\documentclass[twoside]{article}
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\markboth{\hfil Nonlinear elliptic systems with indefinite terms
 \hfil EJDE--2002/83}
{EJDE--2002/83\hfil Ahmed Bensedik \& Mohammed Bouchekif \hfil}

\begin{document}
\title{\vspace{-1in}\parbox{\linewidth}{\footnotesize\noindent
{\sc  Electronic Journal of Differential Equations},
Vol. {\bf 2002}(2002), No. 83, pp. 1--16. \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} \\
 %
  On certain nonlinear elliptic systems with indefinite terms
 %
\thanks{ {\em Mathematics Subject Classifications:} 
35J20, 35J25, 35J60, 35J65, 35J70.
\hfil\break\indent
{\em Key words:} Elliptic systems, p-Laplacian, variational methods,
 mountain-pass Lemma, \hfil\break\indent
 Palais-Smale condition, potential function, Moser iterative method.
\hfil\break\indent
\copyright 2002 Southwest Texas State University. 
\hfil\break\indent
Work supported by research project B1301/02/2000.
\hfil\break\indent
Submitted April 2, 2002. Published October 2, 2002.} }
\date{}
%
\author{Ahmed Bensedik \& Mohammed Bouchekif}
\maketitle

\begin{abstract}
   We consider an elliptic quasi linear system with indefinite 
   term on a bounded domain. 
   Under suitable conditions, existence and positivity results
   for solutions are given.
\end{abstract}

\newtheorem{theorem}{Theorem}[section]
\newtheorem{lemma}[theorem]{Lemma}
\newtheorem{proposition}[theorem]{Proposition}
\numberwithin{equation}{section}


\section{Introduction}

The purpose of this article is to find positive solutions to the system
\begin{equation} \label{Spq}
\begin{gathered}
-\Delta_pu=m( x) \frac{\partial H}{\partial u}( u,v)
\quad \text{in }\Omega \\
-\Delta_qv=m( x) \frac{\partial H}{\partial v}( u,v)
\quad \text{in }\Omega \\
u=v=0 \quad \text{on }\partial \Omega
\end{gathered}
\end{equation}
where $\Omega $ is a bounded regular domain of
$\mathbb{R}^N$, with a smooth boundary $\partial \Omega
$, $\Delta_pu:=\mathop{\rm div}(| \nabla u | ^{p-2}\nabla
u) $ is the $p$-Laplacian with $1<p<N$, $m$ is a continuous
function on $\overline{\Omega }$ which changes sign, and $H$ is a
potential function which will be specified later.

The case where the sign of $m$ does not change has been studied by
F. de Th\'elin and J. V\'elin \cite{TV}. These authors treat the
system \eqref{Spq} with a function $H$ having the
following properties
\begin{itemize}
\item There exists $C>0$, for all $x\in \Omega $, for all
$ (u,v) \in D_3$ such that $0\leq H( x,u,v) \leq C( |u| ^{p'}+
| v| ^{q'})$

\item There exists $C'>0$,  for all $x\in \Omega$,
for all $( u,v) \in D_2$ such that
$H( x,u,v) \leq C'$

\item There exists a positive function $a$ in $L^\infty ( \Omega )$,
  such that for each $x\in \Omega$
and $(u,v) \in D_1\cap \mathbb{R}_{+}^2$,
$H( x,u,v)=a( x)u^{\alpha +1}v^{\beta +1}$,
\end{itemize}
where
\begin{gather*}
D_1=\big\{ ( u,v) \in \mathbb{R}_{}^2 : |
u| \geq A\text{ or }| v| \geq A\big\} ,\\
D_2=\big\{ ( u,v) \in \mathbb{R}_{}^2\backslash
D_{1} : |u| \geq \delta \text{ or }| v| \geq \delta \big\},
\end{gather*}
and $D_3=\mathbb{R}^2\backslash (D_1\cup D_2)$ with $A>\delta >0$,
$1<p'<p^{*}:=\frac{Np}{N-p}$, and $1<q'<q^{*}$.

They established the existence results under the conditions
\[
\frac{\alpha +1}p+\frac{\beta +1}q>1 \quad\mbox{and}\quad
\frac{\alpha +1}{p^{*}}+\frac{\beta +1}{q^{*}}<1
\]
by using a suitable application of the variational method due to
Ambrosetti-Rabinowitz \cite{AR}.
M. Bouchekif \cite{Bo1} generalized the
work of F. de Th\'elin and J.Velin \cite{TV} for the large class of functions
of the form
\[
H( u,v) =a| u| ^\gamma +c| v| ^\delta+b| u| ^{\alpha +1}| v| ^{\beta +1}
\]
where $\alpha $, $\beta \geq 0$; $\gamma $, $\delta >1$ and $a$,
$b$ and $c$ are real numbers.
The case where the system \eqref{Spq} is governed by a single operator
$\Delta_p$ has been studied by Baghli \cite {Ba}.

Our aim is to extend to the system \eqref{Spq} the
results obtained in the scalar case (see \cite{Bo2}). Our
existence results follow from modified quasilinear system in order
to apply the Palais-Smale
condition (P.S.) and then the Moser's Iterative Scheme as in
T. \^{O}tani \cite{O} or in F. de Th\'elin and J. V\'elin \cite{TV}.
We consider only weak solutions, and assume that $H$ satisfies the following
hypothesis.
\begin{enumerate}
\item[(H1)] $H\in C^1( \mathbb{R}^{+}\times \mathbb{R}^{+}) $

\item[(H2)] $H(u,v)=o(u^p+v^q)$ as $(u,v)\to (0^{+},0^{+})$

\item[(H3)] There exists $R_0>0$ and $\mu$, $1<\mu <\min ( p^{*}/p,q^{*}/q)$,
such that
 $$\frac up \frac{\partial H}{\partial u}(u,v)+\frac vq
\frac{\partial H}{\partial v}(u,v)\geq \mu H(u,v)>0
\; \forall (u,v)\in \mathbb{R}_{+}^{*}\times
\mathbb{R}_{+}^{*},\; u^p+v^q\geq R_0.$$
\end{enumerate}

