\documentclass[reqno]{amsart}
\usepackage{hyperref}

\AtBeginDocument{{\noindent\small
\emph{Electronic Journal of Differential Equations},
Vol. 2010(2010), No. 60, pp. 1--13.\newline
ISSN: 1072-6691. URL: http://ejde.math.txstate.edu or http://ejde.math.unt.edu
\newline ftp ejde.math.txstate.edu}
\thanks{\copyright 2010 Texas State University - San Marcos.}
\vspace{9mm}}

\begin{document}
\title[\hfilneg EJDE-2010/60\hfil Maximum principle and existence results]
{Maximum principle and existence results for elliptic systems on
$\mathbb{R}^N$}

\author[L. Leadi, A. Marcos\hfil EJDE-2010/60\hfilneg]
{Liamidi Leadi, Aboubacar Marcos}  % in alphabetical order

\address{Liamidi Leadi \newline
Institut de Math\'ematiques et de Sciences Physiques\\
Universit\'e d'Abomey Calavi\\
01 BP: 613 Porto-Novo, B\'enin (West Africa)}
\email{leadiare@imsp-uac.org, leadiare@yahoo.com}

\address{Aboubacar Marcos \newline
Institut de Math\'ematiques et de Sciences Physiques\\
Universit\'e d'Abomey Calavi\\
01 BP: 613 Porto-Novo, B\'enin (West Africa)}
\email{abmarcos@imsp-uac.org}

\thanks{Submitted June 5, 2009. Published May 5, 2010.}
\subjclass[2000]{35B50, 35J20, 35J55}
\keywords{Principal and nonprincipal eigenvalues; elliptic systems;
\hfill\break\indent  p-Laplacian operator; approximation method}

\begin{abstract}
 In this work we give necessary and sufficient conditions for
 having a maximum principle for cooperative elliptic systems
 involving $p$-Laplacian operator on the whole $\mathbb{R}^{N}$.
 This principle is then used to yield solvability  for the
 cooperative elliptic systems by an approximation method.
\end{abstract}

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

\section{Introduction}

This work is mainly concerned with the elliptic system
\begin{equation}\label{a1}
\begin{gathered}
 -\Delta_pu=am(x)|u|^{p-2}u+bm_1(x)|v|^\beta v + f\quad\text{in }
 \mathbb{R}^N,\\
-\Delta_qv=cn_1(x)|u|^\alpha u+dn(x)|v|^{q-2} v + g\quad\text{in }
 \mathbb{R}^N ,\\
u(x)\to 0, v(x)\to 0 \quad\text{as } |x|\to +\infty.
\end{gathered}
\end{equation}
Here $\Delta_p u:= \mathop{\rm div}(|\nabla u|^{p-2}\nabla u)$,
$1<p<+\infty$, is the so-called $p$-Laplacian operator;
$a,b,c,d,\alpha$ and $\beta$ are reals parameters; $f,g,m, n, m_1$
and $n_1$ are weights whose properties will be specified later.

We are concerned with the existence of positive solutions and with
the following form of maximum principle: If $f,g\geq 0$
in $\mathbb{R}^N$ then $u,v\geq 0$ in $\mathbb{R}^N$ for any
solution $(u,v)$ of \eqref{a1}. It is well known that maximum
principle plays an important role in the theory on nonlinear equations.
For instance, it  is used to access existence results and qualitative
properties of solutions for linear and nonlinear differential equations,
(see for instance \cite{gil} and \cite{pro} for a survey).

Many works have been devoted to the study of linear and nonlinear
elliptic systems either on a bounded domain or an unbounded domain
of $\mathbb{R}^N$ (in particular the whole $\mathbb{R}^{n}$)
(cf.\cite{bouch,fig1,fig2,fig3,fleck,fleck1,sera}). In
\cite{fleck4, fleck5} for the linear case (i.e $p=q=2$), it was
presented necessary and sufficient conditions for having maximum
principle and existence of positive solutions. These results have
been later extended in \cite{fleck1} to the nonlinear system
\begin{gather*}
 {-\Delta_p u_i=\sum_{j=1}^na_{ij}|u_j|^{p-2}u_j + f_i\quad
\text{in }\Omega}\\
u_i=0\quad \text{on }\partial\Omega,\; i=1,2,\dots n
\end{gather*}
where $\Omega$ is a bounded domain of $\mathbb{R}^N$.

