>0$ there exist $R>0$
such that $\liminf\limits_{k\to \infty}\int_{B_{R}(y_{k})}|\nabla
u_m(x)|^{2}+|u_m(x)|^{2}\geq\lambda-\varepsilon$. For each $m$ let $k_m$
be the closest $n-$ple of integers to $y_m$, and let $\varepsilon_m$ be a
vanishing positive sequence. Then for all $\varepsilon>0$ there exist $R>0$
such that
\[
\liminf\limits_{k\to \infty}\int_{B_{R}(0)}|\nabla(k_m\ast
u_m(x))|^{2}+|k_m\ast u_m(x)|^{2}\geq\lambda-\varepsilon.
\]
By Lemma \ref{remark2} $k_m\ast u_m\to  u$ in $H^{1}(B_{R}(0))$,
hence $||u||\geq\lambda-\varepsilon$. By the arbitrariness of $\varepsilon$ we
infer $||k_m\ast u_m||\to ||u||$, hence $k_m\ast u_m\to 
u$ in $H$.
\end{proof}

The dichotomy case is more delicate. Let $\{k_m\}\subset\mathbb{Z}^{n}$,
$\{R_m\}\subset\lbrack0,+\infty)$ and $\alpha_1\in(0,\lambda)$ be such
that
\begin{equation}
\left|  \int_{B_{R_m}}k_m\ast\rho_m(x)dx-\alpha_1\right|  \leq\frac
{1}{m}\text{ and }\left|  \int_{B_{2R_m}^{c}}k_m\ast\rho_m
(x)dx-(\lambda-\alpha_1)\right|  \leq\frac{1}{m} \label{eq:dich}
\end{equation}
(taking a subsequence if necessary).

\begin{lemma}
If the dichotomy case occurs and $\{k_m\}$, $\{R_m\}$ and $\alpha_1$ are
as in (\ref{eq:dich}), then $k_m\ast u_m\rightharpoonup u_1\neq0$ and
$u_1$ is a weak solution of (\ref{equation}).
\end{lemma}

\begin{proof}
By Remark \ref{trans} the PS sequence $k_m\ast u_m$ is also a Palais-Smale
sequence, therefore it weakly converges to some solution of (\ref{equation})
$u_1$. Arguing as in Lemma \ref{lem:conc} we have $||u_1||=\alpha_1>0$.
\end{proof}

\begin{remark}
\label{rem:1ok}So far we have proved that a Palais-Smale sequence yields at
least a single solution of the quasilinear system; indeed vanishing cannot
occur, and both concentration and dichotomy provide us with a nontrivial solution.
\end{remark}

If we want instead to apply the deformation technique developed in the
previous sections in order to obtain a multiplicity result, we need a more
detailed study of Palais-Smale sequences; more precisely we need an accurate
description of the ''missing norm'' $(\lambda-\alpha_1)$. To this purpose we
want to get rid of the part of $\{k_m\ast u_m\}$ which strongly converges
to $u$. The standard technique used in this setting consists in showing that
the sequence $v_m:=k_m\ast u_m-u$ is also Palais-Smale, therefore one
can start over the concentration-compactness procedure. This fails for
quasilinear equations, therefore a different approach must be taken.

Let $\varphi_m:\mathbb{R}^{n}\to \lbrack0,1]$ be a sequence of smooth
functions satisfying $\varphi(x)=0$ for all $x\in B_{R_m}$ and
$\varphi(x)=1$ for all $x\notin B_{2R_m}$. Let $v_m(x)=\varphi
_m(x)(k_m\ast u_m(x))$: we have $||v_m||^{2}\to \lambda
-\alpha>0$ and we can apply again the concentration compactness lemma to the
sequence $\rho_m^{1}(x)=|\nabla v_m(x)|^{2}+|v_m(x)|^{2}$.

\begin{lemma}
Let $\{\rho_m^{1}(x)\}$ be defined as above. Vanishing cannot occur.
\end{lemma}