\section{Preliminaries and existence results}

The values of $H( u,v) $ are irrelevant for $u\leq 0$ or
$v\leq 0$. We set
\[
I( u,v) =\frac 1p\int_\Omega | \nabla u|
^pdx+\frac 1q\int_\Omega | \nabla v|
^qdx-\int_\Omega m( x) H( u,v) dx
\]
defined on $E:=W_0^{1,p}( \Omega ) \times W_0^{1,q}( \Omega)$.
The solutions of the system \eqref{Spq} are
critical points of the functional $I$. Note that the functional
$I$ does not satisfy in general the Palais-Smale condition since
\[
B_\mu H( u,v) :=\frac up\frac{\partial H}{\partial u}(
u,v) +\frac vq\frac{\partial H}{\partial v}( u,v) -\mu
H( u,v)
\]
$\,\,$is not always bounded. In order to apply
Ambrosetti-Rabinowitz Theorem \cite{AR}, we modify $H$ so that the
corresponding $B_\mu H( u,v) $ becomes bounded.
Let
\[
A( R) =\max \Big\{ \frac{H( u,v) }{( u^p+v^q)^\mu } :
R\leq u^p+v^q\leq R+1\Big\}
\]
and
\begin{align*}
C_R =\max &\Big\{ \sup_{u^p+v^q\leq R+1} \big|
\frac{\partial H}{\partial u}( u,v) \big| +2p\mu A(
R) ( R+1) ^{\mu +1-\frac 1p}\sup_{R\leq r\leq R+1}| \eta
_R'( r) | ;\\
&\sup_{u^p+v^q\leq R+1}\big| \frac{\partial H}{\partial v}( u,v) \big|
+2q\mu A( R)( R+1) ^{\mu +1-\frac 1q}\sup_{R\leq r\leq R+1}
| \eta_R'( r) | \Big\}
\end{align*}
where $\eta_R\in C^1( \mathbb{R})$ is a cutting function defined by
\[
\eta_R( r) \begin{cases}
 =1 & \text{if } r\leq R \\
 < 0& \text{if } R<r<R+1 \\
 =0 & \text{if } r\geq R+1.
 \end{cases}
\]
Our main result is\thinspace the following:

\begin{theorem} \label{thm1}
Assume that $(H_i)_{i=1,2,3}$  hold and
$C_R=o( R^{\tfrac{p^{*}q^{*}}{( p^{*}-p) ( q^{*}-q) }\mu})$
for $R$ sufficiently large. Then the system
\eqref{Spq} has at least one nontrivial solution $( u,v)$
in $E\cap [ L^\infty ( \Omega) ] ^2$ with $u$ and $v$
positive.
\end{theorem}
Before proving this theorem, we truncate the potential function $H$.

\subsection*{The modified problem}
Let $R\geq R_0$ be fixed, and set
\[
H_R( u,v) :=\eta_R( u^p+v^q) H( u,v) +( 1-\eta
_R( u^p+v^q) ) A( R) ( u^p+v^q) ^\mu ,
\]
By construction $H_R$ is $C^1$ and nonnegative. Let
\begin{align*}
M_R:=&( R+1) \max_{u^p+v^q\leq R+1} \big[ \eta
_R'( u^p+v^q) ( H( u,v) -A( R) (u^p+v^q) ^\mu )\big] \\
&+\max_{u^p+v^q\leq R+1} B_\mu H( u,v) ,
\end{align*}

\begin{lemma} \label{lm2}
$H_{R}$ satisfies (H1)-(H3) and the following estimates
\begin{gather}
0\leq B_\mu H_R( u,v) \leq M_R,\quad
\forall (u,v) \in \mathbb{R}^{+}\times \mathbb{R}^{+}, \label{2.1} \\
\begin{gathered}
\big| \frac{\partial H_R}{\partial u}( u,v) \big| \leq
C_R+\mu pA( R) u^{p-1}( u^p+v^q) ^{\mu -1}, \\
\big| \frac{\partial H_R}{\partial v}( u,v) \big| \leq
C_R+\mu qA( R) v^{q-1}( u^p+v^q) ^{\mu -1}, \quad
\forall (u,v) \in \mathbb{R}_{+}^2 ,
\end{gathered} \label{2.2} \\
H_R( u,v) \geq \frac{m_{R_0}}{R_0^\mu }( u^p+v^q) ^\mu
\quad \forall ( u,v) \in \mathbb{R}_{+}^{*}\times \mathbb{R}_{+}^{*},
\text{ such that }u^p+v^q\geq R_0, \label{2.3}
\end{gather}
with $m_{R_0}:=\min \{ H(u,v);u^p+v^q=R_0\}$.
\end{lemma}

\paragraph{Proof.}
(H1) and (H2) can be easly verified for $H_{R}$. We verify for (H3)
as follows:
For any $\nu >1,$we have
\begin{multline*}
B_\nu H_R(u,v)\\
=( u^p+v^q) \eta_R'(u^p+v^q) [ H( u,v) -A( R) ( u^p+v^q)
^\mu ] +\eta_R( u^p+v^q) B_\nu H(u,v),
\end{multline*}
for $R_0\leq u^p+v^q\leq R$;
\[
B_\nu H_R(u,v)=B_\nu H(u,v)\geq B_\mu H(u,v)\geq 0\quad
\text{for }1<\nu \leq \mu
\]
for $R\leq $ $u^p+v^q\leq R+1$;
\[
B_\nu H_R(u,v)\geq \eta_R( u^p+v^q) B_\nu H(u,v)\geq \eta
_R( u^p+v^q) B_\mu H(u,v)\geq 0\text{ for }1<\nu \leq \mu ;
\]
finally for $u^p+v^q\geq R+1$,
$B_\nu H_R(u,v)=0$ for any $\nu >1$.
Thus (H3) holds for $H_R$.

Conditions \eqref{2.1} and \eqref{2.2} result from  straightforward
computations. Using (H3), we have
\begin{equation}
H_R( u,v) \geq \frac{m_{R_0}}{R_0^\mu }( u^p+v^q) ^\mu
\quad,\forall ( u,v) \in \mathbb{R}_{+}^{*}\times
\mathbb{R}_{+}^{*}\text{ such that } u^p+v^q\geq R_0.  \label{2.4}
\end{equation}
In fact, put $f(t):=H_R( t^{1/p}u,t^{\frac 1q}v) $ with
$u^p+v^q\geq R_0$ then
\begin{equation}
\begin{aligned}
f'(t)=&\frac 1t \Big[ \frac{t^{1/p}u}p\frac{\partial H_R}{
\partial u}( t^{1/p}u,t^{\frac 1q}v) +\frac{t^{\frac 1q}v}q
\frac{\partial H_R}{\partial v}( t^{1/p}u,t^{\frac 1q}v)
\Big] \\
\geq& \frac \mu tf( t) \quad \text{for all }
t\geq t_0:=\frac{R_0}{u^p+v^q}( \leq 1) .
\end{aligned} \label{2.5}
\end{equation}
Integrating \eqref{2.5} between $t_0$ and $t$, we obtain
\begin{equation}
\frac{f( t) }{f( t_0) }\geq \frac{t^\mu }{t_0^\mu }
\quad\text{for all}\quad t\geq t_0  \label{2.6}
\end{equation}
and taking $t=1$ in \eqref{2.6}, we have
\[
H_R( u,v) =f(1)\geq \frac{( u^p+v^q) ^\mu \,}{R_0^\mu }f( t_0)
\]
and $f( t_0) =H_R( u_1,v_1) =H( u_1,v_1)$, where
$u_1=( \frac{R_0}{u^p+v^q}) ^{1/p}u$, and \break
$v_1=( \frac{R_0}{u^p+v^q}) ^{1/q}v$.
Consequently,
\[
\min_{u^p+v^q\geq R_0} f( t_0( u,v) ) =\min_{u^p+v^q=R_0} H(u,v),
\]
hence \eqref{2.4} follows.
Now, consider the modified system
\begin{equation} \label{SpqH}
\begin{gathered}
-\Delta_pu=m( x) \frac{\partial H_R}{\partial u}(
u,v) \quad \text{in }\Omega \\
-\Delta_qv=m( x) \frac{\partial H_R}{\partial v}(u,v)
\quad \text{in }\Omega \\
u=v=0 \quad \text{on }\partial \Omega
\end{gathered}
\end{equation}
which has an associated functional $I_R$ defined on $E$ as
\[
I_R( u,v) =\frac 1p\int_\Omega | \nabla u|
^pdx+\frac 1q\int_\Omega | \nabla v|
^qdx-\int_\Omega m( x) H_R( u,v) dx.
\]

\begin{lemma} \label{lm3}
Under the hypotheses (H1)-(H3), the functional $I_R$ satisfies
the Palais-Smale condition.
\end{lemma}