For specific interest for our purposes is the work in \cite{sera}
where a study of problems such as \eqref{a1} was carried out in
the case of $\mathbb{R}^N$ in the presence of some weight
functions. In our work we consider problem \eqref{a1} with
coefficients $b,c>0$, and the  weight functions $m(x)$, $n(x)$,
$m_{1}(x)$, $n_{1}(x)$ positive. Here $m$ belongs to
$L^{N/p}(\mathbb{R}^N)\cap L_{\rm loc}^{\infty}(\mathbb{R}^N)$ and $n$
belongs to $L^{N/q}(\mathbb{R}^N)\cap
L_{\rm loc}^{\infty}(\mathbb{R}^N)$. Then we state necessary and
sufficient conditions for  a maximum principle
to hold. Moreover our technique can be developed to get
a related result for the following class of cooperative  systems
\begin{equation}\label{a1'}
   \begin{gathered}
  -\Delta_pu = a m(x)|u|^{p-2}u + b m_{1}(x)|u|^\alpha|v|^\beta v + f
\quad\text{in } \mathbb{R}^N,\\
   -\Delta_qv = cn_1(x)|v|^\beta |u|^\alpha u + dn(x)|v|^{q-2}v + g
\quad\text{in } \mathbb{R}^N, \\
  u(x)\to 0, \quad v(x)\to 0 \quad \text{as }  |x|\to +\infty
\end{gathered}
\end{equation}
where the coefficients $a$, $b$, $c$, $d$, and the  weights
$m(x),n(x),m_{1}(x),n_{1}(x)$ are as above.
When $a= b = c = d = 1$, problem \eqref{a1'} is relaxed to
the particular case of system
considered in \cite{sera} where the necessary condition for the
maximum principle to hold given by the authors is depend on $x$.
The arguments developed in this paper enable us to obtain a non dependance on $x$ necessary condition.

 The remainder of the paper is organized as follows: In Section 3,
the maximum principle for $\eqref{a1}$ is given and is shown to be
proven full enough to yield existence results of solutions for
$\eqref{a1}$ in Section 4. In section 5, we briefly give a
version of our result for the  cooperative systems \eqref{a1'}. In
the preliminary Section 2, we collect some known results relative
to the principal positive eigenvalue and to various Sobolev
imbeddings.

\section{Preliminaries}

Throughout this work, we will assume that $1<p,q<N$ and
\begin{itemize}
\item[(H1)] $m,n> 0; m\in L_{\rm loc}^\infty(\mathbb{R}^N)\cap L^{N/p}
(\mathbb{R}^N)$ and
$n\in L_{\rm loc}^\infty(\mathbb{R}^N)\cap L^{N/q}(\mathbb{R}^N)$

\item[(H2)] $ 0<m_1(x)\leq [m(x)]^{\frac{1}{p}}[n(x)]^{\frac{\beta +1}{q}}$
and $0<n_1(x)\leq [n(x)]^{\frac{1}{q}}[m(x)]^{\frac{\alpha +1}{p}}$
a.e. in $\mathbb{R}^N$

\item[(H3)] $f\geq 0$ and $f\in L^{(p^*)'}(\mathbb{R}^N)$;
$g\geq 0$ and $g\in L^{(q^*)'}(\mathbb{R}^N)$

\item[(H4)] $b,c\geq 0$; $\alpha,\beta\geq 0$;
$\frac{\alpha +1}{p}+\frac{1}{q}=1$ and
$\frac{\beta+1}{q}+\frac{1}{p}=1$

\end{itemize}
Here $p^*=\frac{Np}{N-p}$, $q^*=\frac{Nq}{N-q}$ denote the
critical Sobolev exponent of $p$ and $q$ respectively; $p'$ is
the H\"{o}lder conjugate of $p$. It is clear that
$\frac{1}{p'}=\frac{\beta +1}{q}$ and $\frac{1}{q'}=\frac{\alpha +1}{p}$.

We denote by $D^{1,s}(\mathbb{R}^N)$ (with $1<s<N$) the completion of
$C_0^\infty (\mathbb{R}^N)$ with respect to the norm
$$
\|u\|_{D^{1,s}(\mathbb{R}^N)}
=\Big(\int_{\mathbb{R}^N} |\nabla u|^s\Big)^{1/s}.
$$
It can be shown that (cf \cite{koz})
$$
D^{1,s}(\mathbb{R}^N)=\{u\in L^{s^*}(\mathbb{R}^N):
\nabla u\in (L^s(\mathbb{R}^N))^N\}
$$
and for any positive weight
$g\in L_{\rm loc}^\infty(\mathbb{R}^N)\cap L^{N/s}(\mathbb{R}^N)$
the following embeddings hold (cf.\cite{fleck2,fleck3,goss})
$$
D^{1,s}(\mathbb{R}^N)\hookrightarrow L^{s^*(\mathbb{R}^N)}
\quad\text{and}\quad
D^{1,s}(\mathbb{R}^N)\hookrightarrow\hookrightarrow L^s(g,\mathbb{R}^N)
$$
where $L^s(g,\mathbb{R}^N)$ is the $L^s$ space on $\mathbb{R}^N$
with the weight $g$ (cf. \cite{fleck3}).

