\documentclass[twoside]{article}
\usepackage{amssymb, amsmath} % font used for R in Real numbers
\pagestyle{myheadings}

\markboth{\hfil Elliptic systems with critical Sobolev exponent
\hfil EJDE--2002/49}
{EJDE--2002/49\hfil P. Amster, P. De N\'apoli, \& M. C. Mariani\hfil}
\begin{document}
\title{\vspace{-1in}\parbox{\linewidth}{\footnotesize\noindent
{\sc  Electronic Journal of Differential Equations},
Vol. {\bf 2002}(2002), No. 49, pp. 1--13. \newline
ISSN: 1072-6691. URL: http://ejde.math.swt.edu or http://ejde.math.unt.edu
\newline ftp  ejde.math.swt.edu  (login: ftp)}
 \vspace{\bigskipamount} \\
 %
  Existence of solutions for elliptic systems with critical Sobolev exponent
 %
\thanks{ {\em Mathematics Subject Classifications: 35J50.}
\hfil\break\indent
{\em Key words: Elliptic Systems, Critical Sobolev exponent, variational methods.} 
\hfil\break\indent
\copyright 2002 Southwest Texas State University. \hfil\break\indent
Submitted January 2, 2002. Published June 2, 2002.} }
\date{}
%
\author{Pablo Amster, Pablo De N\'apoli, \& Maria Cristina Mariani}
\maketitle

\begin{abstract}
  We establish conditions for existence and for nonexistence of
  nontrivial solutions to an elliptic system of partial differential
  equations. This system is of gradient type and has a nonlinearity
  with critical growth.
\end{abstract}

\newtheorem{theorem}{Theorem}[section]
\newtheorem{corollary}[theorem]{Corollary}
\newtheorem{lemma}[theorem]{Lemma}
\numberwithin{equation}{section}

\section{Introduction}

The purpose of this work is to extend some results known for the
quasilinear elliptic equation
\begin{equation} \label{e1}
\begin{gathered}
-\Delta u=u^{p-1}+\lambda u \quad\text{in }\Omega
 \\ u=0 \quad \text{on }\partial\Omega
\end{gathered}
\end{equation}
to the general system
\begin{equation} \label{e2}
\begin{gathered}
-\Delta u_{i}=f_{i}(u)+\sum ^{n}_{j=1}a_{ij}u_{j}
 \quad\text{in }\Omega \\
 u_{i}=0 \quad \text{on }\partial\Omega.
\end{gathered}
\end{equation}
First we recall some results for the single equation (\ref{e1})
on a bounded domain $\Omega \subset \mathbb{R}^N$.
If $ 2<p<2^{*}=2N/(N-2)$ (the critical Sobolev exponent),
then (\ref{e1}) has a nontrivial solution if and only if $\lambda
<\lambda _{1}(\Omega )$, the first eigenvalue of $-\Delta$. This
is proved by applying the Mountain Pass Theorem for
finding nontrivial critical points for the following functional
 in the Sobolev space $H_{0}^{1}(\Omega )$. 
\begin{equation}
\varphi (u)=\frac{1}{2}\int _{\Omega }|\nabla u|^{2}
-\frac{\lambda }{2}\int _{\Omega }u^{2}- \frac 1p \int _{\Omega
}F(u) \label{e3}
\end{equation}
where $F(u)= |u|^p$. Then by the compact imbedding
$H_{0}^{1}(\Omega )\hookrightarrow L^{p}(\Omega )$,
$\varphi$ satisfies the Palais-Smale condition $(PS)$.
However when $p=2^{*}$, $\varphi$ may not satisfy the
Palais-Smale condition (PS) due to the lack of compactness
of the above imbedding. For $\lambda \leq 0$, a Pohozaev identity shows
that there are no nontrivial solutions when $\Omega$ is star shaped.
For the case $0<\lambda <\lambda _{1}(\Omega )$, Brezis a nd
Nirenberg \cite{BN} proved the existence of at least one nontrivial
solution when $N \geq 4$. Their proof relies in the fact that
$\varphi$ satisfies $(PS)_{c}$ (Palais-Smale at level $c$) if
$c<c^{*}=S^{N/2}/N$, where
$$
S=\inf _{u\in D_{0}^{1,2}(\mathbb{R}^{N}),
 \|u\|_{2^{*}}=1}\|\nabla u\|^{2}_{2}
$$
which is the best constant in the Sobolev inequality. Moreover, when the
value of $S$ and the optimal functions are explicitly known, it is
possible to prove that if
$$
S_{\lambda } = \inf _{u\in H_{0}^{1}(\Omega ),
\|u\|_{2^{*}}=1}\|\nabla u\|^{2}_{2} +\lambda \|u\|_{2}^{2}
$$
then $S_{\lambda }< S$ for $\lambda >0$. Then,
using the Mountain Pass Theorem a critical value $c<c^{*}$ is
obtained. For a detailed exposition see \cite{W}.

Quasilinear elliptic systems have been studied by several authors
\cite{CMM,F1,F2}.
 For gradient type systems such as (\ref{e2}),
Boccardo and de Figueiredo \cite{BF} used variational
arguments to show the existence of nontrivial solutions.
They proved existence of solutions for the problem
\begin{equation}
\begin{gathered}
 -\Delta _{p}u=F_{u}(x,u,v) \quad\text {in } \Omega \\
-\Delta _{q}v=F_{v}(x,u,v) \quad\text {in } \Omega \\
 u=v=0 \quad \text {on } \partial \Omega,
\end{gathered}
\end{equation}
where $F$ is superlinear and subcritical.
In this article, we study the critical case $p=q=2^*$.

