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\markboth{\hfil Local and global estimates for solutions \hfil EJDE--2001/19}
{EJDE--2001/19\hfil A. Bechah \hfil}
\begin{document}
\title{\vspace{-1in}\parbox{\linewidth}{\footnotesize\noindent
{\sc Electronic Journal of Differential Equations},
Vol. {\bf 2001}(2001), No. 19, pp. 1--14. \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} \\
%
Local and global estimates for solutions of systems involving
the p-Laplacian \\ in unbounded domains
%
\thanks{ {\em Mathematics Subject Classifications:} 35J20, 35J45, 35J50, 35J70.
\hfil\break\indent
{\em Key words:} quasilinear systems, p-Laplacian operator, unbounded domain,
Serrin estimate.
\hfil\break\indent
\copyright 2001 Southwest Texas State University. \hfil\break\indent
Submitted November 23, 2001. Published March 23, 2001.} }
\date{}
%
\author{ A. Bechah }
\maketitle
\begin{abstract}
In this paper, we study the local and global behavior of solutions of
systems involving the p-Laplacian operator in unbounded domains.
We extend some Serrin-type estimates which are known for simple equations
to systems of equations.
\end{abstract}
\newcommand{\leb}[2]{L^{#1}(#2)}
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\newtheorem{prop}[theo]{Proposition}
\newtheorem{remark}[theo]{Remark}
\renewcommand{\theequation}{\thesection.\arabic{equation}}
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\section{Introduction}
We consider the system
\begin{gather}
-\Delta_p u=f(x,u,v) \quad x\in\Omega,\label{sys1}\\
-\Delta_q v=g(x,u,v) \quad x\in\Omega, \label{sys2}\\
u=v=0 \quad x\in\partial\Omega .\label{sys3}
\end{gather}
where $\Omega\subset\mathbb{R}^{N}$ is an exterior domain, $f, g$ are a given
functions depending of the variables $x,u,v$ and $\Delta_p$
is the $p$-Laplacian operator; for $1
1$ is a fixed exponent, $a$ is a
positive constant and $g$ is a measurable function, then for each
$y\in\Omega$ and $R>0$ we have the estimate
\begin{equation}
\label{esv-serrin}
\sup_{B_R(y)} u(x)\leq
cR^{-\frac{N}{p}}\left(\|u\|_{\leb{p}{B_{2R}(y)}}+R^{\frac{N}{p}}(R^{\epsilon}\|g\|_{\leb{\frac{N}{p
-\epsilon}}{B_{2R}(y)}})^{\frac{1}{p-1}} \right)
\end{equation}
for all $0<\epsilon \leq 1$.
In many cases, especially for unbounded domain, when we wish to show that the
solution decay at infinity, the estimate (\ref{esv-serrin}) requires that the
function $f$ belongs to $\leb{\alpha}{\Omega}$ with $ \alpha >N/p$,
which is not trivial to prove in some cases. To avoid this difficulty Yu
\cite{yu}, Egnell \cite{egnell} and others have proved that the solution of
(\ref{mathcalE}) have a regularity $\leb{q}{\Omega}$ for each $q\geq p^*$, and
this for all function $f$ bounded by a sublinear, superlinear or an homogeneous
terms. We note that in the case of a mixed terms this last technique cannot be
adapted.
For the case of an homogeneous system see the paper of Fleckinger,
Man{\`a}sevich, Stavrakakis and de Th{\'e}lin \cite{sta}.
The first part of this paper is devoted to the local behavior of solutions of
System (\ref{sys1})--(\ref{sys3}). We obtain an estimate of
Serrin type in the following cases:\\
1)\quad $f$ and $g$ are bounded by a sum of homogeneous and critical terms.\\
2)\quad $f$ and $g$ are bounded by a sum of homogeneous and constant terms.\\
Thus, we extend the results of \cite{yu}, \cite{egnell} concerning Equation and
those of \cite{sta} concerning System.
In the second part, we obtain a global estimates of solutions of System
(\ref{sys1})--(\ref{sys3}) in the particular case
$f=A|u|^{\alpha-1}u|v|^{\beta +1}$ and $g=B|u|^{\alpha
+1}|v|^{\beta-1}v$ under some conditions on $ \alpha,\beta , p$
and $q$. Also we obtain another global estimate when $f$ and $g$ satisfy 2).
We recall that $\mathcal{D}^{1,p}(\Omega)$ is the closure of
$\mathcal{C}^{\infty}_{0}(\Omega)$ with respect to the norm
$$
\|u\|_{\mathcal{D}^{1,p}(\Omega) }=\|\nabla u\|_{\leb{p}{\Omega}}.
$$
$ p'=\frac{p}{p-1}$ is the conjugate of $p$, $ p*=\frac{Np}{N-p}$ is the Sobolev
exponent and we define $ S_p$ by
$$
\frac{1}{S_p}=\inf\left\{ \frac{\|\nabla u \|_{\leb{p}{\Omega}}^{p}}{\|
u\|_{\leb{p}{\Omega}}^{p}}\quad u\in W^{1,p}(\Omega)\backslash\{0\}
\right\}.
$$
\section{Local estimates for solutions of (\ref{sys1})--(\ref{sys3})}
\begin{theo}
\label{esv-th}
Let $ (u,v)\in
\mathcal{D}^{1,p}(\mathbb{R}^{N})\times\mathcal{D}^{1,q}(\mathbb{R}^{N})$ be a
solution of $(\ref{sys1}) - (\ref{sys3}) $ and
$\tau=\frac{N}{N-p}$, $\bar{\tau}=\frac{N}{N-q} $. Assume that
$ \max\{ p,q \}0$ and $x\in\mathbb{R}^{N}$ satisfying
\begin{equation}
\label{esv}
\begin{split}
C\max\left\{2^{p}S_p\tau^{p-1},2^{2q-p}S_q|B_{1}|^{\frac{q-p}{N}}R^{q-p}
\tau^{q-1}\right\} &\\
\times\left( \|u\|_{\leb{p^*}{B_{2R}(x)}}^{p(\tau-1)}
+\|v\|_{\leb{q\tau}{B_{2R}(x)}}^{q(\tau-1)}\right)&<1
\end{split}
\end{equation}
where $S_p$ and $S_q$ are the Sobolev constants, we have
\begin{eqnarray*}
\lefteqn{ \|u\|_{\leb{\infty}{B_{\frac{R}{2}}(x)}} }\\
&\leq& c \left(
1+R^{q}\right)^{\frac{N(N-p)}{p^{3}}}\max\left\{
R^{\frac{p-N}{p}}\|u\|_{\leb{p^{*}}{B_R(x)}}, R^{\frac{q-N}{p}}
\|v\|_{\leb{q^{*}}{B_R(x)}}^{q/p} \right\}.