\paragraph{Proof.}
Let $( u_n,v_n)$ be an element of $E$ such that $I_R( u_n,v_n) $ is bounded
and $I_R'(u_n,v_n) \to 0$ strongly in
$W_0^{-1,p'}( \Omega ) \times W_0^{-1,q'}( \Omega )$ (dual space of $E$).

\noindent Claim 1.\quad $(u_n,v_n) $ is bounded in $E$. In fact, for any $M$,
we have
\[
-M\leq \frac 1p\int_\Omega | \nabla u_n|
^pdx+\frac 1q\int_\Omega | \nabla v_n|
^qdx-\int_\Omega m( x) H_R( u_n,v_n) dx\leq M;
\]
and for $\varepsilon \in ( 0,1) $, we have again
\begin{eqnarray*}
-\varepsilon &\leq& \frac 1p\int_\Omega | \nabla u_n|^pdx
+\frac 1q\int_\Omega | \nabla v_n|^qdx \\
&&-\int_\Omega m( x) \big[ \frac{u_n}p\frac{\partial H_R
}{\partial u}( u_n,v_n) +\frac{v_n}q\frac{\partial H_R}{\partial v
}( u_n,v_n) \big] dx\leq \varepsilon .\,\,
\end{eqnarray*}
Then we obtain
\begin{eqnarray*}
\frac{\mu -1}p\int_\Omega | \nabla u_n| ^pdx+\frac{\mu -1}
q\int_\Omega | \nabla v_n| ^qdx
&\leq &M\mu-\int_\Omega m( x) B_\mu H_R(u,v)dx \\
&\leq &M\mu +1+| m|_0M_R( \mathop{\rm meas}\Omega )
\end{eqnarray*}
where $| m|_0:=\max_{x\in \stackrel{-}{\Omega }}(
| m( x) | ) $. Hence $( u_n,v_n) $ is
bounded in $E$.

\noindent Claim 2.\quad $( u_n,v_n) $
converges strongly in $E$.  Since $( u_n,v_n) $ is
bounded in $E$, there exists a subsequence denoted again by $(
u_n,v_n) $ which converges weakly in $E$ and strongly in the
space $L^\zeta ( \Omega ) \times L^\eta ( \Omega ) $
for any $\zeta $ and $\eta $ such that, $1<\zeta <p^{*}$ and
$1<\eta <q^{*}$.
>From the definition of $I_R'$, we write
\begin{eqnarray*}
\lefteqn{ \int_\Omega ( | \nabla u_n| ^{p-2}\nabla
u_n-| \nabla u_l| ^{p-2}\nabla u_l) \nabla (u_n-u_l) dx}\\
&=&\langle I_R'( u_n,v_n)-I_R'( u_l,v_l) ,( u_n-u_l,0)\rangle \\
&&+\int_\Omega m(x) \Big[ \frac{\partial H_R}{\partial u}( u_n,v_n)
-\frac{\partial H_R}{\partial u}( u_l,v_l) \Big] (u_n-u_l) dx.
\end{eqnarray*}
By assumptions on $I_R'$,
$\langle I_R'( u_n,v_n)-I_R'( u_l,v_l) ,( u_n-u_l,0)\rangle $
converges to 0 as $n$ and $l$ tend to $+\infty $.
In what follows, $C$ denotes a generic positive constant.
Now, we prove that
$$ C_{n,l}:=\int_\Omega m( x) [ \frac{\partial H_R}{
\partial u}( u_n,v_n) -\frac{\partial H_R}{\partial u}(
u_l,v_l) ] ( u_n-u_l) dx
$$
converges to 0 as $n$ and $l$ tend to $+\infty $.
We have
\[
| C_{n,l}| \leq | m|_0\int_\Omega
[ | \frac{\partial H_R}{\partial u}( u_n,v_n)
| +| \frac{\partial H_R}{\partial u}( u_l,v_l)
| ] | u_n-u_l| dx
\]
and
\begin{eqnarray*}
\lefteqn{\int_\Omega | \frac{\partial H_R}{\partial u}(
u_n,v_n) | | u_n-u_l| dx }\\
&\leq& \int_\Omega ( C_R+\mu pA( R) | u_n|
 ^{p-1}( | u_n| ^p+| v_n| ^q) ^{\mu -1}) |u_n-u_l| dx \\
& \leq& 2^{\mu -1}C_R\int_\Omega ( 1+| u_n| ^{\mu
p-1}+| u_n| ^{p-1}| v_n| ^{q\mu -q})| u_n-u_l| dx\\
&\leq& 2^{\mu -1}C_R\Big[ \int_\Omega | u_n-u_l|dx
+\int_\Omega | u_n| ^{\mu p-1}| u_n-u_l|dx\\
&&+\int_\Omega | u_n| ^{p-1}| v_n| ^{q\mu -q}| u_n-u_l| dx\Big] .
\end{eqnarray*}
Using H\"older's inequality and Sobolev's
embeddings, we obtain
\begin{eqnarray*}
\lefteqn{ \int_\Omega \Big| \frac{\partial H_R}{\partial u}(
u_n,v_n) \Big| | u_n-u_l| dx }\\
&\leq &2^{\mu -1}C_R( \mathop{\rm meas}\Omega ) ^{\frac{p-1}p}\Big[
\int_\Omega |u_n-u_l| ^pdx\Big] ^{1/p} \\
&&+2^{\mu -1}C_R\Big[ \int_\Omega | u_n| ^{\mu
p}dx\Big] ^{\frac{\mu p-1}{\mu p}}\Big[ \int_\Omega |
u_n-u_l| ^{\mu p}dx\Big] ^{\frac 1{\mu p}} \\
&&+2^{\mu -1}C_R\Big[ \int_\Omega |
u_n| ^{\mu p}dx\Big] ^{\frac{p-1}{\mu p}}\Big[ \int_\Omega
| v_n| ^{\mu q}dx\Big] ^{\frac{\mu -1}\mu }\Big[
\int_\Omega | u_n-u_l| ^{\mu p}dx\Big] ^{\frac 1{\mu p}},
\end{eqnarray*}
(because $( u_n) \in W_0^{1,p}( \Omega ) $ and $\mu p<p^{*}$,
$( v_n) \in W_0^{1,q}( \Omega) $ and $\mu q<q^{*}$). Then
\[
\int_\Omega \Big| \frac{\partial H_R}{\partial u}(
u_n,v_n) \Big| | u_n-u_l| dx\leq C\|
u_n-u_l\|_{L^p( \Omega ) }+C\| u_n-u_l\|
_{L^{\mu p}( \Omega ) }.
\]
Similarly, we obtain
\[
\int_\Omega | \frac{\partial H_R}{\partial u}(
u_l,v_l) | | u_n-u_l| dx\leq C\|
u_n-u_l\|_{L^p( \Omega ) }+C\| u_n-u_l\|_{L^{\mu p}( \Omega ) },
\]
and so
$| C_{n,l}| \leq | m|_0( C\| u_n-u_l\|_{L^p}+C\| u_n-u_l\|_{L^{\mu
p}})$.
Hence $C_{n,l}$ converges to $0$ as $n$ and $l$ tend to $+\infty $. \\We
have the following algebraic relation \cite{Si}
\begin{multline}
| \nabla u_n-\nabla u_l| ^p\\
\leq C\big[ ( |\nabla u_n| ^{p-2}\nabla u_n-| \nabla u_l|
^{p-2}\nabla u_l) \nabla ( u_n-u_l) \big] ^{s/2}
\big( | \nabla u_n| ^p+| \nabla u_l|^p\big) ^{1-\frac s2},  \label{2.7}
\end{multline}
where
$s=\begin{cases} p & \text{for }1<p\leq 2 \\
2 &\text{for }2<p\end{cases}$.
Integrating \eqref{2.7} on $\Omega$, and using H\"older's inequality in the right
hand side, we obtain
\begin{multline*}
\| u_n-u_l\|_{1,p}^p \\
\leq C\Big[ \int_\Omega (| \nabla u_n| ^{p-2}\nabla u_n-| \nabla u_l|
^{p-2}\nabla u_l) \nabla ( u_n-u_l) dx\Big] ^{\frac s2}
\big( \| u_n\|_{1,p}^p+\| u_l\|_{1,p}^p\big) ^{1-\frac s2}.
\end{multline*}
Now since
\[
\int_\Omega ( | \nabla u_n| ^{p-2}\nabla
u_n-| \nabla u_l| ^{p-2}\nabla u_l) \nabla (
u_n-u_l) dx\to 0
\]
as $n$ and $l$ tend to $+\infty$, the sequence $( u_n) $ converges
strongly in $W_0^{1,p}(\Omega ) $. Similarly we prove that the sequence
$(v_n) $ converges strongly in $W_0^{1,q}( \Omega ) $.