The general problem of finding a condition on the matrix $A =
(a_{ij})$ for which (\ref{e2}) admits a nontrivial solution is
still an open question. In this paper, we  present some results toward
the solution of this question. For $A$ symmetric with
$\Vert A \Vert <\lambda_{1}(\Omega)$, we prove that the method
presented in \cite{BN} can be applied. More
precisely, we define appropriate numbers $S_{F,A}$ and $S_F$ such
that if $S_{F,A}<S_F$ then (\ref{e2}) admits a solution.
Furthermore, we show cases where this inequality holds.
 We prove also that in some
particular cases the condition $\Vert A \Vert <\lambda_{1}(\Omega )$ is
necessary. We conclude this paper by showing that Pohoazev's
nonexistence result may be generalized to problem (\ref{e2})
when $A$ is symmetric and negative definite.
We remark that the symmetry of $A$ can be considered as a natural
condition, since the proof of  existence is based on the variational
structure of the problem.

Before we state our results, we recall the following definitions \cite{G}.
$$
D^{1,2}(\mathbb{R}^{N},\mathbb{R}^{n}) =\{u=(u_{1},\ldots ,u_{n})\in
L^{2^{*}}(\mathbb{R}^{N},\mathbb{R}^{n}): \nabla u_{i}\in
L^{2}(\mathbb{R}^{N},\mathbb{R}^{N})\}
$$
Let $A=(a_{ij}) \in \mathbb{R}^{n \times n}$. We shall say that:
\\
i) $A$ is nonnegative ($A\geq 0$) if $ a_{ij}\geq 0$ for all
$i,j$.
\\
ii) $A$ is reducible if by a simultaneous permutation of rows and
columns, it may be written in the form
$$
\begin{pmatrix} B & 0\\
C & D \end{pmatrix}
$$
where $B$ and $D$ are square matrices.
Throughout this article, the Euclidean norm in $\mathbb{R}^n$ will be
denoted by $|\cdot |$.

\subsection*{Statement of results}

\begin{theorem} \label{thm1}
Let $p=2^{*}=2N/(N-2)$. Let $[\cdot ]$ be a norm
on $\mathbb{R}^{n}$ such that $F(u)=[u]^{p}$ is differentiable. Define
$f_i =\frac 1p \partial_i F$, and assume that $A\in\mathbb{R}^{n\times n}$
is symmetric, with $\Vert A \Vert <\lambda_{1}(\Omega )$. Set
\begin{gather*}
S_{F}= \inf_{u\in D^{1,2}(\mathbb{R}^{N},\mathbb{R}^{n}),
\int _{\mathbb{R}^{N}}F(u)=1} \sum ^{n}_{i=1}\int _{\mathbb{R}^{N}}|\nabla
u_{i}|^{2}, \\
 S_{F,A}(\Omega )=\inf _{u\in H^{1}(\Omega
,\mathbb{R}^{n}),\int _{\Omega }F(u)=1} \sum ^{n}_{i=1}\int _{\Omega
}|\nabla u_{i}|^{2}- \int _{\Omega }\langle Au,u\rangle
\end{gather*}
Then:
1) $S_{F}$ is attained by a function $u\in
D^{1,2}(\mathbb{R}^{N},\mathbb{R}^{n})$.
\\
2) If $S_{F,A}(\Omega )<S_F$ then (\ref{e2}) admits at least one
nontrivial weak solution.
\end{theorem}
As a consequence of this theorem, we have the existence of solutions
for the following case.

\begin{corollary} \label{coro2}
Let $p$, $A$ and $f_i$ satisfy the conditions of the Theroem \ref{thm1}
with $[u]=|u|_{q} =(\sum^{n}_{i=1}|u_{i}|^{q})^{1/q}$ for
some $q\geq 2$. Moreover, assume that $N \geq 4$ and that
$a_{ii}>0$ for some $i$. Then (\ref{e2}) has a nontrivial weak
solution.
\end{corollary}

\begin{theorem} \label{thm3}
Let us assume that (\ref{e2}) admits a nonnegative nontrivial solution
$u\in H_{0}^{1}(\Omega ,\mathbb{R}^{n})$, and that $f_{i}(u)\geq 0$, with
$f_{i}(u)>0$ for $u>0$. We denote by $\mu_{{\rm min}}$ and
$\mu _{{\rm max}}$
the smallest and the largest eigenvalues of $A$, respectively.
Then \begin{enumerate}

\item[1)] If $A$ is symmetric and positive definite, then
$\mu_{{\rm min}}<\lambda _{1}(\Omega )$.

\item[2)] If $ A\geq 0$ is irreducible, then $\mu _{{\rm max}}<\lambda
_{1}(\Omega )$.

\item[3)] If $ a_{ij}>0$ for every $i,j$, and $A$ is symmetric, then
$\Vert A\Vert <\lambda _{1}(\Omega )$.
\end{enumerate}\end{theorem}

Using a Pohozaev-type identity \cite{P} we shall prove as in
\cite{PS} the following nonexistence result.

\begin{theorem} \label{thm4}
Let $F\in C^{1}(\mathbb{R}^{n})$ be homogeneous of degree
$p=2^{*}=2N/(N-2)$ and define $f_i =\frac 1p \partial_i F$.
Assume that $A$ is symmetric and negative definite, and
that $\Omega$ is star shaped. Then $u=0$ is the unique classical
solution of (\ref{e2}).
\end{theorem}

\section{The Brezis-Lieb Lemma}

We shall use the following version of the Brezis-Lieb lemma \cite{BL}.

\begin{lemma} \label{lm5}
Assume that $F\in C^{1}(\mathbb{R}^{n})$ with $F(0)=0$ and $\left|
\frac{\partial F}{\partial u_{i}}\right| \leq C|u|^{p-1}$.
Let $(u_{k})\subset L^{p}(\Omega )$, ($1\leq p<\infty$).
If $ (u_{k}) $ is bounded in $L^{p}(\Omega )$ and $ u_{k}\to u $ a.e.
 on $\Omega $, then
  $$ \lim _{k\to \infty }\Big( \int_{\Omega
}F(u_{k})-F(u_{k}-u)\Big) =\int _{\Omega }F(u)
$$
\end{lemma}