\end{eqnarray*}
and
\begin{eqnarray*}
\lefteqn{ \|v\|_{\leb{\infty}{B_{\frac{R}{2}(x)}}} }\\
&\leq& c\left(
1+R^{q}\right)^{\frac{N(N-p)}{qp^{2}}}
\max\left\{ R^{\frac{q-N}{q}}\|v\|_{\leb{q^{*}}{B_R(x)}},
R^{\frac{p-N}{q}}\|u\|_{\leb{p^{*}}{B_R(x)}}^{\frac{p}{q}} \right\}.
\end{eqnarray*}
witch $c$ independent of $u, v, x$ and $R$.\\
2) Moreover,
$$
\lim_{|x|\rightarrow +\infty}u(x)=\lim_{|x|\rightarrow +\infty}v(x)=0.
$$
\end{theo}
\begin{remark} \rm
\label{esv-rk}
There exists an $R_{0}$ such that for all $R0$ there exists $\eta >0$ such that for all $R>0$ and
$x\in\mathbb{R}^{N}$
satisfying $ |B_R(x)|\leq \eta$, we have $ \int_{B_R(x)}|u|^{p^*}dx<\epsilon$
and $\int_{B_R(x)}|v|^{q\tau} dx<\epsilon $.
\end{remark}
\paragraph{Proof}
Let $x\in\mathbb{R}^{N}$ be fixed. For $y\in B_{2R}(x)$ and any function $h$
defined on $B_{2R}(x)$ we define
$$
\tilde{h}(t)=h(y),\quad t=\frac{y-x}{R}.
$$
Since $(u,v)$ is a solution for (\ref{sys1})--(\ref{sys3}),
then $(\tilde{u},\tilde{v})$ satisfies
\begin{equation}
\label{esv-equ}
-\Delta_p\tilde{u} =R^{p}f(y,\tilde{u},\tilde{v}),
\end{equation}
\begin{equation}
\label{esv-eqv}
-\Delta_q\tilde{v} =R^{q}g(y,\tilde{u},\tilde{v}).
\end{equation}
In this proof $c$ denotes a positive constant independent of
$u, v, x$ and $R$.
For any ball $B\subset B_{2}(0)$, we have
$$
\forall w\in\mathcal{W}^{1,p}_{0}(B)\quad \|w\|_{\leb{p\tau}{B}}^{p}\leq
S_p\|\nabla w\|_{\leb{p}{B}}^{p},
$$
\begin{equation}
\label{esv-sobolev}
\forall w\in\mathcal{W}^{1,q}_{0}(B)\quad \|w\|_{\leb{q\tau}{B}}^{q}
\leq 2^{q-p}|B_{1}(0)|^{\frac{q-p}{N}}S_q\|\nabla w\|_{\leb{q}{B}}^{q}.
\end{equation}
$S_p$ and $S_q$ are the Sobelev constants.
Let $ (m_n)_n$ be a sequence of positive numbers satisfying
$\sigma<\infty$ where $\sigma$ is defined below and $(r_n)_n$ a decreasing
sequence defined by
$$
r_{0}=2,\quad r_n=2-\frac{1}{\sigma}\sum_{i=0}^{n-1}\left(\frac{m_{i}+p}{p}
\right)^{-1/p'},
$$
where $R$ is positive and $\displaystyle \sigma
=\sum_{i=0}^{\infty}\left(\frac{m_{i}+p}{p}\right)^{-1/p'}$.
We denote by $ B_n=B(0,r_n)$ and we define
$\eta\in\mathcal{C}^{\infty}_{0}(\mathbb{R}^{N})$ so that $0\leq\eta
\leq 1$, $\eta =1$ in $B_{n+1}$, $supp(\eta)\subset B_n$ and
\begin{equation}
\label{esv-a}
|\nabla \eta|\leq c\left(\frac{m_n+p}{p}\right)^{1/p'}.
\end{equation}
We multiply (\ref{esv-equ}) by $ |\tilde{u}|^{m_n}\tilde{u}\eta^{q}$,
and integrate over $B_n$. Using (\ref{esv-hu}), we obtain
\begin{equation}
\label{esv-b}
I_{1}+I_{2}\leq R^{p}\left(I_{3}+I_{4}+I_{5}+I_{6}\right),
\end{equation}
where
\begin{equation*}
\begin{split}
&I_{1}=(1+m_n)\int_{B_n}\eta^{q}|\tilde{u}|^{m_n}|\nabla
\tilde{u}|^{p}dx,\\&
I_{2}=q\int_{B_n}\eta^{q-1}\nabla\eta.\nabla \tilde{u}|\nabla
\tilde{u}|^{p-2}|\tilde{u}|^{m_n}\tilde{u} dx,\\&
I_{3}=C\int_{B_n}|\tilde{u}|^{p+m_n}\eta^{q}dx,\\&
I_{4}=C\int_{B_n}|\tilde{u}|^{p^*+m_n}\eta^{q}dx,\\&
I_{5}=C\int_{B_n}|\tilde{u}|^{m_n}\tilde{u}|\tilde{v}|^{q/p'}\eta^{q}dx,\\&
I_{6}=C\int_{B_n}|\tilde{u}|^{m_n}\tilde{u}|\tilde{v}|^{\frac{\tau q}{(\tau
p)'}}\eta^{q}dx.
\end{split}
\end{equation*}
Since $ 1+m_n=\frac{(p-1)m_n}{p}+\frac{m_n+p}{p}$, we deduce from Young
inequality and the facts $p\leq q$, $|\eta|\leq 1$, that for any $s>0$
\begin{eqnarray*}
|I_{2}|&\leq &\frac{qs^{p'}}{p'}\left(\frac{m_n+p}{p}\right)
\int_{B_n}\eta^{q}|\nabla
\tilde{u}|^{p}|\tilde{u}|^{m_n}dx \\
&&+\frac{q}{ps^{p}}\left(\frac{m_n+p}{p}\right)^{-\frac{p}{p'}}
\int_{B_n}|\nabla \eta|^{p}|\tilde{u}|^{m_n+p}dx
\end{eqnarray*}
Choosing $s$ such that $\frac{qs^{p'}}{p'}\leq
\frac{1}{2} $, and using (\ref{esv-a}), we have
\begin{equation}
\label{esv-c}
|I_{2}|\leq \frac{1}{2}I_{1}+c\int_{B_n}|\tilde{u}|^{m_n+p}dx.
\end{equation}
We deduce from (\ref{esv-b}) and (\ref{esv-c})
\begin{equation}
\label{esv-d}
I_{1}\leq 2 R^{p}\sum_{i=3}^{6}I_{i}+c\int_{B_n}|\tilde{u}|^{m_n+p}dx.
