\documentstyle{amsart} 
\begin{document} 
{\noindent\small {\em Electronic Journal of Differential Equations},
Vol.\ {\bf 1999}(1999), No.~14, pp. 1--12.\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 1999 Southwest Texas State University  and 
University of North Texas.} 
\vspace{1.5cm}

\title[\hfilneg EJDE--1999/14\hfil Quasilinear diagonal elliptic systems]
{Existence of multiple solutions for quasilinear diagonal elliptic systems} 

\author[Marco Squassina\hfil EJDE--1999/14\hfilneg]
{Marco Squassina}

\address{ Marco Squassina \hfill\break
Dipartimento di Matematica, Milan University, Via Saldini 50, 20133 Milano, Italy }
\email{squassin@@ares.mat.unimi.it }

\date{}
\thanks{Submitted January 4, 1999. Published May 10, 1999.}
\subjclass{35D05, 35J20, 35J60}
\keywords{Quasilinear elliptic differential systems, \hfill\break\indent
  Nonsmooth critical point theory.}


\begin{abstract}
Nonsmooth-critical-point theory is applied in proving multiplicity 
results for the quasilinear symmetric elliptic system
$$
-\sum_{i,j=1}^{n}D_j(a^{k}_{ij}(x,u)D_iu_k)+
{\frac 12}\sum_{i,j=1}^{n}\sum_{h=1}^N
D_{s_k}a^{h}_{ij}(x,u)D_iu_hD_ju_h=g_k(x,u)\,, 
$$ 
for $k=1,..,N$.  
\end{abstract}

\maketitle

\newtheorem{theorem}{Theorem}
\newtheorem{lemma}[theorem]{Lemma}
\newtheorem{definition}[theorem]{Definition}

\section{Introduction}
%
Many papers have been published on the study of multiplicity of solutions for 
quasilinear elliptic equations via nonsmooth-critical-point theory;
see e.g. \cite{ab,ag,arioli,c,cvar,cd,cod,pell,s}.
However, for the vectorial case only a few multiplicity results have been
proven: \cite[Theorem 3.2]{s} and recently \cite[Theorem 3.2]{arioli}, 
where systems with multiple identity coefficients are treated. 
In this paper, we consider the following diagonal quasilinear elliptic
system, in an open bounded set $\Omega\subset{\mathbb R}^n$ with $n\ge 3$,
\begin{equation} \label{system}
\begin{gathered} 
-\sum_{i,j=1}^{n}D_j(a^{k}_{ij}(x,u)D_iu_k)+
{\frac 12}\sum_{i,j=1}^{n}\sum_{h=1}^N
D_{s_k}a^{h}_{ij}(x,u)D_iu_hD_ju_h=  \\
= D_{s_k}G(x,u) \quad  \text{in } \Omega\,,
\end{gathered} 
\end{equation}
for $k=1,..,N$, where $u:\Omega\to{\mathbb R}^N$ and $u=0$ on $\partial\Omega.$
To prove the existence of weak solutions, we look for
critical points of the functional 
$f:H^{1}_{0}(\Omega,{\mathbb R}^{N})\to{\mathbb R}$,  
\begin{equation}
f(u)=\frac{1}{2}\int_{\Omega}\sum_{i,j=1}^{n}\sum_{h=1}^{N}
a^{h}_{ij}(x,u)D_iu_hD_ju_h\,dx-
\int_{\Omega}G(x,u)\,dx\,.
\label{Jlambda}
\end{equation}
This functional is not locally Lipschitz if the
coefficients $a^{h}_{ij}$ depend on $u$; however, as pointed out in 
\cite{ab,c}, it is possible to evaluate $f'$,
\begin{eqnarray*}
f'(u)(v)& = &\int_\Omega\sum_{i,j=1}^{n}
\sum_{h=1}^{N}a^{h}_{ij}(x,u)D_iu_hD_jv_h\,dx+ \\
&& +  \frac{1}{2}\int_{\Omega}\sum_{i,j=1}^{n}\sum_{h=1}^{N}D_{s}a^{h}_{ij}(x,u)\cdot vD_iu_hD_ju_h\,dx-
\int_\Omega D_sG(x,u)\cdot v\,dx
\end{eqnarray*}
for all $v\in H^{1}_{0}(\Omega,{\mathbb R}^{N})
\cap L^\infty(\Omega ,{\mathbb R}^N)$. We shall apply the 
nonsmooth-critical-point theory developed in \cite{cdm,dm,ioffe,ka}. 
For notation and related results, the reader is referred to \cite{cd}.
To prove our main result and to provide some regularity of solutions, we 
consider the following assumptions. \smallskip


\noindent{\bf (A1)}
The matrix $\left(a^{h}_{ij}(\cdot,s)\right)$ is measurable
in $x$ for every $s\in{\mathbb R}^N$, and of class $C^1$ in $s$ for a.e.
 $x\in\Omega$ with
$$
a^{h}_{ij}=a^{h}_{ji}\,.
$$
Furthermore, we assume that there exist $\nu >0$ and $C>0$ such that
for a.e. $x\in\Omega$, all $s\in{\mathbb R}^N$
and $\xi\in {\mathbb R}^{nN}$\begin{equation}
\label{elliptic}
\sum_{i,j=1}^{n}\sum_{h=1}^{N}a^{h}_{ij}(x,s){\xi }^{h}_i{\xi }^{h}_j
\ge \nu |{\xi }|^2, \enskip\left|a^{h}_{ij}(x,s)\right|\leq C,
\enskip\left|D_sa^{h}_{ij}(x,s)\right|\leq C
\end{equation}
and\begin{equation}
\sum_{i,j=1}^{n}\sum_{h=1}^{N}s\cdot D_sa^{h}_{ij}(x,s)
{\xi }^{h}_i{\xi }^{h}_j\ge 0.
\label{semipositivity}
\end{equation} \smallskip