\paragraph{Proof}
We first remark that $u\in L^{p}(\Omega )$ and
$\|u\|_{p}\leq \liminf \|u_{k}\|_{p}<\infty $.
We claim that for a fixed $ \varepsilon >0$ there exists $
c(\varepsilon ) $ such that for $ a,b\in \mathbb{R}^{n}$, it holds
\begin{equation}
|F(a+b)-F(a)| \leq \varepsilon |a|^{p}+c(\varepsilon )|b|^{p}
\label{e4}
\end{equation}
Indeed, writing
$$
|F(a+b)-F(a)|=\left| \sum ^{n}_{i=1}\int ^{1}_{0} \frac{\partial
F}{\partial u_{i}}(a+bt)b_{i}dt\right| \leq C\sum ^{n}_{i=1}\int
^{1}_{0}|a+bt|^{p-1}|b_{i}|dt
$$
and using that
$xy\leq c(\widetilde\varepsilon )x^{p} +\widetilde \varepsilon
y^{p'}\, (x,y>0)$ we obtain
$$
|F(a+b)-F(a)|\leq C\sum ^{n}_{i=1}\int ^{1}_{0} \left(
\widetilde\varepsilon |a+bt|^{p}+c(\widetilde\varepsilon
)|b_{i}|^{p}\right) dt
$$
Moreover, as $(x+y)^{p}\leq 2^{p-1}(x^{p}+y^{p})\, (x,y>0) $,
we obtain:
$$
|F(a+b)-F(a)|\leq 2^{p-1}C\sum ^{n}_{i=1}\int ^{1}_{0}
\widetilde \varepsilon(  |a|^{p}+t^{p}|b|^{p})
 +c(\widetilde \varepsilon) |b_{i}|^{p} dt
$$
 and (\ref{e4}) follows. Letting $a=u_{k}(x)-u(x)$, $ b=u(x)$ we
obtain
$$
|F(u_{k})-F(u_{k}-u)|\leq \varepsilon
|u_{k}-u|^{p}+c(\varepsilon)|u|^{p}
$$
We introduce the functions:
$$
 f^{\varepsilon }_{k}=
(|F(u_{k})-F(u_{k}-u)-F(u)|-\varepsilon |u_{k}-u|^{p})^{+}
$$
As $|F(u)|\leq K|u|^{p}$, then $|f^{\varepsilon }_{k}|\leq
(K+c(\varepsilon ))|u|^{p}$. By Lebesgue theorem
$\int_{\Omega}f_{k}^{\varepsilon }\to 0 $. Since
$|F(u_{k})-F(u_{k}-u)-F(u)|\leq f^{\varepsilon }_{k}+ \varepsilon
|u_{k}-u|^{p}$, we obtain
$$
\limsup _{k \to \infty }\int _{\Omega
}|F(u_{k})-F(u_{k}-u)-F(u)| \leq \varepsilon c
$$
 with $ c=\sup_{k}\|u_{k}-u\|^{p}_{p}<\infty $.
 Letting $\varepsilon \to 0$, the result follows.

\paragraph{Remark}
 In particular, this result holds for $F$ is homogeneous of degree $p$.


\section{Proofs of results}

For the proof of part 1) of Theorem \ref{thm1} we shall use the
Lemma \ref{lm6}
below, which is a version of the concentration compactness lemma in
\cite{L}.

Let $ F:\mathbb{R}^{n}\to \mathbb{R}$ be a $C^1$ function homogeneous of degree
$p=2^{*} $, such that $F(u)>0$ if $u\neq 0$. By homogeneity, it is
easy to see that
$$
S_{F}=\inf_{u\in D^{1,2}(\mathbb{R}^{N},\mathbb{R}^{n}),u\neq 0}
\frac{\sum ^{n}_{k=1}\int_{\mathbb{R}^{N}}| \nabla u_{k}|^{2}}{\left( \int
_{\mathbb{R}^{N}}F(u)\right)^{2/2^{*}}}
$$

\begin{lemma} \label{lm6}
Let $(u^{(i)})\subset D_{0}^{1,2}(\mathbb{R}^{N},\mathbb{R}^{n})$ be a sequence
such that:
\begin{enumerate}
\item[i)] $u^{(i)}\to u$ weakly in $D^{1,2}(\Omega )$

\item[ii)] $|\nabla (u^{(i)}_{k}-u_{k})|\to \mu _{k}$ in $M(\mathbb{R}^{n})$
weak ${}^{*}$ for $k = 1,...,n$.

\item[iii)] $F(u^{(i)}-u)\to \nu $ in $M(\mathbb{R}^{n})$ weak${}^{*}$

\item[iv)] $u^{(i)}\to u$ a.e. on $\mathbb{R}^{N}$
\end{enumerate}
and define:
$\mu =\sum ^{n}_{k=1}\mu_{k}$,
\begin{gather*}
 \nu ^\infty = \lim _{R\to \infty } \Big(\limsup _{i\to \infty }
 \int _{|x|\geq R}F(u^{(i)})dx\Big),\\
\mu _{k}^{\infty }=\lim _{R\to \infty }
\Big( \limsup _{i\to \infty } \int _{|x|\geq R}|\nabla
u^{(i)}_{k}|^{2}dx\Big)
\end{gather*}
Then:
\begin{gather}
\left\Vert \nu \right\Vert
^{2/2^{*}}\leq \frac 1{S_{F}} \left\Vert \mu \right\Vert \label{e5.1}\\
(\nu ^{\infty })^{2/2^{*}}\leq \frac 1{S_{F}}\sum ^{n}_{k=1}
\mu ^{\infty }_{k}   \label{e5.2}\\
\limsup _{i\to \infty }|\nabla
u_{k}^{(i)}|_{2}^{2}=|\nabla u|_{2}^{2}+ \left\Vert \mu
_{k}\right\Vert +\mu ^{\infty }_{k} \quad \text{for $k =
1,...,n$} \label{e6.1} \\
\limsup _{i\to \infty }\int_\Omega
F(u^{(i)})=\int_\Omega F(u)+ \left\Vert \nu \right\Vert +\nu
_{\infty }\label{e6.2}
\end{gather}
Moreover, if $ u=0 $ and  equality holds in 
(\ref{e5.1}), then $\mu =0 $ or
$\mu $ is concentrated at a single point.
\end{lemma}