The next lemma shows that $I_R$ satisfies the geometric
assumptions of the Mountain-Pass Theorem.

\begin{proposition} \label{prop4}
Under assumptions (H1)-(H3) we have \begin{enumerate}
\item There exist two positive real numbers $\rho, \sigma $
and a neighborhood $V_\rho $ of the origin of $E$, such that for any
element $( u,v)$ on the boundary of $V_\rho $:
 $I_R( u,v) \geq \sigma >0$.
\item There exist $( \phi ,\theta )$ in $E$
such that $I_R( \phi ,\theta ) <0$.
\end{enumerate}
\end{proposition}

\paragraph{Proof.}
 From (H2) and taking into account that $H_R( u,v) =H( u,v) $
for $u^p+v^q\leq R$, we can write
\[
\forall \varepsilon >0,\exists \delta_\varepsilon >0:u^p+v^q\leq
\delta_\varepsilon \Longrightarrow H_R( u,v) \leq
\varepsilon ( u^p+v^q) ,
\]
and since $H_R( u,v) /( u^p+v^q) ^\mu $ is uniformly
bounded as $u^p+v^q$ tends to $+\infty $
\[
\exists M( \varepsilon ,R) >0:u^p+v^q\geq \delta
_\varepsilon \Longrightarrow H_R( u,v) \leq M(u^p+v^q) ^\mu .
\]
Then for every $( u,v) $ in $\mathbb{R}^{+}\times \mathbb{R}^{+}$ we have
\[
H_R( u,v) \leq \varepsilon ( u^p+v^q) +M(u^p+v^q) ^\mu .
\]
Hence
\begin{eqnarray*}
\lefteqn{\int_\Omega m( x) H_R( u,v) dx }\\
&\leq& |m|_0\Big[ \varepsilon \int_\Omega ( u^p+v^q)
dx+M\int_\Omega ( u^p+v^q) ^\mu dx\Big] \\
&\leq& | m|_0\Big[ \int_\Omega ( \varepsilon
u^p+2^{\mu -1}Mu^{p\mu }) dx+\int_\Omega (
\varepsilon v^q+2^{\mu -1}Mv^{q\mu }) dx\Big]\\
&\leq& C| m|_0\big[ \varepsilon ( \| u\|
_{1,p}^p+\| v\|_{1,q}^q) +M( \| u\|
_{1,p}^{\mu p}+\| v\|_{1,q}^{\mu q}) \big] .
\end{eqnarray*}
For $I_R(u,v)$, we obtain
\begin{eqnarray*}
I_R( u,v) &\geq &\| u\|_{1,p}^p\big[ \frac 1p-C| m|_0( \varepsilon +M\| u\|
_{1,p}^{\mu p-p}) \big] \\
&&+\| v\|_{1,q}^q\big[ \frac 1q-C| m|_0( \varepsilon +M\|
v\|_{1,q}^{\mu q-q})\big]
\geq \sigma >0,
\end{eqnarray*}
for every $( u,v) $ in the sphere $S( 0,\rho ) $ of
$E$ where $\rho $ is such that $0<\rho <\min ( \rho_1,\rho
_2) $ with
\[
\rho_1=\big[ \frac 1{pMC| m|_0}-\frac \varepsilon M\big]
^{\frac 1{\mu p-p}}\quad\text{and}\quad
\rho_2=\big[ \frac 1{qMC|m|_0}-\frac \varepsilon M\big] ^{\frac 1{\mu q-q}}
\]
with $\varepsilon $ sufficiently small.

\noindent 2.\quad  Choose $( \phi ,\theta )\in E$ such that:
$\phi >0$, $\theta >0$,
\[
\mathop{\rm supp}\phi \subset \Omega ^{+},
\quad \mathop{\rm supp}\theta \subset \Omega ^{+},
\]
where $\Omega ^{+}=\{ x\in \Omega ; m( x) >0\}$.
Hence,  for $t$ sufficiently large,
\begin{eqnarray*}
I_R( t^{1/p}\phi ,t^{1/q}\theta ) &=&
\frac tp\| \phi\|_{1,p}^p+\frac tq\| \theta \|
_{1,q}^q-\int_\Omega m( x) H_R( t^{1/p}\phi,t^{1/q}\theta ) dx\\
&\leq& t\big[ \frac{\| \phi \|_{1,p}^p}p+\frac{\|
\theta \|_{1,q}^q}q\big] -t^\mu \frac{m_{R_0}}{R_0^\mu
}\int_\Omega m( x) ( \phi ^p+\theta ^q) ^\mu dx
\end{eqnarray*}
and so $\lim_{t\to +\infty }I_R( t^{1/p}\phi ,t^{1/q}\theta ) =-\infty$,
(because $\mu>1$). By continuity of $I_R$ on $E$, there exists $( \phi
,\theta ) $ in $E\setminus B( 0,\rho ) $ such that
$I_R( \phi ,\theta ) <0$. By the usual Mountain-Pass
Theorem, we know that there exists a critical point of $I_R$ which
we denote by $( u_R,v_R) $, and corresponding to a critical
value $c_R\geq \sigma $. Since $( u_R^{+},v_R^{+}) $, where
$u_R^{+}:=\max ( u_R,0) $, is
also solution for the system $( S_{p,q}^{H_R}) $, we assume $u_R\geq 0$
and $v_R\geq 0$. Positivity of $u_R$ and $v_R$ follows from
Harnack's inequality (see J. Serrin \cite{Se}). We prove now that, under
some conditions, $( u_R,v_R) $ is also solution of the system
\eqref{SpqH}.