\noindent{\bf (A2)} There exists a bounded Lipschitz function 
$\psi:{\mathbb R}\to{\mathbb R}$, such that
for a.e. $x\in \Omega$, for all $\xi\in {\mathbb R}^{nN}$,  
$\sigma\in\{-1,1\}^N$ and $r,s\in{\mathbb R}^{N}$\begin{equation}
\label{key}
\sum_{i,j=1}^{n}\sum_{h=1}^{N}
\left(\frac{1}{2}D_{s}a^{h}_{ij}(x,s)
\cdot\exp_\sigma(r,s)+
a_{ij}^{h}(x,s)D_{s_h}(\exp_\sigma(r,s))_h\right)\xi_i^h\xi_j^h\leq 0,
\end{equation}
where $\left(\exp_{\sigma}(r,s)\right)_i:=
\sigma_i\exp[\sigma_i(\psi(r_i)-\psi(s_i))]$ for each $i=1,..,N$.
\medskip
\medskip

\noindent{\bf (G1)} The function $G(x,s)$ is
measurable in $x$ for all $s\in{\mathbb R}^N$ and of class $C^1$ in $s$ for a.e. $x\in\Omega$, with $G(x,0)=0$. 
Moreover for a.e. $x\in\Omega$ we will denote with
$g(x,\cdot)$ the gradient of $G$ with respect to $s$. \smallskip
\medskip

\noindent{\bf (G2)} For every $\varepsilon>0$ there exists
$a_{\varepsilon}\in L^{2n/(n+2)}(\Omega)$ such that
\begin{equation}
\label{subcritic}
|g(x,s)|\leq a_{\varepsilon}(x) + \varepsilon |s|^{(n+2)/(n-2)}
\end{equation}
for a.e. $x\in\Omega$ and all $s\in{\mathbb R}^N$ and that there exist
$q>2$, $R>0$ such that for all $s\in{\mathbb R}^N$ and for a.e. $x\in \Omega$
\begin{equation}
|s|\geq R\Longrightarrow 0< q G(x,s)\le s\cdot g(x,s).
\label{g}
\end{equation} 

\noindent{\bf (AG)} There exists $\gamma \in(0,q-2)$ such that
for all $\xi\in{\mathbb R}^{nN}$, $s\in{\mathbb R}^N$ and a.e. in 
$\Omega$
\begin{equation}
\sum_{i,j=1}^{n}\sum_{h=1}^{N}s\cdot D_sa^{h}_{ij}(x,s)\xi^{h}_{i}\xi^{h}_{j}\le\gamma
\sum_{i,j=1}^{n}\sum_{h=1}^{N}a^{h}_{ij}(x,s)\xi^{h}_{i}\xi^{h}_{j}\,.
\label{AIJ}
\end{equation} \medskip

Under these assumptions we will prove the following.
\begin{theorem}
\label{main}
Assume that for a.e. $x\in\Omega$ and for each $s\in{\mathbb R}^N$
$$
a_{ij}^{h}(x,-s)=a_{ij}^{h}(x,s),\quad g(x,-s)=-g(x,s)\,.
$$
Then there exists a sequence $(u^{m})\subseteq H^1_0(\Omega,{\mathbb R}^N)$ of 
weak solutions to {\rm (\ref{system})} such that
$$
\lim_{m}f(u^{m})=+\infty\,.
$$
\end{theorem}

The above result is well known for the semilinear scalar problem
$$\begin{gathered}
-\sum_{i,j=1}^{n}D_j(a_{ij}(x)D_iu)=
g(x,u)      \quad \text{in $\Omega$} \\
 u=0   \quad \text{on $\partial\Omega$}\,.
\end{gathered}
$$ 
A. Ambrosetti and P. H. Rabinowitz in \cite{ar,ra} studied this problem 
using techniques of classical critical point theory. 
The quasilinear scalar problem
$$\begin{gathered}
-\sum_{i,j=1}^{n}D_j(a_{ij}(x,u)D_iu)+
{\frac 12}\sum_{i,j=1}^{n}D_sa_{ij}(x,u)D_iuD_ju=
g(x,u)   \quad \text{in }\Omega \\
 u=0   \quad \text{on } \partial\Omega \,,
\end{gathered}$$
was studied in \cite{c,cvar,cd} and in \cite{pell} in a more general setting. 
In this case the functional
$$
f(u)={1\over 2}\int_{\Omega}\sum_{i,j=1}^{n}a_{ij}(x,u)
D_iuD_ju\,dx - \int_{\Omega}G(x,u)\,dx
$$
is continuous under appropriate conditions, but it is not
locally Lipschitz. Consequently, techniques of nonsmooth-critical-point 
theory have to be applied. In the vectorial case, to my knowledge, 
problem (\ref{system})  has only been considered in \cite[Theorem 3.2]{s} and 
recently in \cite[Theorem 3.2]{arioli} for coefficients 
of the type $a^{hk}_{ij}(x,s)=\delta^{hk}\alpha_{ij}(x,s)$. In \cite{arioli}
a new technical condition is introduced to be compared with our (\ref{key}). 
They assume that there exist $K>0$ and an increasing
bounded Lipschitz function $\psi:[0,+\infty[\to[0,+\infty[$, with $\psi(0)=0$, 
$\psi'$ non-increasing, $\psi(t)\to K$ 
as $t\to+\infty$ and such that for all $\xi\in {\mathbb R}^{n}$, 
for a.e. $x\in \Omega$ and for all $r,s\in{\mathbb R}^{N}$\begin{equation}
\label{Arioli}
\sum_{i,j=1}^{n}\sum_{k=1}^{N}
\left|D_{s_k}a_{ij}(x,s)\xi_i\xi_j\right|\leq
2e^{-4K}\psi'(|s|)\sum_{i,j=1}^{n}a_{ij}(x,s)\xi_i\xi_j\,.
\end{equation}
The proof itself of \cite[Lemma 6.1]{arioli} shows that condition
(\ref{Arioli}) implies our assumption \textbf{(A2)}. 
On the other hand, if $N\geq 2$, the two conditions look quite similar. 
However, condition \textbf{(A2)} seems to be preferable, because when
$N=1$ it reduces to the inequality
$$
\left|\sum_{i,j=1}^{n}D_sa_{ij}(x,s)\xi_i\xi_j\right|\leq
2\psi'(s)\sum_{i,j=1}^{n}a_{ij}(x,s)\xi_i\xi_j,
$$
which is not so restrictive in view of (\ref{elliptic}), 
while (\ref{Arioli}) is in this case much stronger.