\paragraph{Proof of Theorem \ref{thm1} Part 1)}
Let $(u^{(i)})\subset D^{1,2}(\mathbb{R}^{N},\mathbb{R}^n)$ be a minimizing
sequence for $S_{F}$, i.e.,
$$
\int _{\mathbb{R}^{N}}F(u^{({i})})=1,\qquad \sum
^{n}_{k=1}\int _{\mathbb{R}^{N}}|\nabla u_{k}^{(i)}|^{2}\to S_{F}
$$
Using (\ref{e5.1})-(\ref{e6.1}), we deduce, as in \cite[Theorem 1.41]{W},
the existence of a sequence $(y_{i},\lambda _{i})\in \mathbb{R}^{N}\times
\mathbb{R}$ such that $ \lambda ^{(N-2)/2}_{i}u(\lambda _{i}x+y_{i})$
 has a convergent subsequence. In particular there exists a minimizer for
$ S_{F}$.

To prove the second part of Theorem 1, we shall use the
following version of the Mountain Pass Lemma \cite{W}.

\begin{theorem}[Ambrosetti-Rabinowitz] \label{thm7}
Let $ X$ be a Hilbert space, $\varphi$ be an element of $C^{1}(X,\mathbb{R})$,
$ e\in X$ and $r>0$ such that $\left\Vert e\right\Vert >r$, $b=\inf
_{\left\Vert u\right\Vert =r}\varphi (u)>\varphi (0)\geq \varphi
(e)$. Then for each $\varepsilon >0$ there exists $u\in X$ such
that $c-\varepsilon \leq \varphi (u)\leq c+\varepsilon$ and
$\left\Vert \varphi'(u)\right\Vert \leq \varepsilon$
where
$$
c=\inf _{\gamma \in \Gamma }\max _{t\in [0,1]}\varphi (\gamma (t)),
$$
with $\Gamma =\left\{ \gamma \in C([0,1],X):\gamma (0)=0,\gamma
(1)= e\right\}$.

Letting $\varepsilon =1/k$, we get a Palais-Smale sequence
at level $c$; i.e., a sequence $(u^{(k)})\subset X$ such that
 $$
\varphi (u^{(k)})\to c, \quad \varphi'(u^{(k)})\to 0
$$
We shall apply this result to the functional
$$
 \varphi (u)=\frac{1}{2}\int _{\Omega }\sum ^{n}_{i=1}|\nabla u_{i}|^{2}-
\frac{1}{2^{*}}\int _{\Omega }F(u)-\frac{1}{2}\int _{\Omega } \sum
^{n}_{i,j=1}a_{ij}u_{i} u_{j}
$$
in the Sobolev space
$X=H_{0}^{1}(\Omega, \mathbb{R}^{n})$. As $\| A \|< \lambda_1(\Omega) $,
we may define on $X$ the norm
$$
\| u \|= \left(\int_{\Omega} \sum_{i=1}^n |\nabla u_i |^2 - \int_{\Omega}
\sum_{i,j=1}^n a_{ij} u_i u_j \right)^{1/2}
$$
which is equivalent to the usual norm. By standard arguments
$\varphi \in C^{1}(X)$ and
$$
\left\langle \varphi'(u),h \right\rangle = \sum
^{n}_{i=1} \int _{\Omega }\nabla u_{i}\cdot \nabla h_{i}-
\sum_{i=1}^n \int _{\Omega }f_{i}(u)h_{i}- \int _{\Omega }\sum
^{n}_{i,j=1}a_{ij}h_{i}u_{j}
$$
It follows that the critical points of $\varphi$ are weak
solutions of the system.
\end{theorem}

To ensure that the value $c$ given by the mountain pass
theorem is indeed a critical value we need to prove the following
lemma.

\begin{lemma} \label{lm7}
Let $F$ be homogeneous of degree $2^{*}$. Then any
$(PS)_{c}$ sequence with $c<c^{*}=\left(
\frac{1}{2}-\frac{1}{2^{*}}\right) \frac{S_{F}^{N/2}}{N}$ has a
convergent subsequence.
\end{lemma}

\paragraph{Proof}
Let $(u^{(k)})\subset H^1_0(\Omega,\mathbb{R}^n)$ be a $(PS)_{c}$
sequence. First we show that it is bounded.
$$
\langle \varphi^{\prime}(u^{(k)}),u^{(k)}\rangle =
\Vert u^{(k)}\Vert ^{2}-  \sum ^{n}_{i=1}\int_{\Omega}
f_{i}(u^{(k)}) u^{(k)}_{i}
$$
Since $F$ is homogeneous of degree $2^{*},$ we have that
$\sum^{n}_{i=1}f_{i}(u^{(k)})u^{(k)}_{i}=F(u^{(k)})$.
Then
$$
\frac{1}{2}\Vert u^{(k)}\Vert ^{2} =\varphi
(u^{(k)})+\frac{1}{2^{*}}\int _{\Omega }F(u^{(k)}) =\varphi
(u^{(k)})+\frac{1}{2^{*}}\big( \Vert u^{(k)}\Vert ^{2} -\langle
\varphi'(u^{(k)}),u^{(k)}\rangle \big)
$$
Hence, for
$k\geq k_{0}$ we have
$$
\big( \frac{1}{2}-\frac{1}{2^*}\big) \Vert u^{(k)}
\Vert ^{2}\leq C+\varepsilon \Vert u^{(k)}\Vert
$$
and we conclude
that $\Vert u^{(k)}\Vert$ is bounded.