\section{Existence results}

We adapt the Moser iteration used in \cite{O,TV} to
construct two strictly unbounded sequences $( \lambda_k)
_{k\in \mathbb{N}}$ and $( \mu_k)_{k\in \mathbb{N}}$ such that
$( u_R,v_R) $ satisfies
\[
\text{if }\left\{\begin{array}{c}
u_R\in L^{\lambda_k}( \Omega ) \\
v_R\in L^{\mu_k}( \Omega )\end{array}\right\}\quad\text{then}\quad
\left\{\begin{array}{c}
u_R\in L^{\lambda_{k+1}}( \Omega ) \\
v_R\in L^{\mu_{k+1}}( \Omega ) .
\end{array} \right\}
\]

\subsection*{Bootstrap argument}

\begin{proposition} \label{prop5}
Under the assumptions of Theorem \ref{thm1}, there exist two sequences
$( \lambda_k)_k$ and $( \mu_k)_k$ such that
\begin{enumerate}
\item For each $k$, $u_R$ and $v_R$
 belong to $L^{\lambda_k}( \Omega ) $ and $L^{\mu_k}( \Omega ) $ respectively

\item There exist two positive constants $C_p$  and $C_q$ such that
\[
\| u_R\|_\infty \leq \limsup_{k\to +\infty }\| u_R\|_{L^{\lambda_k}}
\leq C_p,\quad\text{and}\quad
\|v_R\|_\infty \leq \limsup_{k\to +\infty }\| v_R\|_{L^{\mu_k}}\leq C_q.
\]
\end{enumerate}
\end{proposition}

\begin{lemma} \label{lm6}
Let $( a_k)_{k\in \mathbb{N}}$  and $( b_k)_{k\in \mathbb{N}}$
be two positive sequences satisfying, for each integer $k$,
the relations
\begin{equation}
\frac{p+a_k}{\lambda_k}+\frac{q( \mu -1) }{\mu_k}=1,\quad\text{and}
\quad \frac{q+b_k}{\mu_k}+\frac{p( \mu -1) }{\lambda_k}=1. \label{3.1}
\end{equation}
If $u_R$  and $v_R$ are in $L^{\lambda_k}( \Omega ) $ and
$L^{\mu_k}( \Omega )$ respectively,
$\lambda_{k+1}\leq (1+\frac{a_k}p) \pi_p$,
$\mu_{k+1}\leq ( 1+\frac{b_k}q) \pi_q$ with
 $1<\pi_p<p^{*}$ and $1<\pi_q<q^{*}$, then we have:
\begin{gather}
\| u_R\|_{\lambda_{k+1}}^{\lambda_{k+1}}\leq
K_p\Big\{ \theta_p\big[ 1+\frac{a_k}p\big] \big[ C_R|
m|_0( \| u_R\|_{\lambda_k}^{\lambda_k}+\| v_R\|_{\mu_k}^{\mu
_k}) \big] ^{1/p}\Big\} ^{\frac{\lambda_{k+1}}{1+\frac{a_k}p}
},  \label{3.2} \\
\| v_R\|_{\mu_{k+1}}^{\mu_{k+1}}\leq K_q\Big\{
\theta_q\big[ 1+\frac{b_k}q\big] \big[ C_R| m|_0(
\| u_R\|_{\lambda_k}^{\lambda_k}+\| v_R\|
_{\mu_k}^{\mu_k}) \big] ^{\frac 1q}\Big\} ^{\frac{\mu
_{k+1}}{1+\frac{b_k}q}} \label{3.2'}
\end{gather}
where $\| z\|_\beta $ is $\| z\|_{L^\beta ( \Omega ) }$
and $K_p$, $K_q$, $\theta_p$, and $\theta_q$ are positive constants.
\end{lemma}