\section{Boundedness of concrete Palais-Smale sequences}
%
\begin{definition}
\label{defncpsc} 
Let $c\in{\mathbb R}$. A sequence $(u^m)\subseteq H^1_0(\Omega;{\mathbb R}^N)$ is said to be
{\em a concrete Palais-Smale sequence at level $c$}
({\em $(CPS)_c-$sequence}, in short) for $f$,
if $f(u^m)\to c$,
$$
\sum_{i,j=1}^{n}\sum_{h=1}^ND_{s_k}a^{h}_{ij}(x,u^m)D_iu^m_hD_ju^m_h\in H^{-1}(\Omega;{\mathbb R}^N)
$$
eventually as $m\to\infty$, and
$$
-\sum_{i,j=1}^{n}D_j(a^{k}_{ij}(x,u^m)D_iu^m_k)+
{\frac 12}\sum_{i,j=1}^{n}\sum_{h=1}^N
D_{s_k}a^{h}_{ij}(x,u^m)D_iu^m_hD_ju^m_h-g_k(x,u^m)
$$
converges to zero strongly in $H^{-1}(\Omega;{\mathbb R}^N)$. 
We say that $f$ satisfies
{\em the concrete Palais-Smale condition at level $c$}
($(CPS)_c$ in short),
if every $(CPS)_c-$sequence for $f$ admits a strongly
convergent subsequence in $H^1_0(\Omega;{\mathbb R}^N)$.
\end{definition}

Next we state and prove a vectorial version of Brezis-Browder's Theorem
\cite{bb}.

\begin{lemma}
\label{bbvect}
Let $T\in L_{\rm loc}^1(\Omega,{\mathbb R}^N)\cap H^{-1}(\Omega,{\mathbb R}^N)$, 
$v\in H^1_0(\Omega,{\mathbb R}^N)$ and $\eta\in L^1(\Omega)$ with $T\cdot v\geq\eta$.
Then $T\cdot v\in L^1(\Omega)$ and
$$
\langle T,v\rangle=\int_{\Omega}T\cdot v\,dx
$$
\end{lemma}
\begin{pf}
Let $(v_h)\subseteq C^{\infty}_c(\Omega,{\mathbb R}^N)$ with $v_h\to v$.
Define $\Theta_h(v)\in H^1_0\cap L^{\infty}$ with compact support in $\Omega$ 
by setting
$$
\Theta_h(v)=\min\{|v|,|v_h|\}\frac{v}{\sqrt{|v|^2+\frac{1}{h}}}.
$$
Since 
$$
\min\{|v|,|v_h|\}\frac{T\cdot v}{\sqrt{|v|^2+\frac{1}{h}}}\geq -\eta^-\in L^1(\Omega),
$$
and
$$
\left\langle T,\min\{|v|,|v_h|\}\frac{v}{\sqrt{|v|^2+\frac{1}{h}}}\right\rangle
=\int_\Omega\min\{|v|,|v_h|\}\frac{T\cdot v}{\sqrt{|v|^2+\frac{1}{h}}}\,dx,
$$
a variant of Fatou's Lemma implies
$\int_\Omega T\cdot v\,dx\leq \left\langle T,v\right\rangle$,
so that $T\cdot v\in L^1(\Omega)$. Finally, since
$$
\left|\min\{|v|,|v_h|\}\frac{T\cdot v}{\sqrt{|v|^2+\frac{1}{h}}}\right|\leq |T\cdot v|,
$$
Lebesgue's Theorem yields
$$
\left\langle T,v\right\rangle=\int_\Omega T\cdot v\,dx,
$$
and the proof is complete.
\end{pf}