\end{equation}
Using Sobolev inequality and observing that for any $a\geq 0$ and $b\geq 0$
$(a+b)^{p}\leq
2^{p-1}(a^{p}+b^{p})$, we have
\begin{equation}
\label{esv-e}
\left\| \eta^{q/p} \tilde{u}^{\frac{m_n+p}{p}}\right\|_{L^{p\tau}(B_n)}^{p}
\leq 2^{p-1}S_p \left(I_{7}+I_{8}\right),
\end{equation}
where
$$
I_{7}=(\frac{q}{p})^{p}\int_{B_n}\eta^{q-p}|\nabla
\eta|^{p}|\tilde{u}|^{m_n+p}dx\leq c\left(\frac{m_n+p}{p}\right)^{p-1}
\int_{B_n}|\tilde{u}|^{m_n+p}dx,
$$
and
$$
I_{8}=\left(\frac{m_n+p}{p}\right)^{p}\int_{B_n}\eta^{q}|\tilde{u}|^{m_n}|\nabla
\tilde{u}|^{p}dx\leq \left(\frac{m_n+p}{p}\right)^{p-1}I_{1},
$$
thus we deduce from (\ref{esv-d}) that
\begin{equation}
\label{esv-f}
\left\| \eta^{q/p} \tilde{u}^{\frac{m_n+p}{p}}\right\|_{L^{p\tau}(B_n)}^{p}
\leq \left(\frac{m_n+p}{p}\right)^{p-1}\left(c \int_{B_n}|\tilde{u}|^{m_n+p}dx
+2^{p}S_pR^{p}\sum_{i=3}^{6}I_{i} \right).
\end{equation}
\textbf{First step.}
We construct the sequences $(p_n)_n$ and $(q_n)_n$ by
$$
p_n=p\tau^{n},\quad q_n=q\tau^{n},
$$
and we set
$$
m_n=p(\tau^{n}-1), and\quad l_n=q(\tau^{n}-1).
$$
We show that if the condition
\begin{eqnarray*}
C\max\left\{ 2^{p}S_pR^{p}\tau^{n(p-1)}, 2^{2q-p}|B_{1}|
^{\frac{q-p}{N}}S_qR^{q}\tau^{n(q-1)}\right\} && \\
\times \left(\|\tilde{u}\|_{\leb{p^*}{B_{2}}}^{p(\tau-1)}+ \|\tilde{v}\|
_{\leb{q\tau}{B_{2}}}^{q(\tau-1)} \right)&<&1,
\end{eqnarray*}
is satisfied, the solution $(\tilde{u},\tilde{v})$ belongs to
$\leb{p_{n+1}}{B_{n+1}}\times\leb{q_{n+1}}{B_{n+1}}$.\\
First, we start by estimating the integrals $ (I_{i}), i=3,\dots ,6$.
We have
\begin{equation}
\label{esv-g}
I_{3}=C\int_{B_n}|\tilde{u}|^{p+m_n}\eta^{q} dx \leq
c\|\tilde{u}\|_{\leb{p_n}{B_n}}^{p_n}.
\end{equation}
Remarking that $\frac{m_n+1}{p_n}+\frac{\frac{q}{p'}}{q_n}=1$,
we deduce from H\"older inequality that
\begin{equation}
\label{esv-h}
I_{5}=C\int_{B_n}|\tilde{u}|^{m_n}\tilde{u}|\tilde{v}|^{q/p'}\eta^{q}dx\leq
c\|\tilde{u}\|_{\leb{p_n}{B_n}}^{m_n+1}\|\tilde{v}\|_{\leb{q_n}{B_n}}^{q/p'}.
\end{equation}
We write $m_n+p^{*}=p(\tau-1)+m_n+p\quad ,
q=\tau q(\frac{m_n+p}{p_{n+1}})$. Observing that $
\frac{m_n+p}{p_{n+1}}+\frac{p(\tau -1)}{p^{*}}=1,
$ we deduce from H\"older inequality
\begin{equation}\label{esv-j}
\begin{split}
I_{4}=&C\int_{B_n}|\tilde{u}|^{p^{*}+m_n}\eta^{q}dx\leq
C\int_{B_n}|\tilde{u}|^{p(\tau-1)}|\tilde{u}|^{p+m_n}
\eta^{\tau q(\frac{m_n+p}{p_{n+1}})} dx\\
\leq &C\|\tilde{u}\|_{\leb{p^{*}}{B_n}}^{p(\tau
-1)}\|\eta^{q/p} \tilde{u}^{\tau^{n}}\|_{\leb{p\tau}{B_n}}^{p}.
\end{split}
\end{equation}
Remark that
\begin{equation}
\label{esv-k1}
\frac{\tau q}{(\tau p)'}-\frac{q}{p'}=q(\tau -1), \quad\frac{q(\tau -1)}{\tau
q}+\frac{m_n+1}{p_{n+1}}+\frac{\frac{q}{p'}}{q_{n+1}}=1,
\end{equation}
and
$$
\tau \frac{m_n+1}{p_{n+1}}+\tau\frac{\frac{q}{p'}}{q_{n+1}}=1,
$$
then from H\"older inequality, we have
\begin{equation}
\label{esv-k}
\begin{split}
I_{6}=& C\int_{B_n}|\tilde{u}|^{m_n}\tilde{u}|\tilde{v}|^{\frac{\tau
q}{(\tau
p)'}}\eta^{q}dx \\
\leq & C\int_{B_n}|\tilde{v}|^{q(\tau-1)}\eta^{\tau
q(\frac{m_n+1}{p_{n+1}})}|\tilde{u}|^{m_n+1}\eta^{\tau
q(\frac{\frac{q}{p'}}{q_{n+1}})}|\tilde{v}|^{q/p'}dx\\
\leq&
C\|\tilde{v}\|_{\leb{\tau q}{B_n}}^{q(\tau -1)}\| \eta^{q/p} \tilde{u}^{\tau
^{n}}
\|_{\leb{p\tau}{B_n}}^{p(\frac{1+m_n}{p_n})} \|\eta
\tilde{v}^{\tau^{n}}\|_{\leb{q\tau}{B_n}}^{q(\frac{\frac{q}{p'}}{q_n})}.
\end{split}
\end{equation}
Substituting $m_n$ by $p(\tau^{n}-1)$ in (\ref{esv-f}), we obtain
\begin{equation}
\begin{split}
\|\eta^{q/p}
\tilde{u}^{\tau^{n}}\|_{\leb{p\tau}{B_n}}^{p}-\tau^{n(p-1)}2^{p}S_pR^{p}
(I_{4}+I_{6}) \\
\leq \tau^{n(p-1)}\left( c\int_{B_n}|\tilde{u}|^{p_n}dx+2^{p}S_pR^{p}
(I_{3}+I_{5})\right).
\end{split}
\end{equation}
It follows from (\ref{esv-g}) - (\ref{esv-k}) and the fact $p\leq q $ that
\begin{equation}
\label{esv-l}
\begin{split}
& \|\eta^{q/p}
\tilde{u}^{\tau^{n}}\|_{\leb{p\tau}{B_n}}^{p}-C2^{p}S_pR^{p}\tau^{n(p-1)}\left(\|\tilde{u}\|_{\leb{p^*}{B_n}}^{p(\tau
-1)}\|\eta^{q/p} \tilde{u}^{\tau^{n}}\|_{\leb{p\tau}{B_n}}^{p}\right.\\
&\left.+ \|\tilde{v}\|_{\leb{\tau q}{B_n}}^{q(\tau
-1)} \|\eta^{q/p} \tilde{u}^{\tau
^{n}}\|_{\leb{p\tau}{B_n}}^{p\frac{(1+m_n)}{p_n}}
\|\eta\tilde{v}^{\tau^{n}}\|_{\leb{q\tau}{B_n}}^{q(\frac{\frac{q}{p'}}{q_n})}\right)\\
&\leq
c(1+R^{q})\tau^{n(q-1)}\left(\|\tilde{u}\|_{\leb{p_n}{B_n}}^{p_n}+\|\tilde{u}\|_{\leb{p_n}{B_n}}^{m_n+1}\|\tilde{v}\|_{\leb{q_n}{B_n}}^{q/p'}\right).