\paragraph{Proof.}
Remark that if, for an infinite number of integers $k$,
$\| u_R\|_{\lambda_k}\leq 1$ then $\| u_R\|_\infty \leq 1$ and
proposition 1 is proved. So we suppose that
$\| u_R\|_{\lambda_k}\geq 1$ for all $k\in \mathbb{N}$.
Let $\zeta_n$, $n\in \mathbb{N}$, be $C^1$ functions such that
\[
\begin{array}{ll}
\zeta_n( s) =s & \text{if\ \thinspace }s\leq n\, \\
\zeta_n( s) =n+1 & \text{if \ }s\geq n+2 \\
0<\zeta_n'( s) <1 & \text{if \ }s\in \mathbb{R}^+.
\end{array}
\]
Put $u_n:=\zeta_n( u_R) $, then $u_n^{1+a_k}\in W_0^{1,p}(
\Omega ) \cap L^\infty ( \Omega ) $ and $u_R$
satisfies the first equation of the system \eqref{SpqH}.
Multiply this equation by $u_n^{1+a_k}$ and integrate over $\Omega $
to get
\begin{eqnarray*}
\int_\Omega -\Delta_pu_R.u_n^{1+a_k}dx
&=&\int_\Omega m( x) \frac{\partial H_R}{\partial u}(
u_R,v_R) u_n^{1+a_k}dx \\
&\leq &2^{\mu -1}C_R| m|_0\int_\Omega (
1+u_R^{p\mu -1}+u_R^{p-1}v_R^{q\mu -q}) u_n^{1+a_k}dx.
\end{eqnarray*}
Since $u_n\leq u_R$, we have
\begin{multline*}
\int_\Omega -\Delta_pu_R.u_n^{1+a_k}dx\\
\leq 2^{\mu -1}C_R|m|_0\Big\{ \int_\Omega u_R^{1+a_k}dx+\int_\Omega
u_R^{p\mu +a_k}dx+\int_\Omega u_R^{p+a_k}v_R^{q\mu -q}dx\Big\}.
\end{multline*}
Using H\"older's inequality, we obtain
\begin{eqnarray*}
\int_\Omega -\Delta_pu_R.u_n^{1+a_k}dx
&\leq& 2^{\mu -1}C_R|m|_0\Big\{ ( \mathop{\rm meas}\Omega ) ^{1-\frac{1+a_k}{
\lambda_k}}\| u_R\|_{\lambda_k}^{1+a_k}+\| u_R\|
_{p\mu +a_k}^{p\mu +a_k}\\
&&+\| u_R\|_{\lambda_k}^{p+a_k}\|
v_R\|_{\mu_k}^{q\mu -q}\Big\}.
\end{eqnarray*}
We shall show below that $p\mu +a_k=\lambda_k$. Since
$\| u_R\|_{\lambda_k}\geq 1$, we get
\begin{multline*}
\int_\Omega -\Delta_pu_R.u_n^{1+a_k}dx\\
\leq 2^{\mu-1}C_R| m|_0\max ( 1,\,\mathop{\rm meas}\Omega )
\big[ 2\| u_R\|_{\lambda_k}^{\lambda_k}+\|
u_R\|_{\lambda_k}^{p+a_k}\| v_R\|_{\mu
_k}^{q\mu -q}\big] .
\end{multline*}
Moreover, using the relation \eqref{3.1}, we obtain
\[
\| u_R\|_{\lambda_k}^{p+a_k}\| v_R\|_{\mu_k}^{q\mu
-q}\leq \| u_R\|_{\lambda_k}^{\lambda_k}+\| v_R\|_{\mu_k}^{\mu_k},
\]
so, with $c_0:=3\max( 1,\mathop{\rm meas}\Omega )$,
\begin{equation}
\int_\Omega -\Delta_pu_R.u_n^{1+a_k}dx\leq 2^{\mu-1}c_0C_R
| m|_0\big[ \| u_R\|_{\lambda_k}^{\lambda_k}+\| v_R\|_{\mu_k}^{\mu_k}\big].
\label{3.3}
\end{equation}
On the other hand we have
\begin{eqnarray*}
\int_\Omega -\Delta_pu_R.u_n^{1+a_k}dx
&=&( 1+a_k)\int_\Omega | \nabla u_R| ^p\zeta_n'(u_R) u_n^{a_k}dx \\
&\geq &( 1+a_k) \int_\Omega | \nabla u_R|^p( \zeta_n'( u_R) ) ^pu_n^{a_k}dx \\
&=&( 1+a_k) \int_\Omega | \nabla u_n|^pu_n^{a_k}dx
\end{eqnarray*}
and thus
\begin{equation}
\int_\Omega -\Delta_pu_R.u_n^{1+a_k}dx\geq
\int_\Omega | \nabla u_n| ^pu_n^{a_k}dx\,.  \label{3.4}
\end{equation}
Since  $u_n^{1+\frac{a_k}p}\in W_0^{1,p}( \Omega ) $, the
continuous imbedding of $W_0^{1,p}( \Omega ) $ in
$L^{\pi_p}( \Omega ) $ implies the existence of a positive constant
$c$ such that
\begin{eqnarray}
\Big( \int_\Omega | \,u_n^{1+\frac{a_k}p\pi_p}|
dx\Big) ^{\frac 1{\pi_p}} &\leq &
c\Big( \int_\Omega |\nabla u_n^{1+\frac{a_k}p}| ^p dx\Big) ^{1/p}
\nonumber \\
&=&c\big[ 1+\frac{a_k}p\big] \Big( \int_\Omega
u_n^{a_k}| \nabla u_n| ^pdx\Big) ^{1/p}.\,
\label{3.5}
\end{eqnarray}
By assumption, we have
$\lambda_{k+1}\leq j_k:=\big[ 1+\frac{a_k}p\big] \pi_p$.
Then
\[
\| u_n\|_{\lambda_{k+1}}\leq ( \mathop{\rm meas}\Omega) ^{m_k}
\| u_n\|_{j_k},\quad  \text{where }
m_k:=\frac 1{\lambda_{k+1}}-\frac 1{( 1+\frac{a_k}p) \pi_p}
\]
and thus
\[
\| u_n\|_{\lambda_{k+1}}^{\lambda_{k+1}}\leq K_p\|u_n\|_{j_k}^{\lambda_{k+1}}
\]
where $K_p$ is a positive constant greater than
$(\mathop{\rm meas}\Omega ) ^{m_k\lambda_{k+1}}$ independently of
the integer $k$. By the relation \eqref{3.5},
\begin{equation}
\| u_n\|_{j_k}^{\lambda_{k+1}}\leq \Big[ c[ 1+\frac{a_k}p]
\Big( \int_\Omega u_n^{a_k}| \nabla u_n|
^pdx\Big) ^{1/p}\Big] ^{\frac{\lambda_{k+1}}{1+\frac{a_k}p}}. \label{3.6}
\end{equation}
Combining the inequalities \eqref{3.3}-\eqref{3.6}, we deduce
\[
\| u_n\|_{\lambda_{k+1}}^{\lambda_{k+1}}\leq
K_p\left\{ \theta_p\big[ 1+\frac{a_k}p\big] \big\{ C_R|
m|_0( \| u_R\|_{\lambda_k}^{\lambda
_k}+\| v_R\|_{\mu_k}^{\mu
_k}) \big\} ^{1/p}\right\} ^{\frac{\lambda_{k+1}}{1+\frac{a_k}p
}},
\]
with $\theta_p=2^{\frac{\mu -1}p}c_0^{1/p}c$.
Hence, by letting $n$ $\to +\infty $, we obtain \eqref{3.2}. Similarly we
show \eqref{3.2'}.

\subsection*{Construction and definition of $( \lambda_k)_k$ and
$( \mu_k)_k.$}

Here we construct the sequences $( \lambda_k)_k$ and $( \mu
_k)_k$ using tools similar as those in [O] or [TV].
The first terms of each sequence cannot be determined
directly by using the Rellich-Kondrachov continuous imbedding
result. So, we first construct two other sequences $(
\widehat{\lambda }_k)_k$ and $( \widehat{\mu }_k)
_k$, such that for each $k$, $u_R$ and $v_R$ belong
to $L^{\widehat{\lambda }_k}( \Omega ) $ and
$L^{\widehat{\mu }_k}( \Omega ) $ respectively.
By a suitable choice of $k_0$, $\widehat{\lambda }_{k_0}$ and
$\widehat{\mu }_{k_0}$ determine the first
terms of $( \lambda_k)_k$ and $( \mu_k)_k$

\subsubsection*{Construction of $( \widehat{\lambda }_k)_k$\ and
$( \widehat{\mu }_k)_k$.}
Suppose  $p\leq q$, and fix a number $s$, such that
$c p/p^*<s<1/\mu$. Put
\[
\widehat{C}:=\frac 12+\frac s2\frac{p^{*}}p.
\]
Remark that $\widehat{C}>1$, $1<\mu p\widehat{C}<p^{*}$ and
$1<\mu q\widehat{C}<q^{*}$.
Now, we take
$\widehat{\lambda }_k=\mu p\widehat{C}^k$ and
$\widehat{\mu }_k=\mu q \widehat{C}^k$.
By definition of $( a_k^{}) $, we have
\[
\frac{p+a_k^{}}{\widehat{\lambda }_k}+\frac{\mu -1}{\widehat{\mu }_k}q=1
\]
then
$a_k=\widehat{\lambda }_k-p\mu$. Similarly, we find
$b_k=\widehat{\mu }_k-q\mu$.

\begin{lemma} \label{lm7}
For each integer $k$, $u_R\in L^{\widehat{\lambda }_k}( \Omega )$ and
$v_R\in L^{\widehat{\mu }_k}( \Omega )$.
\end{lemma}

\paragraph{Proof.}
By induction. For $k=0$, $\widehat{\lambda }_0=\mu p<p^{*}$,
$\widehat{\mu }_0=\mu q<q^{*}$, and since $( u_R,v_R) \in
E$, by the Sobolev imbedding theorem, we have
$u_R\in L^{\widehat{\lambda }_0}( \Omega )$ and
$v_R\in L^{\widehat{\mu }_0}( \Omega )$.