We may assume that $u^{(k)}\to u$ weakly in $H_{0}^{1}(\Omega )^{n}$,
$u^{(k)}\to u$  in $L^{2}(\Omega)^{n}$, and $u^{(k)}\to u$
a.e..

Since $ (u^{(k)})$ is bounded in $ L^{2^{*}}(\Omega )$,
$f(u^{(k)})$ is bounded in $L^{2N/(N+2)}(\Omega ) $.
So we may assume that $f(u^{(k)})\to f(u)$ weakly in $L^{2N/(N+2)}$.
It follows that $u$ is a critical point of $ \varphi$, i.e. $ u$
is a weak solution of the system. We deduce that
$$
\langle \varphi'(u),u\rangle =\Vert u
\Vert ^{2}-\int _{\Omega }\sum ^{n}_{i=1}f_{i}(u)u_{i}=0
$$
Moreover,
$$
\varphi (u)=\frac{\Vert u\Vert ^{2}}{2}- \frac{1}{2^*} \int
_{\Omega }F(u)=\frac{\Vert u\Vert ^{2}}{2}- \frac{1}{2^{*}}\int
_{\Omega }\sum ^{n}_{i=1}f_{i}(u)u_{i}= (
\frac{1}{2}-\frac{1}{2^{*}}) \left\Vert u\right\Vert^{2}\geq 0
$$
Writing $ v^{(k)}=u-u^{(k)}$, by Lemma \ref{lm5} we have
$$
\int _{\Omega}F(u^{(k)})=\int _{\Omega }F(u)+\int _{\Omega}F(v^{(k)})+o(1)
$$
and then
$$
\varphi (u^{(k)})=\frac{1}{2}\Vert u-v^{(k)}
\Vert^{2}-\frac 1{2^*} \int _{\Omega }F(u^{(k)})- \frac 1{2^*}\int
_{\Omega }F(v^{(k)})+o(1)
$$
As $v^{(k)} \to 0$ weakly,
$$
\Vert u^{(k)}\Vert ^{2} =
\Vert u-v^{(k)}\Vert ^{2}= \Vert u\Vert ^{2}+ \Vert
v^{(k)}\Vert^{2}+o(1)
$$
and then we obtain
\begin{equation}
\frac{1}{2}\Vert u\Vert ^{2} +\frac{1}{2}\Vert
v^{(k)}\Vert ^{2}- \frac 1{2^*}\int _{\Omega }F(u) - \frac 1{2^*}
\int _{\Omega}F(v^{(k)})\to c \label{e7}
\end{equation}
On the other hand we also know that
$ \langle \varphi '(u^{(k)}),u^{(k)}\rangle \to 0$ and
\begin{align*}
\langle \varphi'(u^{(k)}),u^{(k)}\rangle
=&\Vert u^{(k)}\Vert ^{2}-\int _{\Omega } \sum
^{n}_{i=1}f_{i}(u^{(k)})u_i^{(k)} =\Vert u^{(k)}\Vert
^{2}-2^{*}\int _{\Omega }F(u^{(k)})\\
=&\Vert u\Vert ^{2}+\Vert v^{(k)}\Vert
^{2}-2^{*} \int _{\Omega }F(u)-2^{*}\int _{\Omega}
F(v^{(k)})+o(1)\to 0
\end{align*}
Hence,
$$
\Vert v^{(k)}\Vert^{2}-2^{*}\int _{\Omega }F(v^{(k)})
\to 2^{*}\int _{\Omega}F(u)-\Vert u\Vert ^{2}=0
$$
We may therefore assume that $\Vert v^{(k)}\Vert ^{2}\to b$,
$2^{*} \int _{\Omega }F(v^{(k)})\to b$.
From (\ref{e7}), we deduce that
$$
\varphi (u)+\Big(\frac{1}{2}-\frac{1}{2^{*}}\Big) b=c
$$
and since $\varphi(u)\geq 0$,
$$
\Big( \frac{1}{2}-\frac{1}{2^{*}}\Big) b\leq c
$$
We claim that $b=0$. Indeed, since $u^{(k)}\to 0$ in
$L^{2}(\Omega,\mathbb{R}^{n})$, it follows that
$\sum ^{n}_{i=1}\int_{\Omega }|\nabla(v^{(k)})_{i}|^{2}\to b$.
On the other hand,
$$
\sum ^{n}_{i=1} \int _{\Omega }|\nabla (v^{(k)})_{i}|^{2} \geq
S_{F}\Big( \int _{\Omega }F(v^{(k)})\Big) ^{2/2^{*}}
$$
and, letting $k \to \infty$,
$b\geq S_{F}b^{2/2^{*}}$.
It follows that $ b=0 $ or $b\geq S^{N/2}_{F}$. In this last case,
$$
c^{*}=\big( \frac{1}{2}-\frac{1}{2^{*}}\big) \frac
{S^{N/2}_{F}}{N}\leq \big( \frac{1}{2}-\frac{1}{2^{*}}\big)
\frac bN\leq c,
$$
a contradiction. Hence, $b = 0$ and $v^{(k)}\to 0$ strongly.

\paragraph{Proof of Theorem \ref{thm1} part 2)}
In the same way of \cite[Theorem 1.45]{W}, it suffices to apply the
Mountain Pass Theorem with a value $c<c^{*}$. We shall find the
maximum of the function $h:[0,1]\to \mathbb{R}$ given by
$$
h(t)= \varphi (tv)= \left( \frac{t^{2}}{2}\left\Vert
v\right\Vert^{2}- \frac {t^{2^{*}}}{2^{*}} \int _{\Omega
}F(v)\right) = A\frac{t^{2}}{2}-Bt^{2^{*}}
$$
Since $2^{*}-2=\frac{4}{N-2}$, we obtain the critical point
$$
t_{0}=\big( \frac{A}{2^{*}B}\big) ^{(N-2)/4}
$$
with
$$
h(t_{0}) =\big( \frac{A}{2^{*}B}\big) ^{N/2}
\big(\frac{2^{*}}{2}-1\big) B>0
$$
Then, it is easy to conclude that $c< c^*$.