\end{split}
\end{equation}
Similarly, we have
\begin{equation}
\begin{split}
& \|\eta\tilde{v}^{\tau^{n}}\|_{\leb{q\tau}{B_n}}^{q}
-C2^{2q-p}S_q|B_{1}|^{\frac{q-p}{N}}R^{q}\tau^{n(q-1)}
\Big(\|\tilde{v}\|_{\leb{q\tau}{B_n}}^{q(\tau
-1)}\|\eta \tilde{v}^{\tau^{n}}\|_{\leb{q\tau}{B_n}}^{q}\\
&+ \|\tilde{u}\|_{\leb{\tau p}{B_n}}^{p(\tau
-1)} \|\eta \tilde{v}^{\tau ^{n}}\|_{\leb{q\tau}{B_n}}^{q\frac{(1+l_n)}{q_n}}
\|\eta^{q/p} \tilde{u}^{\tau^{n}}\|_{\leb{p\tau}{B_n}}
^{p(\frac{\frac{p}{q'}}{p_n})}\Big)\\
&\leq c(1+R^{q})\tau^{n(q-1)}\|\tilde{v}\|_{\leb{q_n}{B_n}}^{q_n}
+cR^{q}\tau^{n(q-1)}\|\tilde{v}\|_{\leb{q_n}{B_n}}^{l_n+1}
\|\tilde{u}\|_{\leb{p_n}{B_n}}^{p/q'}.
\end{split}
\end{equation}
Next, we define
$ \theta_{n+1}=\max\{ \|\eta^{q/p}
\tilde{u}^{\tau^{n}}\|_{\leb{p\tau}{B_n}}^{p},\|\eta
\tilde{v}^{\tau^{n}}\|_{\leb{q\tau}{B_n}}^{q} \}$,
and \\
$E_n=\max\{\|\tilde{u}\|_{\leb{p_n}{B_n}}^{p_n},
\|\tilde{v}\|_{\leb{q_n}{B_n}}^{q_n}
\}^{1/p_n}$.
Simple computations using H\"older inequality and the definition of $E_n$
and $\theta_n$, show that
\begin{equation}
\label{esv-r}
\begin{split}
\theta_{n+1}-C\max\left\{ 2^{p}S_pR^{p}\tau^{n(p-1)}, 2^{2q-p}|B_{1}|
^{\frac{q-p}{N}}S_qR^{q}\tau^{n(q-1)}\right\} \\
\times \left(\|\tilde{u}\|_{\leb{p^*}{B_n}}^{p(\tau-1)}+
\|\tilde{v}\|_{\leb{q\tau}{B_n}}^{q(\tau-1)} \right)\theta_{n+1}
\leq c(1+R^{q})\tau^{n(q-1)} E_n^{p_n}.
\end{split}
\end{equation}
We know that there exists $R_{0}>0$ such that for any $R1$.
We construct a sequences $(s_n)_n$ and $(t_n)_n$ by
$$
s_n=p\chi^{n},\quad t_n=q\chi^{n}.
$$
In this step $m_n$ and $r_n$ are defined by
$$
m_n=p\left(\frac{\chi^{n}}{\delta}-1 \right),
$$
and
$$
r_{0}=1,\quad r_n=1-\frac{1}{2\sigma}\sum_{i=0}^{n-1}\left(\frac{m_{i}+p}{p}
\right)^{-1/p'},
$$
which implies $m_n+p=s_n/\delta$.
Now, we estimate the integrals $(I_{i})_{i=3,\dots ,6}$.
We have
\begin{equation}
\label{esv-aa}
I_{3}\leq c\|\tilde{u}\|_{\leb{\frac{s_n}{\delta}}{B_n}}^{s_n/\delta}\leq
c\|\tilde{u}\|_{\leb{s_n}{B_n}}^{s_n/\delta}.
\end{equation}
Remarking that $\frac{m_n+1}{s_n/\delta}+\frac{q/p'}{t_n/\delta}=1$,
it follows from H\"older inequality that
\begin{equation}
\label{esv-ab}
I_{5}\leq
c\|\tilde{u}\|_{\leb{\frac{s_n}{\delta}}{B_n}}^{m_n+1}
\|\tilde{v}\|_{\leb{\frac{t_n}{\delta}}{B_n}}^{q/p'}
\leq\|\tilde{u}\|_{\leb{s_n}{B_n}}^{m_n+1}
\|\tilde{v}\|_{\leb{s_n}{B_n}}^{q/p'}.
\end{equation}
We have $\frac{p(\tau -1)}{p\tau^{2}}+\frac{m_n+p}{s_n}=1$, thus
from H\"older inequality we have
\begin{equation}
\label{esv-ad}
I_{4}\leq
c\|\tilde{u}\|_{\leb{p\tau^{2}}{B_n}}^{p(\tau-1)}\|\tilde{u}\|_{\leb{s_n}{B_n}}^{s_n/\delta}
\leq c\|\tilde{u}\|_{\leb{s_n}{B_n}}^{s_n/\delta}.
\end{equation}
Observing that $\frac{q(\tau-1)}{q\tau^{2}}+\frac{m_n+1}{s_n}
+\frac{q/p'}{t_n}=1$, it follows from H{\"o}lder inequality that
\begin{equation}
\label{esv-af}
\begin{split}
I_{6}&\leq
c\int_{B_n}|\tilde{v}|^{q(\tau-1)}|\tilde{u}|^{m_n+1}|\tilde{v}|^{q/p'}dx\\&
\leq c\|\tilde{v}\|_{\leb{q\tau^{2}}{B_n}}^{q(\tau
-1)}\|\tilde{u}\|_{\leb{s_n}{B_n}}^{m_n+1}\|\tilde{v}\|_{\leb{t_n}{B_n}}^{q/p'}\\&
\leq
c\|\tilde{u}\|_{\leb{s_n}{B_n}}^{m_n+1}\|\tilde{v}\|_{\leb{t_n}{B_n}}^{q/p'}.