\begin{proof}
If this were the case, then $\int_{\mathbb{R}^{n}}F(x,v_m)\to 0$ and
$\int_{\mathbb{R}^{n}}\nabla F(x,u_m)v_m\to 0$. We have
\[
2J(v_m)-J'(u_m)[v_m]=I_1-I_2-I_{3}+o(1)
\]
where
\begin{align*}
I_1  &  =\int_{\mathbb{R}^{n}}a_{ij}(x,v_m)\langle D_iv_m,D_{j}
v_m\rangle,\\
I_2  &  =\int_{\mathbb{R}^{n}}a_{ij}(x,u_m)\langle D_iu_m,D_{j}
v_m\rangle,\\
I_{3}  &  ={\frac{1}{2}}\int_{\mathbb{R}^{n}}\langle\nabla a_{ij}
(x,u_m),v_m\rangle\langle D_iu_m,D_{j}u_m\rangle.
\end{align*}
Note that $v_m(x)=0$ for all $x\in B_{R_m},$ $v_m(x)=u_m(x)$ for all
$x\notin B_{2R_m}$ and by the dichotomy definition there exists $c>0$ such
that
\[
\int_{R\leq|x|\leq2R}a_{ij}(x,v_m)\langle D_iv_m,D_{j}v_m\rangle
\leq\frac{c}{m}
\]
and 
\[\int_{R\leq|x|\leq2R}a_{ij}(x,u_m)\langle
D_iu_m,D_{j}v_m\rangle\leq\frac{c}{m},
\]
therefore $|I_1-I_2|$ $\leq\frac{2c}{m}$. $I_{3}\geq0$ because of
(\ref{semipositivity}) and $v_m$ is defined as $u_m$ times a positive
scalar function. Finally $J'(u_m)[v_m]\to 0$ because
$\{v_m\}$ is bounded, therefore $J(v_m)\to  c\leq0$. On the other
hand
\[
J(v_m)=\int_{\mathbb{R}^{n}}{\frac{1}{2}}a_{ij}(x,v_m)\langle D_i
v_m,D_{j}v_m\rangle+o(1)\geq c||v_m||^{2}+o(1),
\]
therefore $v_m\to 0$ contradicting $||v_m||^{2}\to 
\lambda-\alpha>0$.
\end{proof}

\begin{lemma}
If concentration or dichotomy holds, then $v_m\rightharpoonup v\not \equiv0$
up to translations and subsequences and $v$ is a weak solution.
\end{lemma}

\begin{proof}
Assume that dichotomy holds (the concentration case is simpler) and let
$\{k_m^{1}\}\subset\mathbb{Z}^{n}$ and $R_m^{1}$ be such that $\left|
\int_{B_{R_m^{1}}}k_m^{1}\ast\rho_m^{1}(x)dx-\alpha_2\right|
\leq\frac{1}{m}$. By the definition of dichotomy we have $k_m^{1}
\to \infty$, therefore $(k_m+k_m^{1})\ast u_m\rightharpoonup v$
and $v$ is a weak solution by Remark \ref{trans}$.$ Furthermore, arguing as in
Lemma \ref{lem:conc} we have $||v||=\alpha_1$.
\end{proof}

Since there exists $c>0$ such that $||u||\geq c$ for all $u\in K$, then by
iterating this procedure for a finite number of times we finally end up with a
sequence strongly convergent to $0$. The following theorem collects all the
results we obtained so far.

\begin{theorem}
\label{th:repr}All Palais-Smale sequences $\{u_m\}\subset H$ at level $c>0$
are \emph{representable}.
\end{theorem}

The following proposition is a trivial consequence of the definition of the
action of $\mathbb{Z}^{m}$ on $H$ and of diverging sequence:

\begin{theorem}
\label{pr:split}For all $l\in\mathbb{N}$, all diverging sequences $\{{\bar{k}
}_m\}\subset(\mathbb{Z}^{m})^{l}$ and all $\bar{u}=(u^{1},\dots,u^{l})\in
H^{l}$
\[
\lim_{m\to \infty}\Vert\bar{k}_m\ast\bar{u}\Vert^{2}=\sum_{j}\Vert
u^{j}\Vert^{2}.
\]
\end{theorem}

\section{ Proofs of the theorems}
%
We first show that the functional $J$ has a mountain pass structure.

\begin{proposition}
\label{pr:mps}

\begin{enumerate}
\item $J(0)=0$

\item  For all finite dimensional subspaces $V$ of $H$ there exists $R>0$ such
that $J(x)\leq0$ for all $x\in V$, $||x||\geq R$

\item  There exists $\rho>0$ such that $J(x)>0$ for all $x,$ $||x||=\rho$
\end{enumerate}
\end{proposition}

\begin{proof}
\begin{enumerate}
\item [1.]Obvious.

\item[2.] Since the function $F$ is superquadratic at $+\infty$, then for all
$x\in H\setminus\{0\}$ we have $\lim\limits_{t\to \infty}J(tx)=-\infty
$; the result follows by compactness.