By solution $(u,v)$ of \eqref{a1} (or related equations),
we mean a weak solution; i.e.,
$(u,v)\in D^{1,p}(\mathbb{R}^N)\times D^{1,q}(\mathbb{R}^N)$ with
\begin{equation}\label{a2}
\begin{gathered}
 {\int_{\mathbb{R}^N}|\nabla u|^{p-2}\nabla u.\nabla w
=\int_{\mathbb{R}^N} [am(x)|u|^{p-2}uw+bm_1(x)|v|^\beta vw
+fw]}
\\
{\int_{\mathbb{R}^N}|\nabla v|^{q-2}\nabla v.\nabla
z=\int_{\mathbb{R}^N} [cn_1(x)|u|^\alpha uz+dn(x)|v|^{q-2}
vz+ gz]}
\end{gathered}
\end{equation}
for all $(w,z)\in D^{1,p}(\mathbb{R}^N)\times D^{1,q}(\mathbb{R}^N)$.
Note that by the above embeddings, every integral in \eqref{a2}
is well-defined. Regularity results from \cite{ser,tolk} on
general quasilinear equations imply that such a weak solution $(u,v)$
belong to $C^1(\mathbb{R}^N)\times C^1(\mathbb{R}^N)$.
It is also known that a weak solution of \eqref{a1} decays to zero
at infinity (cf. \cite{dra, fleck2}).

To conclude this introduction, let us briefly recall some properties
 of the spectrum of $-\Delta_p$ with weight to be used later
(cf. \cite{all,fleck3}). We denote by
\begin{equation}\label{a3'}
\lambda_1(m,p):=\min\big\{\int_{\mathbb{R}^N}|\nabla u|^p:
u\in D^{1,p}(\mathbb{R}^N)\text{ and }\int_{\mathbb{R}^N}m|u|^p=1\big\}
\end{equation}
the unique principal eigenvalue of
\begin{equation}\label{a3}
\begin{gathered}
 -\Delta_pu=\lambda m(x)|u|^{p-2}u\quad \text{in }\mathbb{R}^N\\
u(x)\to 0\quad \text{as } |x|\to +\infty;\; u>0\text{ in }\mathbb{R}^N
\end{gathered}
\end{equation}
and by
$\varphi_1(m)=\varphi_1(m,p)\in D^{1,p}(\mathbb{R}^N)
\cap C^1(\mathbb{R}^N)$ the associated positive eigenvalue such that
$\int_{\mathbb{R}^N} m|\varphi_1(m)|^p=1$.
It is well known that $\lambda_1(m,p)$ is simple and isolated.

Here and henceforth, we will denote by $\Phi=\varphi_1(m,p)$
(respectively by $\Psi=\varphi_1 (n,q)$) the positive eigenfunction
associated to $\lambda_1(m,p)$ (respectively $\lambda_1(n,q)$)
and normalized by
\begin{equation}
\int_{\mathbb{R}^N}m\Phi(x)^p=\int_{\mathbb{R}^N}n\Psi(x)^q=1
\end{equation}


\section{Maximum principle}

We assume that $1<p$, $q<N$ and that hypothesis (H1), (H2), (H3) and (H4)
 are satisfied.
We begin by consider the  problem
\begin{equation}\label{a4}
 \begin{gathered}
 -\Delta_pu=\mu m(x)|u|^{p-2}u+ h(x)\quad \text{in }\mathbb{R}^N\\
u(x)\to 0\quad \text{as } |x|\to +\infty
\end{gathered}
\end{equation}
The following results were proved in \cite{fleck2, fleck3}

\begin{proposition} \label{prop1}
(1) Let $h\in L^{(p^*)'}(\mathbb{R}^N)$ and assume that {\rm (H1)}
 is satisfied. If $\mu <\lambda_1(m,p)$ then \eqref{a4} admits
a solution in $D^{1,p}(\mathbb{R}^N)$.