The first step for the $(CPS)_c$ to hold is the boundedness of $(CPS)_c$ sequences.
\begin{lemma}
\label{bound}
Assume {\rm\bf{(A1)}}, {\rm\bf{(G1)}}, {\rm\bf{(G2)}} and {\rm\bf{(AG)}}. 
Then for all $c\in{\mathbb R}$ each $(CPS)_{c}$ sequence of $f$ 
is bounded in $H^{1}_{0}(\Omega,{\mathbb R}^{N})$. 
\end{lemma}
\begin{pf}
Let $a_{0}\in L^{1}(\Omega)$ be such that
for a.e. $x\in \Omega$ and all $s\in{\mathbb R}^N$ 
$$
qG(x,s)\leq s\cdot g(x,s)+a_{0}(x).
$$
Now let $(u^{m})$ be a $(CPS)_{c}$ sequence for $f$ and let $w^m\to 0$ in 
$H^{-1}(\Omega,{\mathbb R}^N)$ such that
for all $v\in C^{\infty}_{c}(\Omega,{\mathbb R}^{N})$,
\begin{eqnarray*}
 \langle w^{m},v\rangle &=&\int_{\Omega}\sum_{i,j=1}^{n}\sum_{h=1}^{N}
 a_{ij}^{h}(x,u^{m})D_{i}u_{h}^{m}D_{j}v_{h}\,dx+ \\
& &+  {1\over 2}\int_{\Omega}\sum_{i,j=1}^{n}\sum_{h=1}^{N}D_{s}a_{ij}^{h}(x,u^{m})
\cdot vD_{i}u_{h}^{m}D_{j}u_{h}^{m}\,dx-
\int_{\Omega}g(x,u^{m})\cdot v\,.
\end{eqnarray*}
Taking into account the previous Lemma, for every 
$m\in{\mathbb N}$ we obtain
\begin{eqnarray*}
\lefteqn{ -\| w^{m}\|_{H^{-1}(\Omega,{\mathbb R}^{N})}\| u^{m}\|_{H^{1}_{0}
(\Omega,{\mathbb R}^{N})}\leq } \\
& \leq & \int_{\Omega}\sum_{i,j=1}^{n}\sum_{h=1}^{N}a_{ij}^{h}(x,u^{m})
D_{i}u_{h}^{m}D_{j}u_{h}^{m}\,dx+ \\
&& + {1\over 2}\int_{\Omega}\sum_{i,j=1}^{n}\sum_{h=1}^{N}D_{s}a_{ij}^{h}
  (x,u^{m})\cdot u^m D_{i}u_{h}^{m}D_{j}u_{h}^{m}\,dx 
  -  \int_{\Omega}g(x,u^{m})\cdot u^{m}\,dx\leq \\
&\leq& 
\int_{\Omega}\sum_{i,j=1}^{n}\sum_{h=1}^{N}a_{ij}^{h}(x,u^{m})D_{i}u_{h}^{m}D_{j}u_{h}^{m}\,dx+ \\
& &+ {1\over 2}\int_{\Omega}\sum_{i,j=1}^{n}\sum_{h=1}^{N}D_{s}a_{ij}^{h}(x,u^{m})
\cdot u^m D_{i}u_{h}^{m}D_{j}u_{h}^{m}\,dx+ \\
& &-  q\int_{\Omega}G(x,u^{m})\,d x+\int_{\Omega}a_{0}\,dx\,.
\end{eqnarray*}
Taking into account the expression of $f$ and assumption {\rm\bf{(AG)}}, 
we have that for each $m\in{\mathbb N}$,
\begin{eqnarray*}
\lefteqn{ -\| w^{m}\|_{H^{-1}(\Omega,{\mathbb R}^{N})}\| u^{m}\|_{H^{1}_{0}
(\Omega,{\mathbb R}^{N})}\leq  }\\
& \leq & -\left({q\over 2}-1\right)\int_{\Omega}\sum_{i,j=1}^{n}\sum_{h=1}^{N}a_{ij}^{h}(x,u^{m})D_{i}u_{h}^{m}
D_{j}u_{h}^{m}\,dx+ \\
& &+ {1\over 2}\int_{\Omega}\sum_{i,j=1}^{n}
\sum_{h=1}^{N}D_{s}a_{ij}^{h}(x,u^{m})\cdot u^m D_{i}u_h^{m}D_{j}u_h^{m}\,d x+qf(u^{m})+\int_{\Omega}a_{0}\,dx\leq \\\ 
& \leq & -\left({q\over 2}-1-{{\gamma}\over 2}\right)
\int_{\Omega}\sum_{i,j=1}^{n}\sum_{h=1}^{N}a_{ij}^{h}(x,u^{m})D_{i}u_{h}^{m}D_{j}u_{h}^{m}\,dx+ \\
& &+  qf(u^{m})+\int_{\Omega}a_{0}\,dx\,.
\end{eqnarray*}
Because of {\rm\bf{(A1)}},  for each $m\in{\mathbb N}$,
\begin{eqnarray*}
\nu(q-2-\gamma)\| Du^{m}\|^2_2
& \leq &  (q-2-\gamma)\int_{\Omega}\sum_{i,j=1}^{n}\sum_{h=1}^{N}a_{ij}^{h}(x,u^{m})D_{i}u_{h}^{m}D_{j}u_{h}^{m}\,dx \\
& \leq &  2\| w^{m}\|_{H^{-1}(\Omega,{\mathbb R}^{N})}\| u^{m}\|_{H^{1}_{0}(\Omega,{\mathbb R}^{N})}+
2qf(u^{m})+2\int_{\Omega}a_{0}\,dx\,.
\end{eqnarray*}
Since $(w^{m})$ converges to $0$ in $H^{-1}(\Omega,{\mathbb R}^{N})$, we conclude that
$(u^{m})$ is a bounded sequence in $H^{1}_{0}(\Omega,{\mathbb R}^{N})$.
\end{pf}

\begin{lemma}
\label{gcc}
If condition {\rm(\ref{subcritic})} holds, then the map 
\begin{eqnarray*}
H^{1}_{0}(\Omega,{\mathbb R}^{N}) & \longrightarrow & 
L^{2n/(n+2)}(\Omega,{\mathbb R}^{N})  \\
u & \longmapsto & g(x,u) 
\end{eqnarray*}
is completely continuous.
\end{lemma}
\begin{pf}
This is a direct consequence of \cite[Theorem 2.2.7]{cd}.
\end{pf}

\section{Compactness of concrete Palais-Smale sequences}
%
The next result is crucial for the $(CPS)_c$ condition to hold for our elliptic system.
\begin{lemma}
\label{comp}
Assume {\rm\bf (A1)} and {\rm\bf(A2)}, let $(u^m)$ be a bounded sequence in
$H^{1}_{0}(\Omega,{\mathbb R}^{N})$, and set
\begin{eqnarray*}
\langle w^m,v\rangle &=& \int_{\Omega}\sum_{i,j=1}^{n}
  \sum_{h=1}^{N}a^{h}_{ij}(x,u^m)D_iu_{h}^mD_jv_{h}\,dx+  \\
&& +\frac 12\int_{\Omega}\sum_{i,j=1}^{n}
 \sum_{h=1}^{N}D_{s}a^{h}_{ij}(x,u^m)\cdot vD_iu_h^mD_ju_h^m\,dx
\end{eqnarray*}
for all $v\in C^{\infty}_c(\Omega,{\mathbb R}^N)$. If $(w^m)$ is strongly 
convergent to some $w$ in $H^{-1}(\Omega,{\mathbb R}^{N})$,
then $(u^m)$ admits a strongly convergent subsequence in 
$H^{1}_{0}(\Omega,{\mathbb R}^{N})$.
\end{lemma}