\item[3.] For all $\varepsilon>0$ there exists $C_{\varepsilon}>0$ such that
$0\leq F(x,s)\leq\varepsilon s^{2}+C_{\varepsilon}s^{2^{\ast}}$, hence, by
(\ref{elliptic}) we have $J(u)\geq C_1\Vert u\Vert^{2}-C_2\Vert
u\Vert^{2^{\ast}}$. The proof follows.
\end{enumerate}
\end{proof}

\begin{proposition}
\label{pr:mp}Choose a function $e\in H\setminus\{0\}$ such that $e\in
{C_{c}^{\infty}}$ and $J(te)\leq0$ for all $t\geq1$ (such function exists by
Proposition \ref{pr:mps}), set
\[
\Gamma=\{\gamma\in C([0,1];H),\ \gamma(0)=0,\ \gamma(1)=e\}
\]
and
\[
b=\inf_{\gamma\in\Gamma}\max_{t\in\lbrack0,1]}J(\gamma(t));
\]
then $b>0$ and $J$ admits a Palais-Smale sequence $\{u_m\}$ at level $b$.
\end{proposition}

\begin{proof}
It follows by Proposition \ref{pr:mps} and the nonsmooth mountain pass Lemma
in \cite{dm}.
\end{proof}

We can now prove Theorem \ref{th:1}: by Propositions \ref{pr:mp} we have a
Palais-Smale sequence at positive level, and by Remark \ref{rem:1ok} it weakly
converges to a nontrivial solution of (\ref{equation}) (up to translations and
a subsequence).

In order to prove our multiplicity result Theorem \ref{th:2} we apply the
symmetric critical point theory presented developed by Bartsch. The following
theorem is an adaptation of Theorem 2.25 in \cite{b}$.$

\begin{theorem}
\label{th:bartsch}Let $G$ be a compact Lie group and let $X$ be a $G-$Hilbert
space. Consider an admissible representation of $G$ on $\mathbb{R}^{m}$, and
assume that $X=\bigoplus_iX_i$ where $X_i\simeq\mathbb{R}^{m}$ and each
$X_i$ is isomorphic to $\mathbb{R}^{m}$ as a representation of $G$. Let
$I:X\to \mathbb{R}$ be a continuous $G-$invariant functional, $I(0)=0$;
assume moreover that there exists an integer $p$ such that

\begin{enumerate}
\item [(i)]There exist $\rho,\beta>0$ such that $I(x)\geq\beta$ for all
$x\in\partial B_{{\rho}}$.

\item[(ii)] For every integer $k$ there exists $R_{k}>\rho$ such that
$I(x)\leq0$ for all $x\in\partial B_{R_{k}}\bigcap\bigoplus\limits_{i=1}
^{k}X_i$.
\end{enumerate}

Let
\[
B_{k}=\left\{  x\in\bigoplus\limits_{i=1}^{k}X_i:||x||\leq R_{k}\right\}  ,
\]

let $\Gamma_{k}$ be the set of $G-$maps of the form $h:B_{k}\to  X$
satisfying $h(x)=x$ if $||x||=R_{k}$ and let
\[
c_{k}=\inf_{h\in\Gamma_{k}}\sup_{x\in B_{k}}I(x)
\]

\noindent Then $c_{k}\to +\infty$.
\end{theorem}

By Proposition \ref{pr:mps} the assumptions of Theorem \ref{th:bartsch} are
satisfied. We need to prove that $\{c_{k}\}$ is a sequence of critical values.
First note that, if (\ref{eq:noncompact}) does not hold, then there is nothing
to prove (see also Remark \ref{rem:nothing}). Condition (\ref{eq:split}) is
satisfied, (see Proposition \ref{pr:split}), and by Theorem \ref{th:repr} all
Palais-Smale sequences are representable. Then all results of Section 3 hold,
in particular Theorem \ref{th:def}. By the existence of an equivariant
deformation and the definition of $c_{k}$ it is a standard procedure to show
that the levels $c_{k}$ are critical and Theorem \ref{th:2} is proved.

The proof of Theorem \ref{th:3} follows exactly the same steps: only recall
that the action of $\mathbb{Z}_2$ on $H$ considered has no fixed points,
therefore it is admissible (see the following section), and assumption (A2) is
not required for a single equation.