Suppose that the proposition is true for all integers $k'$
such that $0\leq k'\leq k$. Take
\[
\pi_p=\mu p\widehat{C}\quad \text{and}\quad \pi_q=\mu q\widehat{C}.
\]
Since $u_R\in L^{\widehat{\lambda }_k}( \Omega ) $ and
\[
[ 1+\frac{a_k}p] \pi_p =\big[ 1+\frac{\widehat{\lambda }_k-\mu p}p\big]
\mu p\widehat{C}
= \mu p\widehat{C}+\mu ^2p\widehat{C}^{k+1}-\mu ^2p\widehat{C}\geq \mu p
\widehat{C}^{k+1}
\]
i.e. $[ 1+\frac{a_k}p] \pi_p\geq \widehat{\lambda }_{k+1}$,
Lemma 3 allows us to write
$u_R\in L^{\widehat{\lambda }_{k+1}}( \Omega )$ and
$v_R\in L^{\widehat{\mu }_{k+1}}( \Omega )$.
\paragraph{Construction of $( \lambda_k)_k$ and $( \mu_k)_k$}
Put
\[
C=\frac N{N-p},\quad \text{and}\quad \delta =\big[ \frac pN\mu \widehat{C}
^{k_0}-( \mu -1)\big] C,
\]
where the integer $k_0$ is chosen so as to have $\delta >0$. The sequences $%
( \lambda_k)_k$ and $( \mu_k)_k$ are defined by
$\lambda_k=pf_k$  and $\mu_k=qf_k$,
where
\[
f_k=\frac C{C-1}[ \delta C^{k-1}+( \mu -1) ] .
\]
We remark that the three last sequences are strictly increasing and
unbounded. Furthermore $( f_k) $ satisfies the relation
$f_{k+1}=C[ f_k-( \mu -1) ]$.

\paragraph{Proof of Proposition 2.}
1. We show by induction that for all integer $k$,
$u_R\in L^{\lambda_k}( \Omega )$ and $v_R\in L^{\mu_k}( \Omega )$.
For $k=0$,
\[
\lambda_0=pf_0=\frac{pC}{C-1}\big[ \frac \delta C+( \mu-1)\big]
=p\frac Np\big[ \frac pN\mu \widehat{C}^{k_0}\big]
=\widehat{\lambda }_{k_0},
\]
and similarly, $\mu_0=\widehat{\mu }_{k_0}$.

By Lemma 4, $u_R\in L^{\lambda_0}( \Omega )$ and $v_R\in L^{\mu_0}( \Omega )$.
Suppose that $( u_R,v_R) \in L^{\lambda_k}( \Omega )
\times L^{\mu_k}( \Omega ) $. First we establish that
$\lambda_k=a_k+p\mu$. By  condition \eqref{3.1},
\[
1=\frac{p+a_k}{\lambda_k}+q\frac{\mu -1}{\mu_k}=\frac p{\lambda_k}-\frac
q{\mu_k}+\frac{a_k}{\lambda_k}+\mu \frac q{\mu_k},
\]
thus
\[
\frac{a_k}{pf_k}+\frac \mu {f_k}=1
\]
which implies
$a_k=p( f_k-\mu ) =\lambda_k-p\mu$, and  similarly
$\mu_k=b_k+q\mu =q( f_k-\mu )$.
Now when we take $\pi_p=Cp$ and $\pi_q=Cq$, we then have
\[
\big[ 1+\frac{a_k}p\big] \pi_p=( 1+f_k-\mu )Cp=pf_{k+1}=\lambda_{k+1}.
\]
and similarly
$[ 1+\frac{b_k}q] \pi_q=\mu_{k+1}$.
Since $( u_R,v_R) \in L^{\lambda_k}( \Omega ) \times
L^{\mu_k}( \Omega ) $, we conclude, according to Lemma 3,
that
\[
( u_R,v_R) \in L^{\lambda_{k+1}}( \Omega ) \times
L^{\mu_{k+1}}( \Omega ) .
\]
So $u_R\in L^{\lambda_k}( \Omega ) $, and $v_R\in L^{\mu_k}( \Omega ) $,
for all integer $k$.