Now we consider the special case
$$
[u]=|u|_{q}=\Big( \sum ^{n}_{i=1}|u_{i}|^{q}\Big) ^{1/q}
\quad\text{and} \quad
F_{q}(u)=\Big( \sum^{n}_{i=1}|u_{i}|^{q}\Big) ^{2^{*}/q}
$$
for proving Corollary \ref{coro2}.

\begin{lemma} \label{lm8}
$ S_{F}(\Omega) =S$ for $q\geq 2$ where $S$ is the best constant
for the Sobolev inequality with $n=1$.
\end{lemma}

\paragraph{Proof}
Suppose first that $q\geq 2^{*}$, then we have the estimate
\begin{align*}
\Big[ \int _{\Omega }\Big( \sum ^{n}_{i=1}|u_{i}|^{q}\Big) ^{2^{*}/q}
\Big] ^{2/2^{*}}
\leq& \Big[ \int _{\Omega } \sum
^{n}_{i=1}|u_{i}|^{2^{*}}\Big]^{2/2^{*}}= \Big[ \sum
^{n}_{i=1}\int _{\Omega }|u_{i}|^{2^{*}}\Big] ^{2/2^{*}}\\
\leq& \Big[ \sum ^{n}_{i=1}(S^{-1}\int _{\Omega } |\nabla
u_{i}|^{2}_{2})^{2^{*}/2}\Big] ^{2/2^{*}} \leq \sum
^{n}_{i=1}S^{-1}\int _{\Omega }|\nabla u_{i}|^{2}
\end{align*}
It follows that $S_{F}\geq S$.
For $2\leq q\leq 2^{*}$ we use Minkowski inequality:
\begin{align*}
\Big[ \int _{\Omega }\Big( \sum ^{n}_{i=1}|u_{i}|^{q}\Big)
^{2^{*}/q}\Big] ^{2/2^{*}}= &\left\{ \Big[ \int _{\Omega }\Big(
\sum ^{n}_{i=1}|u_{i}|^{q}\Big)^{2^{*}/q}\Big]^{q/2^{*}}
\right\}^{2/q}\\
\leq&
\Big[ \Big( \sum ^{n}_{i=1} \int_{\Omega }|u_{i}|^{2^{*}}
\Big)^{q/2^{*}}\Big]^{2/q} \\
=& \sum ^{n}_{i=1}
\Big( \int _{\Omega }|u_{i}|^{2^{*}}\Big)^{2/2^{*}} \leq
\sum^{n}_{i=1} S^{-1}\int_{\Omega }|\nabla u_{i}|^{2}
\end{align*}
The inequality $S_{F}\leq S$ is verified easily taking functions of
the form $u = (u_1,0,...,0)$.


\paragraph{Proof of Corollary \ref{coro2}}
First we note that by the $2^{*}$-homogeneity of $F$, $S_{F}$ does
not depend on $\Omega$.
Taking $u(x)=U(x)e_{i}$, where
$$
U(x)=\frac{[N(N-2)]^{(N-2)/4}}{(1+|x|^{2})^{(N-2)/2}}
$$
is the function that attains Sobolev's best constant in one
dimension \cite[Theorem 1.42]{W}, it follows that $S_{F}$ is achieved
when $\Omega =\mathbb{R}^{N}$ (where $N\geq 4$).
By translation invariance of the problem, $S_{F}$ is also achieved
with $u_{\varepsilon }(x)=U_{\varepsilon }(x)\cdot e_{i}$, for
$$
U_{\varepsilon }(x)=\varepsilon ^{(2-N)/2}U(x/\varepsilon )
$$

We shall see that $S_{F,A}<S_{F}$. Indeed, we may assume that
$0\in \Omega$ and choose $i$ such that $a_{ii}>0$. Then, if we
define $u(x)=v_{\varepsilon }(x)e_{i}$, with $v_{\varepsilon
}(x)=\psi (x)U_{\varepsilon }(x)$, and $ \psi$ a smooth function
with compact support in $\Omega$ such that $\psi \equiv 1$ in
$B(0,\rho )$, we obtain as in \cite[Lemma 1.46]{W}:
$$
\frac {\int _{\Omega }\sum ^{n}_{i=1} |\nabla
u_{i}|^{2}-\int_{\Omega }\left\langle Au,u \right\rangle }{\Big(
\int_{\Omega }F(u)\Big)^{2/p} } = \frac {\int _{\Omega } |\nabla
u_{\varepsilon }|^{2}-a_{ii}\int _{\Omega } u^{2}_{\varepsilon}
}{\Big( \int _{\Omega }|u_{\varepsilon }|^p \Big)} <S
$$
 for $\varepsilon$ small enough.


\paragraph{Proof of Theorem \ref{thm3}:
Necessary conditions for the existence of nonnegative solutions}
We recall the following theorem in \cite{G}.

\begin{lemma}[Perron-Frobenius Theorem]
Let $A\ge 0$ be irreducible. Then $A$ has a positive simple
eigenvalue $ \mu _{{\rm max}}$ such that $|\mu |\le \mu_{{\rm max}}$ for any
$\mu$ eigenvalue of $A$. Furthermore, there exists an eigenvector
of $\mu _{{\rm max}}$ with positive coordinates.
\end{lemma}

Now we are able to prove Theorem \ref{thm3}.