\begin{pf}
Since $(u^m)$ is bounded, we have $u^m\rightharpoonup u$ for some $u$ up to a
subsequence. Each component $u_k^m$ satisfies (2.5) in \cite{bm}, so we may suppose
that $D_iu_k^m\rightarrow D_iu_k$ a.e. in $\Omega$ for all $k=1,\dots,N$
(see also \cite{dmm}). We first prove that
\begin{multline}
\int_\Omega\sum_{i,j=1}^{n}\sum_{h=1}^{N}a^{h}_{ij}(x,u)D_iu_hD_ju_h\,dx+  \\
+{\frac 12}\int_\Omega\sum_{i,j=1}^{n}\sum_{h=1}^{N}D_{s}a^{h}_{ij}(x,u)
\cdot uD_iu_hD_ju_h\,dx=\langle w,u\rangle.
\label{tech}
\end{multline}
Let $\psi$ be as in assumption {\rm\bf{(A2)}} and consider the following test 
functions
\begin{equation*}
v^m=\varphi (\sigma _1\exp [\sigma _1(\psi (u_1)-\psi (u_1^m))],\ldots
,\sigma_N\exp [\sigma_N(\psi (u_N)-\psi (u_N^m))]),
\end{equation*}
where $\varphi\in C^\infty_c(\Omega)$, $\varphi \ge 0$ and
$\sigma_l=\pm 1$ for all $l$. Therefore, since we have
$$
D_jv_k^m=\left(\sigma_kD_j\varphi+
(\psi'(u_k)D_ju_k-\psi'(u_k^m)D_ju_k^m)\varphi\right)
\exp[\sigma_k(\psi(u_k)-\psi(u_k^m))],
$$
we deduce that for all $m\in{\mathbb N}$,
\begin{multline*}
\int_\Omega\sum_{i,j=1}^{n}
 \sum_{h=1}^{N}a^{h}_{ij}(x,u^m)D_iu_h^m(\sigma_hD_j\varphi
 +\psi '(u_h)D_ju_h\varphi) \exp [\sigma_{h}(\psi (u_h)-\psi (u_h^m))]\,dx+ \\
+ \int_\Omega\sum_{i,j=1}^{n}\sum_{h,l=1}^{N}
  \frac{\sigma_l}{2}D_{s_l}a^{h}_{ij}(x,u^m)\exp [\sigma_l(\psi (u_l)-\psi(u_l^m))]D_iu_h^mD_ju_h^m\varphi\,dx+ \\
- \int_\Omega\sum_{i,j=1}^{n}\sum_{h=1}^{N}a^{h}_{ij}(x,u^m)
   D_iu_h^mD_ju_h^m\psi'(u_h^m)\exp[\sigma_h(\psi(u_h)-\psi (u_h^m))]
   \varphi\,dx=\\
=\langle w^m,v^m\rangle\,. 
\end{multline*}