\noindent 2. Now we prove that $u_R$ and $v_R$ are bounded. By Lemma 3,
we have
\begin{gather*}
\| u_R\|_{\lambda_{k+1}}^{\lambda_{k+1}}\leq
K_p\Big\{ \theta_p\big[ 1+\frac{a_k}p\big] \big\{ C_R|
m|_0( \| u_R\|_{\lambda_k}^{\lambda_k}+\| v_R\|_{\mu_k}^{\mu_k})
\big\} ^{1/p}\Big\} ^{\frac{\lambda_{k+1}}{1+\frac{a_k}p}},
\\
\| v_R\|_{\mu_{k+1}}^{\mu_{k+1}}\leq K_q\Big\{
\theta_q\big[ 1+\frac{b_k}q\big] \big\{ C_R| m|
_0( \| u_R\|_{\lambda_k}^{\lambda_k}+\|
v_R\|_{\mu_k}^{\mu_k}) \big\} ^{\frac 1q}\Big\}
^{\frac{\mu_{k+1}}{1+\frac{b_k}q}}.
\end{gather*}
We remark that
\[
\frac{\lambda_{k+1}}{1+\frac{a_k}p}=pC\quad\text{and}\quad
\frac{\mu_{k+1}}{1+\frac{b_k}q}=qC.
\]
Consequently,
\begin{gather*}
\| u_R\|_{\lambda_{k+1}}^{\lambda_{k+1}}\leq
2^CK_p\theta_p^{pC}\big[ 1+\frac{a_k}p\big]_\infty ^{pC}(
| m|_0^{}C_R^{}) ^C\max \big( \| u_R\|
_{\lambda_k}^{\lambda_kC},\| v_R\|_{\mu_k}^{\mu_kC}\big),
\\
\| v_R\|_{\mu_{k+1}}^{\mu_{k+1}}\leq 2^CK_q\theta
_q^{qC}\big[ 1+\frac{b_k}q\big] ^{qC}( | m|
_0^{}C_R^{}) ^C\max \big( \| u_R\|_{\lambda
_k}^{\lambda_kC},\| v_R\|_{\mu_k}^{\mu_kC}\big) .
\end{gather*}
We have
\[
1+\frac{a_k}p=1+\frac{b_k}q=1+f_k-\mu <\frac C{C-1}
\big[ \frac \delta C+\mu-1\big] C^k.
\]
Take
\[
A:=\frac C{C-1}\big[ \frac \delta C+\mu -1\big] [ K_p+K_q]
\]
and $\theta :=2| m|_0\max ( \theta_p^p,\theta_q^q)$,
then we can write
\[
\max \Big( \| u_R\|_{\lambda_{k+1}}^{\lambda
_{k+1}},\| v_R\|_{\mu_{k+1}}^{\mu_{k+1}}\Big)
\leq ( A^q\theta ) ^CC^{kqC}C_R^C\max \Big( \| u_R\|
_{\lambda_k}^{C\lambda_k},\| v_R\|_{\mu_k}^{C\mu_k}\Big) .
\]
We construct an iterative relation
\[
E_{k+1}\leq r_k+CE_k
\]
where
$E_k=\ln \max ( \| u_R\|_{\lambda_k}^{\lambda_k},
\|v_R\|_{\mu_k}^{\mu_k}) $, and $r_k=ak+b$, with
$a=\ln C^{qC}$ and $b=\ln [ A^q\theta C_R] ^C$.
Proceeding step by step, we find
\begin{eqnarray*}
E_{k+1} &\leq &r_k+Cr_{k-1}+C^2r_{k-2}+\cdots +C^kr_0+C^{k+1}E_{0,} \\
E_{k+1} &\leq &C^{k+1}E_0+\sum_{i=0}^kC^ir_{k-i}.
\end{eqnarray*}
Let us evaluate
\[
\sigma_k:=\sum_{i=0}^kC^ir_{k-i}.
\]
We have
$r_{k-i}=a( k-i) +b=ak+b-ai$,
then
\begin{eqnarray*}
\sigma_k&=&( ak+b) \sum_{i=0}^kC^i-a\sum_{i=0}^kiC^i\\
&=& \frac{bC^{k+2}+( a-b) C^{k+1}+( 1-C) ak-[ C(
a+b) -b] }{( C-1) ^2}.
\end{eqnarray*}
Since $C>1$, and $a$, $b$ are positive, we have
\[
\sigma_k\leq \frac{bC^{k+2}+( a-b) C^{k+1}}{( C-1)^2}
\]
then
\[
E_{k+1}\leq \frac{bC^{k+2}}{( C-1) ^2}+C^{k+1}
\big[ \frac{a-b}{( C-1) ^2}+E_0\big] .
\]
By an appropriate choice for the constants $K_p$ and $K_q$, we ensure that
\[
\frac{b-a}{( C-1) ^2}\geq E_0.
\]
Recall that
\[
b-a=C\ln \frac{A^q\theta C_R}{C^q}\quad\text{with}\quad
A=\frac C{C-1}\big[ \frac \delta C+\mu -1\big] [K_p+K_q] ;
\]
hence $E_{k+1}\leq bC^{k+2}/ ( C-1) ^2$.
By the definition of $E_{k+1}$ and the last inequality, we obtain
\[
\lambda_{k+1}\ln \| u_R\|_{\lambda_{k+1}}\leq
E_{k+1}\leq \frac{bC^{k+2}}{( C-1) ^2},
\]
thus
\[
\ln \| u_R\|_{\lambda_{k+1}}\leq
\frac{bC^{k+2}}{\lambda_{k+1}( C-1) ^2}.
\]
Letting $k\to +\infty $, we find
\[
\ln \| u_R\|_\infty \leq \frac{bC}{p\delta (C-1) },
\quad\text{or}\quad
\ln \| u_R\|_\infty \leq \frac N{\delta p^2}b.
\]
Similarly
\[
\ln \| v_R\|_\infty \leq \frac N{\delta q^2}b.
\]
We deduce the existence of constants $C_p$ and $C_q$ such that:
\[
\| u_R\|_\infty \leq C_p\quad \text{and}\quad
\| v_R\|_\infty \leq C_q.
\]
Take
\[
C_p=\exp \frac N{\delta p^2}b,\quad\text{and}\quad
C_q=\exp \frac N{\delta q^2}b.
\]
Then $C_p$ and $C_q$, are greater than $1$, which is compatible with the
remark noted at the beginning of the proof of Lemma 3.
This completes the proof of proposition 1.

\paragraph{Proof of Theorem \ref{thm1}.}
If $\| u_R\|_\infty ^p+\| v_R\|_\infty^q<R$, then $( u_R,v_R) $ furnishes
a solution of the system \eqref{Spq}. We have
\[
\| u_R\|_\infty ^p+\| v_R\|_\infty ^q\leq
C_p^p+C_q^q\leq 2\exp \frac N{\delta p}b;
\]
so it is sufficient to have $2\exp \frac N{\delta p}b<R$
for $R$ large enough, to get $( u_R,v_R) $ solution of
the initial system \eqref{Spq}. Replacing $b$ by its
expression, we obtain
$( A^q\theta C_R) ^{\frac{CN}{\delta p}}<\frac R2$
i.e.
\[
C_R<\frac{R^{\frac{\delta p}{CN}}}{2^{\frac{\delta p}{CN}}\theta A^q}.
\]
But $\delta $ can be chosen such that
\[
\frac{\delta p}{CN}>\frac{N^2}{pq}\mu =\frac{p^{*}q^{*}}{(
p^{*}-p) ( q^{*}-q) }\text{ }\mu
\]
and we can take $C_R<\frac{R^{\frac{p^{*}q^{*}}{( p^{*}-p)
( q^{*}-q) }\text{ }\mu }}{2^{\frac{\delta p}{CN}}\theta A^q}.\;$%
Then $( u_R,v_R) $ is solution of system \eqref{Spq} if
\[
C_R=o\Big( R^{\frac{p^*q^*}{( p^*-p) (q^*-q)}\mu }\Big)
\]  for $R$ sufficiently large.


\subsection*{Examples}
Now, we present functions satisfying the hypotheses in
our main result.

For $1<\gamma <\min ( \frac{p^{*}}p,\frac{q^{*}}q) $, let
\[
H( u,v) =( u^p+v^q) ^\gamma
\]
be defined on $\mathbb{R}_{+}^2$. Then $H$ satisfies the hypotheses
of Theorem \ref{thm1}.

For $\alpha ,\beta \geq 0$, $\frac{\alpha +1}p+\frac{\beta +1}q>1$ and
$\frac{\alpha +1}{p^{*}}+\frac{\beta +1}{q^{*}}<1$, let
\[
H( u,v) =u^{\alpha +1}v^{\beta +1}\,.
\]
be defined on $\mathbb{R}_{+}^2$. Then $H$ satisfies the hypotheses
of Theorem \ref{thm1}.


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\noindent\textsc{Ahmed Bensedik} 
(e-mail: ahmed\_benseddik2002@yahoo.fr)\\
\textsc{Mohammed Bouchekif} 
(e-mail: m\_bouchekif@mail.univ-tlemcen.dz)\\[3pt]
D\'epartement de Math\'ematiques, Universit\'e de Tlemcen \\
B. P. 119, 13000 Tlemcen, Alg\'erie.
 




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