Suppose that the system has a nonnegative nontrivial
solution. If $e_{1}$ is the first eigenfunction of $-\Delta$ in
$H_{0}^{1}(\Omega )$, then $ e_{1}\in C^{\infty
}(\overline{\Omega})$ and $ e_{1}(x)>0\, \forall x\in \Omega$.
Then
$$
\lambda _{1}\int _{\Omega }u_{i}e_{1}= \int _{\Omega }-\Delta
u_{i}e_{1} =\int _{\Omega }f_{i}(u)e_{1} +\sum
^{n}_{j=1}a_{ij}\int _{\Omega }u_{j}e_{1}
$$
If $z_{i}=\int _{\Omega }u_{i}e_{1}$, then
$$\lambda _{1}z\geq Az,$$
and the inequality between the $i$-th components is strict if
$u_{i}\neq 0$ for some $i$.
Since $z\geq 0$, and $z_{i}>0$ for some $i$, we obtain
$$
\lambda _{1}|z|^{2}>\left\langle Az,z\right\rangle.
$$
Since $A$ is symmetric and positive definite,
$$
\lambda _{1}|z|^{2}>\mu_{{\rm min}}|z|^{2}
\quad{and}\quad \lambda _{1}>\mu_{{\rm min}}.
$$
This proves the first claim of the theorem.

For $A\geq 0$ and irreducible, let $v$ be the eigenvector of
$A^{t}$ corresponding to $\mu _{{\rm max}}$, then from the
Perron-Frobenius Theorem \cite{G}, $v_i >0$ for any $i$ and
$$
\lambda _{1}\left\langle z,v\right\rangle > \left\langle
Az,v\right\rangle =\left\langle z,A^{t}v\right\rangle = \mu
_{{\rm max}}\left\langle z,v\right\rangle$$ and since $\left\langle
z,v\right\rangle >0$, it follows that $\lambda _{1}>\mu _{{\rm max}}$
and the second claim is proved.

Finally when $A\geq 0$ is symmetric, we have $\mu
_{{\rm max}}=\left\Vert A\right\Vert$, and the proof is complete.
\hfill$\Box$

\paragraph{Proof of Theorem \ref{thm4}}
The proof of Theorem \ref{thm4} consists of the next lemma and
the next corollary.

\begin{lemma} \label{lm9}
Suppose that $u\in C^{2}(\overline{\Omega },\mathbb{R}^{n})$ is a
classical solution of the gradient elliptic system
\begin{gather*}
-\Delta u_{i} = g_{i}(u) \quad\text { in } \Omega \\
u = 0 \text { on } \quad \partial \Omega
\end{gather*}
where $g_{i}=\frac{\partial G}{\partial u_{i}}$, $G\in
C^{1}(\mathbb{R}^{n})$, $G(0)=0$ and $\Omega \subset \mathbb{R}^{N}$ is a
bounded open set with smooth boundary. Then  for a fixed $y$,
$$
\sum ^{n}_{k=1} \int _{\partial \Omega } |\nabla
u_{k}|^{2}(x-y)\cdot n(x)dS =2N\int _{\Omega }G(u)dx-(N-2) \sum
^{n}_{k=1}\int _{\Omega }g_{k}(u)u_{k}dx
$$
\end{lemma}

\paragraph{Proof}
Multiply the $k$-th equation by  $(x-y)\cdot \nabla u_{k}
=\sum ^{N}_{i=1}(x_{i}-y_{i})\frac{\partial u_{k}}{\partial
x_{i}}$ and integrate by parts, then we have
\begin{multline*}
\int _{\Omega } \sum ^{N}_{i=1}
(x_{i}-y_{i})\frac{\partial u_{k}}{\partial x_{i}}g_{k}(u) \\
=\int_{\Omega }|\nabla u_{k}|^{2}+\int _{\Omega }\sum ^{n}_{i,j=1}
(x_{i}-y_{i})\frac{\partial u_{k}}{\partial x_{j}}\frac{\partial
^{2} u_{k}}{\partial x_{i}x_{j}}-\int _{\partial \Omega }|\nabla
u_k|^{2}(x-y) \cdot n(x)dS
\end{multline*}
Hence,
$$
\int _{\Omega }\sum ^{N}_{i=1}
(x_{i}-y_{i})\frac{\partial u_{k}}{\partial x_{i}}g_{k}(u)
=\big(1-\frac{N}{2}\big) \int _{\Omega }|\nabla u_{k}|^{2}-\frac{1}{2}
\int _{\partial \Omega }|\nabla u_{k}|^{2}(x-y)\cdot n(x)dS
$$
Adding this identities for $k=1,2,\ldots n$,
\begin{multline*}
\int _{\Omega }\sum ^{N}_{i=1}(x_{i}-y_{i})
\sum ^{n}_{k=1}\frac{\partial u_{k}}{\partial x_{i}}g_{k}(u)\\
=\big( 1-\frac{N}{2}\big) \sum ^{n}_{k=1} \int _{\Omega
}|\nabla u_{k}|^{2}-\frac{1}{2} \sum ^{n}_{k=1}\int _{\partial
\Omega }|\nabla u|^{2}(x-y)\cdot n(x)dS
\end{multline*}
By the chain rule we have
\begin{align*}
\int _{\Omega }\sum ^{N}_{i=1}(x_{i}-y_{i})\sum ^{N}_{k=1}
\frac{\partial u_{k}}{\partial x_{i}}g_{k}(u)
=&\int _{\Omega } \sum
^{N}_{i=1}(x_{i}-y_{i})\frac{\partial G(u)}{\partial x_{i}}\\
=&-N\int _{\Omega }G(u)+\sum ^{n}_{i=1} \int_{\partial \Omega }
G(u)(x_{i}-y_{i}) \cdot n_{i}(x)dS\,.
\end{align*}
Since $G(u)=0$ on $\partial \Omega$,
$$-N\int _{\Omega }G(u)=( 1-\frac{N}{2})
\sum ^{N}_{k=1}\int _{\Omega }|\nabla u_{j}|^{2}- \frac{1}{2}\sum
^{N}_{k=1}\int _{\partial \Omega }|\nabla u|^{2}(x-y) \cdot
n(x)dS
$$
Finally
$$
\int _{\Omega }|\nabla u_{k}|^{2}=\int _{\Omega }g_{k}(u)u_{k}
$$
and
$$
\sum ^{N}_{k=1}\int _{\partial \Omega }|\nabla u_{k}|^{2}(x-y)
\cdot n(x)=2N\int _{\Omega }G(u)-(N-2)\sum ^{N}_{k=1} \int
_{\Omega }g_{k}(u)u_{k}
$$
\quad\hfill$\Box$