Let us study the behavior of each term of the previous equality as $m\to\infty$.
First of all, if $v=(\sigma_1\varphi,\dots ,\sigma_N\varphi)$, we have that
$v^m\rightharpoonup v$ implies\begin{equation}
\lim_m\langle w^m,v^m\rangle=\langle w,v\rangle.
\label{passlim}
\end{equation}
Since $u^m\rightharpoonup u$, by Lebesgue's Theorem we obtain
\begin{eqnarray}
\lefteqn{ \lim_m\int_\Omega\sum_{i,j=1}^{n}\sum_{h=1}^{N}a^{h}_{ij}(x,u^m)
D_iu_h^m(D_j(\sigma_h\varphi)+  } \nonumber\\
&& \hspace{15mm}+\varphi\psi'(u_h)
D_ju_h)\exp[\sigma_{h}(\psi (u_h)-\psi (u_h^m))]\,dx=  \label{doppia}\\
& = & \int_\Omega\sum_{i,j=1}^{n}
\sum_{h=1}^{N}a^{h}_{ij}(x,u)D_iu_h(D_jv_h+\varphi\psi '(u_h)D_ju_h)\,dx\,.\nonumber
\end{eqnarray}
Finally, note that by assumption {\rm\bf{(A2)}} we have
\begin{eqnarray*}
\sum_{i,j=1}^{n}\sum_{h=1}^{N}\Big(\sum_{l=1}^{N}
\frac{\sigma_l}{2}D_{s_l}a^{h}_{ij}(x,u^m)\exp [\sigma_l(\psi (u_l)
-\psi(u_l^m))]+ && \\
 -a^{h}_{ij}(x,u^m)\psi'(u_h^m)\exp[\sigma_h(\psi(u_h)-\psi (u_h^m))]\Big)
 D_iu_h^mD_ju_h^m&\leq& 0\,.
\end{eqnarray*}
Hence, we can apply Fatou's Lemma to obtain
\begin{eqnarray*}
\lefteqn{ \limsup_m\Big\{\frac{1}{2}\int_\Omega
\sum_{i,j=1}^{n}\sum_{h,l=1}^{N}D_{s_l}a^{h}_{ij}(x,u^m)\exp [\sigma_l(\psi (u_l)-\psi(u_l^m))]
D_iu_h^mD_ju_h^m(\sigma_l\varphi)\,dx+  }\\
\lefteqn{ -\int_\Omega\sum_{i,j=1}^{n}\sum_{h=1}^{N}a^{h}_{ij}(x,u^m)D_iu_h^mD_ju_h^m
\psi '(u_h^m)\exp[\sigma_h(\psi (u_h)-\psi (u_h^m))]\varphi\,dx\Big\}\leq   }\\
&\leq& \frac{1}{2}\int_\Omega\sum_{i,j=1}^{n}\sum_{h,l=1}^{N}D_{s_l}a^{h}_{ij}
(x,u)D_iu_hD_ju_h(\sigma_l\varphi)\,dx+ \\
&&-\int_\Omega\sum_{i,j=1}^{n}\sum_{h=1}^{N}a^{h}_{ij}(x,u)D_iu_hD_ju_h\psi'
(u_h)\varphi\,dx \,, \hspace{48mm}
\end{eqnarray*}
which, together with (\ref{passlim}) and (\ref{doppia}), yields
\begin{multline*}
\int_\Omega\sum_{i,j=1}^{n}\sum_{h=1}^{N}a^{h}_{ij}(x,u)D_iu_hD_jv_h\,dx+ \\
+\frac{1}{2}\int_\Omega\sum_{i,j=1}^{n}\sum_{h=1}^{N}
D_{s}a^{h}_{ij}(x,u)\cdot vD_iu_hD_ju_h\,dx\ge \langle w,v\rangle
\end{multline*}
for all test functions $v=(\sigma_1\varphi ,\dots ,\sigma_N\varphi )$ with
$\varphi\in C_c^\infty(\Omega,{\mathbb R}^N)$, $\varphi \ge 0$.
Since we may exchange $v$ with $-v$ we get 
\begin{multline*}
\int_\Omega\sum_{i,j=1}^{n}\sum_{h=1}^{N}a^{h}_{ij}(x,u)D_iu_hD_jv_h\,dx+ \\
+{\frac 12}\int_\Omega\sum_{i,j=1}^{n}\sum_{h=1}^{N}
D_{s}a^{h}_{ij}(x,u)\cdot vD_iu_hD_ju_h\,dx=\langle w,v\rangle
\end{multline*}
for all test functions $v=(\sigma _1\varphi ,\dots ,\sigma_N\varphi )$, and since
every function $v\in C_c^\infty(\Omega,{\mathbb R}^N)$ can be written as a linear combination
of such functions, taking into account Lemma \ref{bbvect}, 
we infer (\ref{tech}). Now, let us prove that\begin{equation}
\limsup_m\int_\Omega
\sum_{i,j=1}^{n}\sum_{h=1}^{N}a^{h}_{ij}(x,u^m)D_iu_h^mD_ju_h^m\,dx\leq
\int_\Omega\sum_{i,j=1}^{n}\sum_{h=1}^{N}a^{h}_{ij}(x,u)D_iu_hD_ju_h\,dx.
\label{limsup}
\end{equation}
Because of~(\ref{semipositivity}), Fatou's Lemma implies that
\begin{eqnarray*}
\lefteqn{ \int_{\Omega}\sum_{i,j=1}^n\sum_{h=1}^{N}u\cdot 
D_sa^{h}_{ij}(x,u)D_iu_hD_ju_h\,dx\leq  }\\
&\leq&  \liminf_m \int_{\Omega}\sum_{i,j=1}^n\sum_{h=1}^{N}u^m\cdot 
D_sa^{h}_{ij}(x,u^m)D_iu^m_hD_ju^m_h\,dx\,.
\end{eqnarray*}
Combining this fact with~(\ref{tech}), we deduce that
\begin{eqnarray*}
\lefteqn{ \limsup_m \int_{\Omega}\sum_{i,j=1}^n\sum_{h=1}^{N}
a^{h}_{ij}(x,u^m)D_iu^m_hD_ju^m_h\,dx=  }\\
&=& \limsup_m\Big[-{1\over 2}\int_{\Omega}\sum_{i,j=1}^n
\sum_{h=1}^{N}u^m\cdot D_sa^{h}_{ij}(x,u^m)
D_iu^m_hD_ju^m_h\,dx+\langle w^m,u^m \rangle \Big]\leq \\
&\leq& -{1\over 2}
\int_{\Omega}\sum_{i,j=1}^n\sum_{h=1}^{N}u\cdot D_sa^{h}_{ij}
(x,u)D_iu_hD_ju_h\,dx+\langle w,u\rangle= \\
&=&\int_{\Omega}\sum_{i,j=1}^n\sum_{h=1}^{N}a^{h}_{ij}(x,u)
D_iu_hD_ju_h\,dx\,,
\end{eqnarray*}
so that (\ref{limsup}) is proved. Finally, by (\ref{elliptic}) we have
\begin{eqnarray*}
\lefteqn{ \nu \| Du^m-Du\|_2 ^2\leq }\\
&\le& \int_\Omega\sum_{i,j=1}^{n}\sum_{h=1}^{N}
a^{h}_{ij}(x,u^m)\left(D_iu_h^mD_ju_h^m-2D_iu_h^mD_ju_h+D_iu_hD_ju_h\right)\,dx.
\end{eqnarray*}
Hence, by (\ref{limsup}) we obtain\begin{equation*}
\limsup_m\Vert Du^m-Du\Vert_2\le 0
\end{equation*}
which proves that $u^m\to u$ in $H^{1}_{0}(\Omega,{\mathbb R}^{N})$.
\end{pf}

We now come to one of the main tools of this paper, the $(CPS)_c$
 condition for system (\ref{system}).
\begin{theorem}
\label{exp}
Assume {\rm\bf{(A1)}}, {\rm\bf{(A2)}}, {\rm\bf{(G1)}}, {\rm\bf{(G2)}},
{\rm\bf{(AG)}}. Then $f$ satisfies $(CPS)_c$ condition for each 
$c\in{\mathbb R}$.
\end{theorem}
\begin{pf}
Let $(u^m)$ be a $(CPS)_c$ sequence for $f$.
Since $(u^m)$ is bounded in $H^{1}_{0}(\Omega,{\mathbb R}^{N})$, from Lemma \ref{gcc} we deduce that, up
to a subsequence, $(g(x,u^m))$ is strongly convergent in $H^{-1}(\Omega,{\mathbb R}^{N})$.
Applying Lemma \ref{comp}, we conclude the present proof.
\end{pf}

\section{Existence of multiple solutions for elliptic systems}
%
We now prove the main result, which is an extension of
theorems of \cite{c,cd} and a generalization of \cite[Theorem 3.2]{arioli} 
to systems in diagonal form.