\end{split}
\end{equation}
We deduce from (\ref{esv-f}), (\ref{esv-aa})--(\ref{esv-af}) and the fact $p\leq
q$ that
\begin{equation}
\label{esv-ag}
\left\| \eta^{q/p} \tilde{u}^{\chi^n/\delta}\right\|_{\leb{p\tau}{B_n}}^{p}\leq
c\chi^{n(q-1)}\left(1+R^{q}
\right)\left(\|\tilde{u}\|_{\leb{s_n}{B_n}}^{s_n/\delta}+\|\tilde{u}\|_{\leb{s_n}{B_n}}^{m_n+1}\|\tilde{v}\|_{\leb{t_n}{B_n}}^{q/p'}
\right)
\end{equation}
Similarly, we have
\begin{equation}
\label{esv-ah}
\begin{split}
\left\| \eta \tilde{v}^{\chi^n/\delta}\right\|_{\leb{q\tau}{B_n}}^{q}&
\leq c\chi^{n(q-1)}\left(1+R^{q}
\right)\left(\|\tilde{v}\|_{\leb{t_n}{B_n}}^{t_n/\delta}
+\|\tilde{v}\|_{\leb{t_n}{B_n}}^{l_n+1}\|\tilde{u}
\|_{\leb{s_n}{B_n}}^{p/q'}
\right)
\end{split}
\end{equation}
As in the first step, we let
$\Lambda_n=\max\left\{\|\tilde{u}\|_{\leb{s_n}{B_n}}^{s_n},
\|\tilde{v}\|_{\leb{t_n}{B_n}}^{t_n} \right\}^{1/s_n}
$\\ $\Gamma_n=\max\left\{ \|\eta^{q/p}
\tilde{u}^{\chi^n/\delta}\|_{\leb{p\tau}{B_n}}^{p}, \|\eta
\tilde{v}^{\chi^n/\delta}\|_{\leb{q\tau}{B_n}}^{q} \right\}$
and\\
$\Upsilon_n=\max\left\{ \|\tilde{u}\|_{\leb{s_n}{B_n}}^{s_n},
\|\tilde{v}\|_{\leb{t_n}{B_n}}^{t_n} \right\}^{\frac{1}{t_n}}$.
Simple computations show that
\begin{equation}
\label{esv-ai}
\|\tilde{u}\|_{\leb{s_n}{B_n}}^{m_n+1}\|\tilde{v}\|_{\leb{t_n}{B_n}}^{q/p'}\leq
\min\left\{\Lambda_n^{s_n/\delta}, \Upsilon_n^{t_n/\delta}\right\},
\end{equation}
and
\begin{equation}
\label{esv-aj}
\|\tilde{v}\|_{\leb{t_n}{B_n}}^{l_n+1}\|\tilde{u}\|_{\leb{s_n}{B_n}}^{p/q'}\leq\min\left\{\Lambda_n^{s_n/\delta},
\Upsilon_n^{t_n/\delta}\right\}.
\end{equation}
Also, remark that
\begin{equation}
\label{esv-ak}
\Gamma_n\geq
\max\left\{\|\tilde{u}\|_{\leb{s_{n+1}}{B_{n+1}}}^{s_n/\delta} ,
\|\tilde{v}\|_{\leb{t_{n+1}}{B_{n+1}}}^{t_n/\delta} \right\}
=\Lambda_{n+1}^{s_n/\delta}= \Upsilon_n^{t_n/\delta}.
\end{equation}
Thus, we deduce from (\ref{esv-ag})--(\ref{esv-ak}) that
$$
\Lambda_{n+1}^{s_n/\delta}\leq c\chi^{n(q-1)}\left(
1+R^{q}\right)\Lambda_n^{s_n/\delta},
$$
and so
$$
\Lambda_{n+1}\leq c^{\delta/s_n}\chi^{\frac{n(q-1)\delta}{s_n}}\left(
1+R^{q}\right)^{\delta/s_n}\Lambda_n.
$$
Which implies that
$$
\|\tilde{u}\|_{\leb{s_n}{B_n}}\leq \Lambda_n\leq
c^{\sum_{i=0}^{\infty}\frac{\delta}{s_{i}}}\chi^{\sum_{i=0}^{\infty}\frac{i(q-1)\delta}{s_{i}}}\left(
1+R^{q}\right)^{\sum_{i=0}^{\infty}\frac{\delta}{s_{i}}}\Lambda_{0}.
$$
Since $\sum_{i=0}^{\infty}\frac{\delta}{s_{i}}=\frac{\delta\tau}{p(\tau
-\delta)},
$ and $\sum_{i=0}^{\infty}\frac{i(q-1)\delta}{s_{i}}<\infty,$
then
\begin{equation*}\begin{split}
\|\tilde{u}\|_{\leb{\infty}{B_{\frac{1}{2}}}}&\leq \lim_{n\rightarrow
+\infty}\sup\|\tilde{u}\|_{\leb{s_n}{B_n}}\\
&\leq c \left(1+R^{q}\right)^{\frac{\delta\tau}{p(\tau
-\delta)}}\max\left\{ \|\tilde{u}\|_{\leb{p}{B_{1}}},
\|\tilde{v}\|_{\leb{q}{B_{1}}}^{q/p} \right\}.
\end{split}\end{equation*}
Similarly, we have
$$
\Upsilon_{n+1}\leq c^{\frac{\delta}{t_n}}\chi^{\frac{n(q-1)\delta}{t_n}}
\left(
1+R^{q}\right)^{\frac{\delta}{t_n}}\Upsilon_n
$$
As $n$ tends to infinity, we obtain
\begin{equation*}\begin{split}
\|\tilde{v}\|_{\leb{\infty}{B_{\frac{1}{2}}}}&\leq \lim_{n\rightarrow
+\infty}\sup\|\tilde{v}\|_{\leb{t_n}{B_n}}\\
&\leq c \left(1+R^{q}\right)^{\frac{\delta\tau}{q(\tau
-\delta)}}\max\left\{ \|\tilde{v}\|_{\leb{p}{B_{1}}},
\|\tilde{u}\|_{\leb{q}{B_{1}}}^{\frac{p}{q}} \right\}.
\end{split}\end{equation*}
By the imbeddings
$$
\leb{p^*}{B_{1}}\subset\leb{p}{B_{1}}\quad\text{and}\quad
\leb{q^*}{B_{1}}\subset\leb{q}{B_{1}},
$$
and the fact
$$
\frac{\delta\tau}{\tau
-\delta}=\frac{\tau}{(\tau-1)^{2}}=\frac{N(N-p)}{p^{2}},$$
we have
$$
\|\tilde{u}\|_{\leb{\infty}{B_{\frac{1}{2}}}}\leq c \left(
1+R^{q}\right)^{\frac{N(N-p)}{p^{3}}}\max\left\{
\|\tilde{u}\|_{\leb{p^{*}}{B_{1}}}, \|\tilde{v}\|_{\leb{q^{*}}{B_{1}}}^{q/p}
\right\},
$$
and
$$
\|\tilde{v}\|_{\leb{\infty}{B_{\frac{1}{2}}}}\leq c \left(
1+R^{q}\right)^{\frac{N(N-p)}{qp^{2}}}\max\left\{
\|\tilde{v}\|_{\leb{p^{*}}{B_{1}}},
\|\tilde{u}\|_{\leb{q^{*}}{B_{1}}}^{\frac{p}{q}} \right\}.
$$
Coming back to $(u,v)$ by a simple change of variables, we find
\begin{eqnarray*}
\lefteqn{ \|u\|_{\leb{\infty}{B_{\frac{R}{2}(x)}}} }\\
&\leq& c \left(
1+R^{q}\right)^{\frac{N(N-p)}{p^{3}}}\max\left\{
R^{\frac{p-N}{p}}\|u\|_{\leb{p^{*}}{B_R(x)}}, R^{\frac{q-N}{p}}
\|v\|_{\leb{q^{*}}{B_R(x)}}^{q/p} \right\}.