With the following corollary, we complete the proof of Theorem \ref{thm4}.

\begin{corollary} \label{coro10}
Assume that $ F\in C^{1}(\mathbb{R}^{n})$ is homogeneous of degree
$p=2^{*}=2N/(N-2)$, with $F(0)=0$. Further, assume that $A$
is symmetric and negative definite, and that $\Omega $ is star
shaped. Then the system
\begin{gather*}
-\Delta u_{j} = f_{k}(u)+\sum ^{k}_{j=1}a_{jk}u_{j}  \quad\text {in } \Omega \\
u = 0 \quad\text {on } \partial \Omega
\end{gather*}
with $f_{k}=\frac{\partial F}{\partial u_{k}}$ admits only the
trivial solution.
\end{corollary}

\paragraph{Proof}
Let
$G(u)=F(u)+\frac{1}{2}\langle Au,u\rangle$.
Since $F$ is homogeneous of degree $ p $,
$$ \sum ^{N}_{k=1}f_{k}(u)u_{k}=pF(u)  $$
and
$$ \sum ^{N}_{k=1}\int _{\partial
\Omega }|\nabla u_{k}|^{2}(x-y)\cdot n(x) =[2N-p(N-2)]\int
_{\Omega }F(u)+2\sum ^{N}_{k=1}\int _{\Omega } \left\langle
Au,u\right\rangle
$$
Since  $p=2N/(N-2)$,
$$
\sum ^{N}_{k=1}\int _{\partial \Omega }|\nabla u_{k}|^{2}(x-y)
\cdot n(x)=2\sum ^{N}_{k=1}\int _{\Omega }\left\langle
Au,u\right\rangle
$$
Now, because $A$ is negative definite, $\left\langle Au,u\right\rangle
\leq M |u|^{2}$ where $M<0$ and then
$$
\sum ^{N}_{k=1}\int _{\partial \Omega }|\nabla
u_{k}|^{2}(x-y)\cdot n(x) \leq 2M \sum ^{n}_{k=1}\int _{\Omega
}|u|^{2}
$$
Since $\Omega$ is star shaped, $(x-y)\cdot n(x)>0$ on
$\partial \Omega $, and we conclude that $u=0$.

\begin{thebibliography}{00} \frenchspacing

\bibitem{BN} H. Brezis and L. Nirenberg, {\it Positive Solutions of
Nonlinear Elliptic Equations Involving
 Critical Sobolev Exponents,} Communications on Pure and Applied Mathematics.
 Vol XXXVI , 437-477 (1983)

\bibitem{BF} L. Boccardo and D. de Figueiredo, {\it Some Remarks on a
system of quasilinear elliptic equations}, to appear.

\bibitem{BL} H.  Brezis and  E.
 Lieb, {\it A relation between point wise
 convergence of functions and convergence
 of functionals,} Proc.  A.M.S. vol. 48, No 3 (1993), pp.  486-499

\bibitem{CMM} P. Cl\'ement, E. Mitidieri and R., Man\'asevich, {\it
Positive solutions for a quasilinear system via blow up,} Comm. in
Part. Diff. Eq., Vol. 18, No. 12, (1993).

\bibitem{F1} D. de Figueiredo, {\it Semilinear Elliptic Systems},
 Notes of the course on semilinear elliptic systems
 of the EDP Chile-CIMPA Summer School
 (Universidad de la Frontera, Temuco, Chile, January 1999).

\bibitem{F2} D. de Figueiredo, {\it Semilinear Elliptic Systems: A
survey of superlinear problems.}
 Resenhas IME-USP 1996, vol.  2, No.  4, pp. 373-391.


\bibitem{GT} D. Gilbarg, and N. S.  Trudinger,  {\it  Elliptic
partial differential equations of second order}, Springer- Verlag
(1983).

\bibitem{G} F. R. Grantmacher, {\it The Theory of matrices}, Chelsea
(1959).

\bibitem{L} P. L. Lions, {\it  The concentration compactness
principle in the calculus of variations. The limit case.}  Rev.
Mat. Iberoamericana (1985), pp. 145-201 and 45-121.


\bibitem{P} S. Pohozaev, {\it Eigenfunctions of the equation $\Delta
u+\lambda f(u)=0$.} Nonlinearity Doklady Akad. Nauk SSRR 165,
(1965), pp. 36-9.


\bibitem{PS} P. Pucci and J. Serrin, {\it A general variational
identity,} Indiana University Journal, Vol. 35, No. 3, (1986),
681-703.


\bibitem{W} M.  Willem, {\it Minimax Theorems},  Birkhauser (1986)

\end{thebibliography}

\noindent\textsc{Pablo Amster} (e-mail: pamster@dm.uba.ar)\\
\textsc{Pablo De N\'apoli} (e-mail: pdenapo@dm.uba.ar)\\
\textsc{Maria Cristina Mariani} (e-mail: mcmarian@dm.uba.ar)\\[2pt]
Departamento. de Matem\'atica \\
Facultad de Ciencias Exactas y Naturales\\
Universidad de Buenos Aires.\\
Pabell\'on I, Ciudad Universitaria (1428) \\
Buenos Aires, Argentina

\end{document}