\vskip5pt
\noindent{\it Proof of Theorem} \ref{main}.  
We want to apply \cite[Theorem 2.1.6]{cd}. First of all, because of
Theorem \ref{exp}, $f$ satisfies $(CPS)_{c}$ for all $c\in{\mathbb R}$.
Whence, $(c)$ of \cite[Theorem 2.1.6]{cd} is satisfied. Moreover we have
\begin{eqnarray*}
{\nu\over 2}\int_{\Omega}\vert Du\vert^{2}\,dx-\int_{\Omega}G(x,u)\,dx 
&\leq& f(u)\leq \\
&\leq& {1\over 2}nNC\int_{\Omega}\vert Du\vert^{2}\,dx-
\int_{\Omega}G(x,u)\,dx.
\end{eqnarray*}
We want to prove that assumptions (a) and (b) of \cite[Theorem 2.1.6]{cd} are also satisfied.
Let us observe that, instead of (b) of \cite[Theorem 2.1.6]{cd}, it is enough to find a sequence 
$\left(W_n\right)$ of finite dimensional subspaces with $\dim(W_n)\to+\infty$ satisfying the inequality
of $(b)$ (see also  \cite[Theorem 1.2]{marzo}). 
Let $W$ be a finite dimensional subspace of $H^{1}_{0}(\Omega;{\mathbb R}^{N})\cap L^\infty(\Omega,{\mathbb R}^N)$. 
From (\ref{g}) we deduce that for all $s\in{\mathbb R}^N$
with $|s|\geq R$  
$$ G(x,s)\geq {{G\left(x,R{s\over{\vert s\vert}}\right)}
\over R^{q}}\vert s\vert^{q}\geq b_0(x)|s|^q,
$$
where
$$
b_0(x)=R^{-q}\inf\{G(x,s): |s|=R\}>0
$$
a.e. $x\in\Omega$. 
Therefore there exists $a_0\in L^1(\Omega)$ such that\begin{equation}
\label{a0}
G(x,s)\geq b_0(x)|s|^q-a_0(x)
\end{equation}
a.e. $x\in\Omega$ and for all $s\in{\mathbb R}^N$. Since $b_0\in L^1(\Omega),$
we may define a norm $\|\cdot\|_G$ on $W$ by
$$
\|u\|_G=\left(\int_{\Omega}b_0|u|^q\,dx\right)^{1/ q}.
$$
Since $W$ is finite dimensional and $q>2$, from (\ref{a0}) it follows
$$
\lim_{\|u\|_G\to+\infty\atop u\in W}f(u)=-\infty
$$
and condition (b) of \cite[Theorem 2.1.6]{cd} is clearly fulfilled too for
a sufficiently large $R>0$. Let now $(\lambda_{h},u_{h})$ be the sequence of eigenvalues
and eigenvectors for the problem
\begin{gather*}
\Delta u=-\lambda u \quad \text{in }\Omega \\
 u=0 \quad \text{on } \partial\Omega \,.
\end{gather*}
Let us prove that there exist $h_0,\alpha>0$ such that
$$
\forall u\in V^{+}: \| Du\|_{2}=1
\Longrightarrow f(u)\geq\alpha,
$$
where $V^{+}=\overline{{\rm span}}\left\{u_{h}\in H^1_0(\Omega,{\mathbb R}^N): h\geq h_{0}\right\}$. In fact, given
$u\in V^{+}$ and $\varepsilon>0$, we find 
$$
a_{\varepsilon}^{(1)}\in C^{\infty}_{c}(\Omega),\enskip a_{\varepsilon}^{(2)}
\in L^{{2n}\over{n+2}}(\Omega),
$$
such that $\| a_{\varepsilon}^{(2)}\|_{{2n}\over{n+2}}\leq\varepsilon$ and
$$
|g(x,s)|\leq a_{\varepsilon}^{(1)}(x)+a_{\varepsilon}^{(2)}(x)
+\varepsilon |s|^{n+2\over n-2}.
$$
If $u\in V^{+}$, it follows that\begin{eqnarray*}
f(u) & \geq & {\nu\over 2}\| Du\|_2^2-\int_{\Omega}G(x,u)\,dx\\
& \geq & {\nu\over 2}\| Du\|_{2}^{2}-\int_{\Omega}
\left(\left(a_{\varepsilon}^{(1)}+a_{\varepsilon}^{(2)}\right)
\vert u\vert+{n-2\over 2n}\varepsilon\vert u\vert^{{2n}\over{n-2}}\right)\,dx \\
& \geq & {\nu\over 2}\| Du\|_{2}^{2}-\|a_{\varepsilon}^{(1)}\|_2\|u\|_2
-c_1\|a_{\varepsilon}^{(2)}\|_{2n\over n+2}\|Du\|_2 -\varepsilon c_2
\|Du\|_2^{2n\over n-2} \\
& \geq & {\nu\over 2}\| Du\|_{2}^{2}-\|a_{\varepsilon}^{(1)}\|_2\|u\|_2
-c_1\varepsilon\|Du\|_2 -\varepsilon c_2\|Du\|_2^{2n\over n-2}.
\end{eqnarray*}

Then if $h_0$ is sufficiently large, from the fact that $(\lambda_h)$
diverges, for all $u\in V^{+}$, $\| Du\|_{2}=1$  implies
$$
\|a_{\varepsilon}^{(1)}\|_2\|u\|_2\leq {\nu\over 6}\,.
$$
Hence, for $\varepsilon>0$ small enough, $\| Du\|_{2}=1$
implies that $f(u)\geq {\nu/ 6}$. \medskip

Finally, set $V^{-}=\overline{{\rm span}}\left\{u_{h}\in H^{1}_{0}(\Omega,{\mathbb R}^N): h<h_{0}\right\},$
we have the decomposition
$$
H^{1}_{0}(\Omega;{\mathbb R}^{N})=V^{+}\oplus V^{-}.
$$
Therefore, since the hypotheses for \cite[Theorem 2.1.6]{cd} are fulfilled, 
we can find a sequence $(u^{m})$ of weak solution of system~(\ref{system})
such that
$$
\lim_{m}f(u^{m})=+\infty,
$$
and the theorem is now proven.