\end{eqnarray*}
and
\begin{eqnarray*}
\lefteqn{ \|v\|_{\leb{\infty}{B_{\frac{R}{2}(x)}}} }\\
&\leq& c \left(
1+R^{q}\right)^{\frac{N(N-p)}{qp^{2}}}\max\left( R^{\frac{q-N}{q}}
\|v\|_{\leb{q^{*}}{B_R(x)}},
R^{\frac{p-N}{q}}\|u\|_{\leb{p^{*}}{B_R(x)}}^{\frac{p}{q}} \right\}.
\end{eqnarray*}
The proof of 2) follows from 1) and Remark \ref{esv-rk}
\hfill$\diamondsuit$
\begin{prop}
\label{esv-lm-y}
Let $ (u,v)\in
\mathcal{D}^{1,p}(\mathbb{R}^{N})\times\mathcal{D}^{1,q}(\mathbb{R}^{N})$ a
solution of (\ref{sys1})--(\ref{sys3}). We assume $ q\geq p$,
\begin{equation}
\label{esv-ya}
|f(x,u,v)|\leq C\left
(|u|^{p-1}+|v|^{q/p'}+1\right),
\end{equation}
and
\begin{equation}
|g(x,u,v)|\leq C\left( |v|^{q-1}+|u|^{p/q'}+1\right),
\end{equation}
where $ m'$ is the conjugate of $m$.
Then
\begin{equation}
\label{esv-sous-a}
\|u\|_{\leb{\infty}{B_{1}}}\leq c\left(1+R^{q}
\right)^{\frac{N}{p^2}}\max\left\{
1,R^{\frac{p-N}{p}}\|u\|_{\leb{p^*}{B_{2}}},R^{\frac{q-N}{p}}\|v\|_{\leb{q^*}{B_{2}}}^{q/p}\right\},
\end{equation}
and
\begin{equation}
\label{esv-sous-b}
\|v\|_{\leb{\infty}{B_{1}}}\leq c\left(1+R^{q} \right)^{\frac{N}{pq}}\max\left\{
1,R^{\frac{p-N}{q}}\|u\|_{\leb{p^*}{B_{2}}}^{\frac{p}{q}},R^{\frac{q-N}{q}}\|v\|_{\leb{q^*}{B_{2}}}\right\}.
\end{equation}
\end{prop}
\paragraph{Proof}
We use the same change of variables as in the proof of Theorem \ref{esv-th}.
Thus, we obtain that $(\tilde{u},\tilde{v})$ satisfies (\ref{esv-equ}) and
(\ref{esv-eqv}).
Also we keep the same sequences $(m_n)_n$, $(r_n)_n$,
$(B_n)_n$ and the same function $\eta$.
We multiply Equation (\ref{esv-equ}) by $ |\tilde{u}|^{m_n}\tilde{u}\eta^{q}$,
and integrate over
$B_n$. Using (\ref{esv-ya}), we have
\begin{equation}
\label{esv-yb}
I_{1}+I_{2}\leq R^{p}\left(I_{3}+I_{4}+I_{5}\right),
\end{equation}
where
\begin{equation*}
\begin{split}
&I_{1}=(1+m_n)\int_{B_n}\eta^{q}|\tilde{u}|^{m_n}|\nabla
\tilde{u}|^{p}dx,\\&
I_{2}=q\int_{B_n}\eta^{q-1}\nabla\eta.\nabla \tilde{u}|\nabla
\tilde{u}|^{p-2}|\tilde{u}|^{m_n}\tilde{u} dx,\\&
I_{3}=C\int_{B_n}|\tilde{u}|^{p+m_n}\eta^{q}dx,\\&
I_{4}=C\int_{B_n}|\tilde{u}|^{m_n}\tilde{u}|\tilde{v}|^{q/p'}\eta^{q}dx,\\&
I_{5}=C\int_{B_n}|\tilde{u}|^{m_n}\tilde{u}\eta^{q}dx.
\end{split}
\end{equation*}
The integrals $I_{1}, I_{2}, I_{3}$ and $I_{4}$ are the same to those obtained
in Theorem \ref{esv-th}. Simple computations used before show that
\begin{equation}
\label{esv-yc}
\left\| \eta^{q/p} \tilde{u}^{\frac{m_n+p}{p}}\right\|_{L^{p\tau}(B_n)}^{p}
\leq \left(\frac{m_n+p}{p}\right)^{p-1}\left(c \int_{B_n}|\tilde{u}|^{m_n+p}dx
+2^{p}S_pR^{p}\sum_{i=3}^{5}I_{i} \right).
\end{equation}
Now, we define $(p_n)_n$ and $(q_n)_n$ by
$$
p_n=p\tau^{n},\quad q_n=q\tau^{n},
$$
and let $m_n=p(\tau^{n}-1),\text{and }$, $l_n=q(\tau^{n}-1)$.
Then we estimate the integrals $I_{i}, i=3,\dots ,5$.
It is clear from (\ref{esv-g}) and (\ref{esv-h}) that
\begin{equation}
\label{esv-yd}
I_{3} \leq c\|\tilde{u}\|_{\leb{p_n}{B_n}}^{p_n}\quad\text{and}\quad I_{4}\leq
c\|\tilde{u}\|_{\leb{p_n}{B_n}}^{m_n+1}\|\tilde{v}\|_{\leb{q_n}{B_n}}^{q/p'}.
\end{equation}
On the other hand
\begin{equation}
\label{esv-yf}
\begin{split}
I_{5}&\leq C\int_{B_n}|\tilde{u}|^{m_n+1}dx
=c\|\tilde{u}\|_{\leb{m_n+1}{B_n}}^{m_n+1}
\leq c|B_n|^{(\frac{1}{m_n}
-\frac{1}{p_n})(m_n+1)}\|\tilde{u}\|_{\leb{p_n}{B_n}}^{m_n+1}\\
&\leq c|B_{2}|^{\frac{p-1}{p\tau^{n}}} \|\tilde{u}\|_{\leb{p_n}{B_n}}^{m_n
+1}\\&\leq c\|\tilde{u}\|_{\leb{p_n}{B_n}}^{m_n+1}.