\section{Representation theory}
%
For the convenience of the reader we recall here some definitions and known
results from representation theory which are used in this paper; more
information can be found in \cite{b,btd,cp}. In the following, $G$ denotes
some compact Lie group.

\begin{definition}
A compact Lie group $G$ is \emph{solvable} if there exists a sequence
$G_0\subset G_1\subset\ldots\subset G_{r}=G$ of subgroups of $G$ such that
$G_0$ is a maximal torus of $G$, $G_{i-1}$ is a normal subgroup of $G_i$
and $G_i/G_{i-1}\cong\mathbb{Z}/p_i$, $1\leq i\leq r$. Here the $p_i$'s
are prime numbers.
\end{definition}

\begin{remark}
\emph{If }$G$\emph{\ is abelian, then }$G$\emph{\ is isomorphic to the product
of a torus with a finite abelian group \cite[Corollary I.3.7]{btd}. In
particular, all abelian compact Lie groups are solvable.}
\end{remark}

\begin{definition}
A $G$-space is a topological space $E$ together with a continuous action
\[
G\times E\to  E\hspace{1cm}(g,x)\mapsto gx
\]
satisfying $(gh)x=g(hx)$ and $ex=x$ for all $g,h\in G$ and all $x\in E$, where
$e\in G$ denotes the unit element.
\end{definition}

\begin{definition}
Let $E$ and $\tilde{E}$ be two $G$-spaces. A subset $B$ of $E$ is said to be
\emph{invariant} if $gB\subset B$ for all $g\in G$. A functional
$I:E\to \mathbb{R}$ is said to be \emph{invariant} if $I(gx)=I(x)$ for
all $g\in G$ and all $x\in E$. A map $f:E\to \tilde{E}$ is said to be
\emph{equivariant} if $f(gx)=gf(x)$ for all $g\in G$ and all $x\in E$. A
continuous equivariant map between two $G$-spaces is called a $G$-map.
\end{definition}

\begin{definition}
Let $E$ be a $G$-space. A homotopy $\eta:[0,1]\times E\to  E$ is called
a $G$-homotopy if $\eta(t,\cdot)$ is a $G$-map for all $t\in\lbrack0,1]$.
\end{definition}

\begin{definition}
Let $E$ be a $G-$space provided with a Hilbert structure given by the scalar
product $\langle\cdot,\cdot\rangle$. If the group action is linear and
preserves the scalar product, i.e. $\langle gx,gy\rangle=\langle x,y\rangle$
for all $x,y\in E$ and all $g\in G$, then $E$ is called a $G-$Hilbert space.
\end{definition}

\begin{definition}
The space $E^{G}:=\{x\in E:gx=x\ $for all $g\in G\}$ is called the \emph{fixed
point space} of (the representation of) $G.$
\end{definition}

\begin{definition}
The \emph{orbit} of $x$ is defined by $\mathcal{O}_{G}(x):=\{gx:g\in G\}$. We
say that $x,y\in E$ are \emph{geometrically distinct} if $y\notin
\mathcal{O}_{G}(x)$.
\end{definition}

First we choose a representation of a compact Lie group $G$ in $H$: given a
representation of $G$\ in $\mathbb{R}^{m}$\ (i.e. a finite dimensional
$G$-space which we identify with $\mathbb{R}^{m}$) the natural choice is
$g(u)(x):=g(u(x))$. We have to restrict our choice of representations as follows:

\begin{definition}
Let $V$ be a finite-dimensional $G$-space. $V$ is called \emph{admissible} if
for each open, bounded and invariant neighborhood $\mathcal{U}$ of $0$ in
$V^{k}$ ($k\geq1$) and each equivariant map $f:\overline{\mathcal{U}
}\to  V^{k-1}$, $f^{-1}(0)\cap\partial\mathcal{U}\neq\emptyset$.
\end{definition}

\begin{remark}
\label{admissible}\emph{The admissibility of a representation space consists
substantially in requiring that a generalized Borsuk-Ulam theorem holds. In
\cite[Theorem 3.7]{b} it is proved that }$V$\emph{\ is admissible if and only
if there exist subgroups }$K\subset H$\emph{\ of }$G$\emph{\ such that }
$K$\emph{\ is normal in }$H$\emph{, }$H/K$\emph{\ is solvable, }$V^{K}\neq
0$\emph{\ and }$V^{H}=0$\emph{. It follows that, if }$G$\emph{\ is solvable,
then any finite-dimensional representation space }$V$\emph{\ with }
$V^{G}=\{0\}$\emph{\ is admissible. Furthermore, all admissible
representations have a trivial fixed point space.}
\end{remark}

{\bf Acknowledgements:} The author is very grateful to Marco Degiovanni for many
useful discussions.

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\end{document}