\section{Regularity of weak solutions for elliptic systems}
%
Assume conditions {\rm\bf{(A1)}} and {\rm\bf{(G1)}}, and consider the
nonlinear elliptic system
\begin{equation}
\label{resist}
\int_{\Omega}\sum_{i,j=1}^{n}\sum_{h,k=1}^{N}
a_{ij}^{hk}(x,u)D_{i}u_{h}D_{j}v_{k}\,dx=\int_{\Omega}b(x,u,Du)\cdot v\,dx
\end{equation}
for all $v\in H^{1}_{0}(\Omega;{\mathbb R}^{N}).$ For $l=1,..,N$, we choose
$$
b_l(x,u,Du)=\left\{-\sum_{i,j=1}^{n}\sum_{h,k=1}^{N}
D_{s_l}a^{hk}_{ij}(x,u)D_iu_hD_ju_k+g_l(x,u)\right\}.
$$
Assume that there exist $c>0$ and $q<\frac{n+2}{n-2}$ such that for all $s\in{\mathbb R}^N$ and
a.e. in $\Omega$ 
\begin{eqnarray}
\label{subcr}
|g(x,s)|\leq c\left(1+|s|^{q}\right).
\end{eqnarray}
Then it follows that for every $M>0$, there exists $C(M)>0$ such that
for a.e. $x\in\Omega$, for all $\xi\in{\mathbb R}^{nN}$ and $s\in{\mathbb R}^N$
with $|s|\leq M$ \begin{equation}
\label{subcrit}
\vert b(x,s,\xi)\vert\leq c(M)\left(1+|\xi|^2\right)\,.
\end{equation}
A nontrivial regularity theory for quasilinear systems 
(see, \cite[Chapter VI]{giaquinta}) yields the following :

\begin{theorem}
\label{general}
For every weak solution $u\in H^{1}(\Omega,{\mathbb R}^N)\cap L^{\infty}(\Omega,{\mathbb R}^N)$ of 
the system {\rm(\ref{system})} there exist an open subset 
$\Omega_{0}\subseteq\Omega$ and $s>0$ such that
\begin{gather*}
\forall p\in(n,+\infty): u\in C^{0,1-\frac{n}{p}}(\Omega_{0};
{\mathbb R}^{N}), \\
{\cal H}^{n-s}(\Omega\backslash\Omega_{0})=0\,.
\end{gather*}
\end{theorem}

\begin{pf}
For the proof, see \cite[Chapter VI]{giaquinta}.
\end{pf}

We now consider the particular case when 
$a_{ij}^{hk}(x,s)=\alpha_{ij}(x,s)\delta^{hk},$
and provide an almost everywhere regularity result.

\begin{lemma}
\label{partic}
Assume condition {\rm (\ref{subcrit})}. Then the weak 
solutions $u\in H^{1}_0(\Omega,{\mathbb R}^N)$ of the system
\begin{multline}
\int_{\Omega}\sum_{i,j=1}^{n}\sum_{h=1}^{N}a_{ij}(x,u)D_iu_hD_jv_h\,dx+ \\
+{\frac 12}\int_{\Omega}\sum_{i,j=1}^{n}\sum_{h=1}^{N}D_{s}a_{ij}(x,u)
\cdot vD_iu_hD_ju_h\,dx= \int_{\Omega}g(x,u)\cdot v\,dx
\end{multline}
for all $v\in C^\infty_c(\Omega,{\mathbb R}^N)$, belong to $L^{\infty}(\Omega,{\mathbb R}^N)$.
\end{lemma}
\begin{pf}
By \cite[Lemma 3.3]{s}, for each $(CPS)_c$ sequence $(u^m)$ there exist
$u\in H^{1}_0\cap L^{\infty}$ and a subsequence $(u^{m_k})$ with $u^{m_k}\rightharpoonup u$.
Then, given a weak solution $u$, consider the sequence $(u^m)$ such that each element is equal to $u$
and the assertion follows.
\end{pf}

We can finally state a partial regularity result for our system.

\begin{theorem}
Assume condition {\rm (\ref{subcrit})} and let
$u\in H^{1}_0(\Omega,{\mathbb R}^N)$ be a weak solution of the system
\begin{multline}
\int_{\Omega}\sum_{i,j=1}^{n}\sum_{h=1}^{N}a_{ij}(x,u)D_iu_hD_jv_h\,dx+ \\
+{\frac 12}\int_{\Omega}\sum_{i,j=1}^{n}\sum_{h=1}^{N}D_{s}a_{ij}(x,u)
\cdot vD_iu_hD_ju_h\,dx= \int_{\Omega}g(x,u)\cdot v\,dx
\end{multline}
for all $v\in C^\infty_c(\Omega,{\mathbb R}^N)$. Then there exist an 
open subset $\Omega_{0}\subseteq\Omega$ and $s>0$ such that
\begin{gather*}
\forall p\in(n,+\infty): u\in C^{0,1-\frac{n}{p}}(\Omega_{0}
;{\mathbb R}^{N}), \\
{\cal H}^{n-s}(\Omega\backslash\Omega_{0})=0.
\end{gather*}
\end{theorem}

\begin{pf}
It suffices to combine the previous Lemma with Theorem \ref{general}. 
\end{pf}
\vskip8pt
\noindent{\bf Acknowledgment.\ } The author wishes to thank Professor
Marco Degiovanni for providing helpful discussions.
            
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\end{document}