\end{split}
\end{equation}
We deduce from (\ref{esv-yc})--(\ref{esv-yf}) that
\begin{equation}
\label{esv-yg}
\begin{split}
\|\tilde{u}\|_{\leb{p_{n+1}}{B_{n+1}}}^{p_n}
&\leq \|\eta^{q/p}
\tilde{u}^{\tau^{n}}\|_{\leb{p\tau}{B_n}}^{p}\\
&\leq c\tau^{n(p-1)}\Big(
\|\tilde{u}\|_{\leb{p_n}{B_n}}^{p_n}\\
&+R^{p} \left( \|\tilde{u}\|_{\leb{p_n}{B_n}}^{p_n}+\|\tilde{u}
\|_{\leb{p_n}{B_n}}^{m_n+1}\|\tilde{v}\|_{\leb{q_n}{B_n}}^{q/p'}
+\|\tilde{u}\|_{\leb{p_n}{B_n}}^{m_n+1} \right) \Big).
\end{split}
\end{equation}
Similarly, we have
\begin{equation}
\label{esv-yh}
\begin{split}
\|\tilde{v}\|_{\leb{q_{n+1}}{B_{n+1}}}^{q_n}
&\leq \|\eta
\tilde{v}^{\tau^{n}}\|_{\leb{q\tau}{B_n}}^{q}\\
&\leq c\tau^{n(q-1)}\Big(
\|\tilde{v}\|_{\leb{q_n}{B_n}}^{q_n} \\
&+R^{q}\left( \|\tilde{v}\|_{\leb{q_n}{B_n}}^{q_n}+\|\tilde{v}
\|_{\leb{q_n}{B_n}}^{l_n+1}\|\tilde{u}\|_{\leb{p_n}{B_n}}^{p/q'}
+\|\tilde{v}\|_{\leb{q_n}{B_n}}^{l_n+1} \right) \Big).
\end{split}
\end{equation}
Following the proof of Theorem \ref{esv-th} we let \\
$E_n=\max\left\{
1,\|\tilde{u}_n\|_{\leb{p_n}{B_n}}^{p_n},\|\tilde{v}_n\|_{\leb{q_n}{B_n}}^{q_n}
\right\}^{1/p_n}$ and \\
$F_n=\left\{ 1,\|\tilde{u}_n\|_{\leb{p_n}{B_n}}^{p_n},
\|\tilde{v}_n\|_{\leb{q_n}{B_n}}^{q_n} \right\}^{\frac{1}{q_n}}$.
We obtain
\begin{equation}
\label{esv-sous-u}
\begin{split}
\|\tilde{u}\|_{\leb{\infty}{B_{1}}}&\leq \lim_{n\rightarrow
+\infty}\sup\|\tilde{u}\|_{\leb{p_n}{B_n}}\leq E_n\\&
\leq c\left(1+R^{q} \right)^{\frac{N}{p^2}}E_{0}\\&
=c\left(1+R^{q} \right)^{\frac{N}{p^2}}\max\left\{
1,\|\tilde{u}\|_{\leb{p}{B_{2}}},\|\tilde{v}\|_{\leb{q}{B_{2}}}^{q/p}\right\}.
\end{split}
\end{equation}
\begin{equation}
\label{esv-sous-v}
\begin{split}
\|\tilde{v}\|_{\leb{\infty}{B_{1}}}&\leq
\lim_{n\rightarrow+\infty}\sup\|\tilde{v}\|_{\leb{q_n}{B_n}}
\leq F_n\\&
\leq c\left(1+R^{q} \right)^{\frac{N}{pq}}F_{0}\\&
=c\left(1+R^{q} \right)^{\frac{N}{pq}}\max\left\{
1,\|\tilde{u}\|_{\leb{p}{B_{2}}}^{\frac{p}{q}},\|\tilde{v}\|_{\leb{q}{B_{2}}}\right\}.
\end{split}
\end{equation}
Using a simple change of variables in (\ref{esv-sous-u}) and
(\ref{esv-sous-v}) we obtain (\ref{esv-sous-a}) and (\ref{esv-sous-b}).
\section{Global estimates for solutions of (\ref{sys1})--(\ref{sys3})}
\begin{prop}
Let $(u,v)\in\mathcal{D} ^{1,p}(\Omega )\times\mathcal{D}^{1,q}(\Omega) $ a
solution of (\ref{sys1})--(\ref{sys3}).
We assume that there exist a functions
$a, b\in L^{1}(\Omega)\cap L^{\infty}(\Omega)$ and a constant $C$ such that
\begin{equation}
\left |f(x,u,v)\right |\leq a(x)+C(|u|^{p-1}+|v|^{q/p'}),
\end{equation}
\begin{equation}
\left |g(x,u,v)\right |\leq b(x)+C(|v|^{q-1}+|v|^{p/q'}),
\end{equation}
where $p>1$, $q>1$.
Then\\
1) $(u,v)\in L^{\sigma}(\Omega)\times L^{\eta}(\Omega)$ for all
$(\sigma ,\eta)\in [p^*,+\infty)\times [q^*,+\infty)$.\\
2) $\displaystyle\lim_{|x|\rightarrow +\infty}u(x)=\lim_{|x|\rightarrow +\infty}v(x)=0$.
\end{prop}
\paragraph{Proof}
1) Let $p_n=p\tau^{n}$, $q_n=q\tau^{n}$, $m_n=\tau^{n}-1$,
$t_n=\tau^{n}-1$, $T_{k}(u) = \max\{-k,\min\{k,u\} \}$ and
$w=|T_{k}(u)|^{pm_n}T_{k}(u)$, with $k>0$. Multiplying the equation
$(\ref{sys1})$ by $w$ and integrating over $\Omega$, we obtain
\[
\begin{split}
(pm_n+1)\int _{\Omega }| \nabla T_k(u)|^p |T_k(u)|^{pm_n} dx=
\int_{\Omega}f(x,u,v)w\; dx.
\end{split}
\]
Observing that
\begin{equation}
\label{esl-sobo}
\begin{split}
(\frac{1}{m_n +1})^p |\nabla(T_k (u)) ^{m_n +1}|^p =T_k(u)^{pm_n}
|\nabla T_{k}(u)|^p,
\end{split}
\end{equation}
we deduce from H\"older and Sobolev inequalities that for any $0<\gamma
<1$,
we have
\begin{equation}
\label{esl-e} \begin{split}
&\int_{\Omega } |T_k(u)| ^{\tau (pm_n
+p)})^{1/\tau} \\
&\leq c\left(\|a\|_{\infty}^{1-\gamma}\|a\|^{\gamma}_{L^{1}(\Omega)}
\|u\|_{L^{p_n}(\Omega)}^{pm_n+1} +\|u\|_{L^{p_n}(\Omega)}^{p_n}
+\|v\|_{L^{q_n}(\Omega)}^{q/p'}\|u\|_{L^{p_n}(\Omega)}^{m_n+1}\right).
\end{split} \end{equation}
with $c$ depending from $n$.
Letting $k$ tend to infinity in (\ref{esl-e}), we obtain
\begin{equation}
\label{esl-f}
\|u\|_{L^{p_{n+1}}(\Omega)}^{p_n}\leq
c\left(
\|u\|_{L^{p_n}(\Omega)}^{pm_n+1}+\|u\|_{L^{p_n}(\Omega)}^{p_n}+\|v\|_{L^{q_n}(\Omega)}^{q/p'}\|u\|_{L^{p_n}(\Omega)}^{m_n+1}\right).
\end{equation}
We derive from (\ref{esl-f}) that
$u\in L^{p_n}(\Omega)$ for all $n\in\mathbb{N}$.
Similarly, we prove that $v\in L^{q_n}(\Omega)$ for all $n\in\mathbb{N}$.
By interpolation inequality (see \cite{gilbarg})
we prove that $(u,v)\in L^{\sigma}(\Omega)\times L^{\eta}(\Omega)$, for all
$(\sigma,\eta)\in [p^*,+\infty )\times [q^*,+\infty )$.
The proof of 2) follows from Serrin inequality \cite{serrin2} and 1).
\hfill$\diamondsuit$\smallskip
Next, we study the sub-homogeneous system
\begin{gather}
\label{cas-a}
-\Delta_pu=B(x)|u|^{\alpha -1}u|v|^{\beta +1}, \\
\label{cas-b}
-\Delta_qv=C(x)|u|^{\alpha +1}|v|^{\beta -1}v,
\end{gather}
in $\Omega $ an exterior domain or $\mathbb{R}^{N}$.
\begin{prop}
Assume that $ B,C\in\leb{\infty}{\Omega}$ and
\begin{equation*}
\frac{\alpha +1}{p^*}+\frac{\beta +1}{q^*}<1,\quad
p>1, \quad q>1\,.
\end{equation*}
Then each solution $ (u,v)\in\mathcal{D} ^{1,p}(\Omega
)\times\mathcal{D}^{1,q}(\Omega) $ of the system (\ref{cas-a}), (\ref{cas-b})
satisfies \\
\textbf{1.}\quad $(u,v)\in L^{\sigma }(\Omega )
\times L^{\eta }(\Omega )$ for all $(\sigma,\eta )
\in [p^* ,+\infty [\times [q^* ,+\infty [ .$
\\
\textbf{2.}\quad $\lim_{|x|\rightarrow +\infty}u(x)=0$ and
$\lim_{|x|\rightarrow +\infty}v(x)=0 $.
\end{prop}
\paragraph{Proof}
Let $ \tau =\frac{N}{N-p}, \bar{\tau}=\frac{N}{N-q}$ and
$ L=1-\frac{\alpha +1}{p^{*}} -\frac{\beta +1}{q^{*}}$.
Assume $ q\geq p$, which implies that $ \bar{\tau} \geq \tau$.
We define the sequences $ (p_n)_n$, $ (q_n)_n$ and $(f_n)_n$ by
\begin{equation*}
\begin{split}
&f_{n+1}=\tau (f_n+L-1)+1,\quad f_{0}=1,\\&
p_n=p^{*}f_n,\quad q_n=q^*f_n.
\end{split}
\end{equation*}
Let $\displaystyle T_{k}(u) =max\{ -k,\min \{k,u \} \} $ for $k>0$ and
$\displaystyle \omega =|T_k (u)|^{pm_n} T_k(u) $, with
\begin{equation}
m_n=(1-\frac{\alpha +1}{p_n} -\frac{\beta +1}{q_n})\frac{p_n}{p}=f_{n+1}-1
\end{equation}
Multiplying (\ref{cas-a}) by $ \omega $ and integrating over $\Omega$,
we obtain from (\ref{esl-sobo}) and Sobolev inequality
\[
\begin{split}
\frac{1}{S_p} (pm_n +1) (\frac{1}{m_n +1})^p (\int_{\Omega } |T_k(u)| ^{\tau
(pm_n
+p)})^{1/\tau} \leq \|B\|_{L^{\infty}(\Omega )} \int_{\Omega
}|u|^{\alpha }|v|^{\beta +1}\omega
dx.
\end{split}
\]
From the definition of $m_n$ and H\"older inequality, we deduce that
\[
\begin{split}
(\int_{\Omega } |T_k(u)| ^{p^*(m_n
+1)})^{1/\tau} \leq S_p\frac{ (m_n +1)^p}{(pm_n
+1)}\|B\|_{L^{\infty} (\Omega )} \|u\|_{L^{p _n}(\Omega)}^{\alpha +1+pm_n
}\|v\|_{L^{q_n}(\Omega )}^{\beta +1}.
\end{split}
\]
Let $k$ tends to infinity, we have
\[
\begin{split}
(\int_{\Omega } |u| ^{p^* (m_n
+1)})^{1/\tau} \leq S_p\frac{ (m_n +1)^p}{(pm_n
+1)}\|B\|_{L^{\infty} (\Omega )} \|u\|_{L^{p_n}(\Omega)}^{\alpha +1+pm_n
}\|v\|_{L^{q_n}(\Omega )}^{\beta +1}.
\end{split}
\]
$ p^{*}(m_n +1) =p^{*}(f_{n+1} )=p_{n+1} $, therefore $ u\in
L^{p_{n+1}}(\Omega )$.
To show that $ v\in L^{q_{n+1}}(\Omega )$, We consider $\bar{w
} =|T_{k}(v)|^{qt_n}T_{k}(v)$, with
\begin{equation}
\begin{split}
t_n&=(1-\frac{\alpha+1}{p_n}-\frac{\beta
+1}{q_n})\frac{q_n}{q}\\&=\bar{\tau}(f_n+L-1).
\end{split}
\end{equation}
Proceeding as above, we obtain
\[
\begin{split}
(\int_{\Omega } |v| ^{q^* (t_n
+1)})^{\frac{1}{\bar{\tau} }} \leq S_q\frac{ (t_n +1)^q}{(qt_n
+1)}\|C\|_{L^{\infty} (\Omega )} \|u\|_{L^{p_n}(\Omega)}^{\alpha
+1}\|v\|_{L^{q_n}(\Omega )}^{\beta +1+qt_n}.
\end{split}
\]
Let $ \bar{q}_n=q^{*}(t_n +1)$. It is clear that $ v\in
L^{\bar{q}_n}$, and since
\[
\begin{split}
\bar{q}_n&=q^{*}(t_n +1) \\&=q^{*}(\bar{\tau} (f_n+L-1)+1)\\& \geq
q_{n+1},
\end{split}
\]
then $q_n \leq q_{n+1}\leq \bar{q}_n$.
By interpolation inequality (see \cite{gilbarg}), we deduce that $v\in
L^{q_{n+1}}(\Omega )$.
2) follows from Serrin inequality \cite{serrin2} and 1).
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\noindent{\sc Abdelhak Bechah} \\
Laoratoire MIP\\
UFR MIG, Universit\'e Paul Sabatier Toulouse 3\\
118 route de Narbonne, 31 062\\
Toulouse cedex 4, France\\
e-mail: bechah@mip.ups-tlse.fr or abechah@free.fr\\
web page: http://mip.ups-tlse.fr/$\sim$bechah
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