(2) Let $h\in L^{(p^*)'}(\mathbb{R}^N)$ with $h\geq 0$ a.e.
in $\mathbb{R}^N$ and $h\not\equiv 0$.
\begin{itemize}
\item[(a)]  If $\mu\in[0,\lambda_1(m,p)[$, then any solution $u$
of \eqref{a4} is positive in $\mathbb{R}^N$.
\item[(b)]  If $\mu=\lambda_1(m,p)$ then
 \eqref{a4} has no solution
\item[(c)]  If $\mu>\lambda_1(m,p)$ then \eqref{a4} has no
 positive solution.
\end{itemize}
\end{proposition}

Using \cite{ser,tolk}, one also has  a regularity result.

\begin{proposition} \label{prop2}
 For all $r>0$, any solution $(u,v)$ of \eqref{a1}) belongs to
$C^{1,\gamma}(B_r)\times C^{1,\gamma}(B_r)$, where
 $\gamma=\gamma(r)\in]0,1[$ and $B_r$ is the ball of radius
 $r$ centered at the origin.
\end{proposition}


Let
\begin{equation}
 a_1(r):=\inf_{B_r}k_1(x),\quad
a_2(r):=\sup_{B_r}k_2(x),
\end{equation}
where
\begin{gather*}
 k_1(x):=\big[\frac{n_1(x)}{n(x)}\big]^{\frac{\beta +1}{q}}
\big[\frac{\Phi(x)^p}{\Psi(x)^q}\big]^{\frac{\alpha +1}{p}\,
\frac{\beta +1}{q}},
\\
 k_2(x):=\big[\frac{m(x)}{m_1(x)}\big]^{\frac{\alpha +1}{p}}
\big[\frac{\Phi(x)^p}{\Psi(x)^q}\big]^{\frac{\alpha +1}{p}\,
\frac{\beta +1}{q}}.
\end{gather*}
We denote ${a_{1\infty}=\lim_{r\to +\infty}a_1(r)}$ and
${a_{2\infty}=\lim_{r\to +\infty}a_2(r)}$. Let
\begin{equation}
 \Theta=\frac{a_{1\infty}}{a_{2\infty}}.
\end{equation}
One can easily prove that
\begin{equation}
\Theta\leq \frac{a_1(r)}{a_2(r)}\quad \text{for all } r>0\quad
\text{and}\quad 0\leq \Theta\leq 1
\end{equation}
We say that \eqref{a1} satisfies the maximum principle
(in short (MP)) if for $f,g\geq 0$ a.e in $\mathbb{R}^N$, any
solution $(u,v)$ of \eqref{a1} is such that $u>0,v>0$ a.e.
 in $\mathbb{R}^N$.

We now turn to our first main result, i.e., the validity of the
(MP) which is stated as follows

\begin{theorem}\label{theo1}
Assume that hypothesis {\rm (H1)--(H)} are satisfied.
Then the (MP) holds for \eqref{a1} if
\begin{itemize}
\item[(C1)]  $\lambda_1(m,p)>a$
\item[(C2)] $\lambda_1(n,q)>d$
\item[(C3)] $[\lambda_1(m,p)-a]^{\frac{\alpha +1}{p}}
[\lambda_1(n,q)-d]^{\frac{\beta +1}{q}}
 >b^{\frac{\alpha +1}{p}}c^{\frac{\beta +1}{q}}$
\end{itemize}
Conversely, if the (MP) holds, then
(C1), (C2) and (C4) are satisfied, where
\begin{itemize}
\item[(C4)] $[\lambda_1(m,p)-a]^{\frac{\alpha +1}{p}}
[\lambda_1(n,q)-d]^{\frac{\beta +1}{q}}
>\Theta b^{\frac{\alpha +1}{p}}c^{\frac{\beta +1}{q}}$.
\end{itemize}
\end{theorem}

\begin{corollary} \label{cor4}
If $p=q$ and $m\equiv n$ a.e. in $\mathbb{R}^N$, then the (MP)
holds for \eqref{a1} if only if (C1), (C2) and (C4) are satisfied
\end{corollary}


\begin{proof}[Proof of Theorem \ref{theo1}]
The condition is necessary.
The proof of (C1) or (C2) is standard
(cf. for instance \cite{boc,bouch,sera}).
We give here the sketch of this proof.

If $\lambda_1(m,p)\leq a$, then the functions
$f:=[a-\lambda_1(m,p)]m\Phi^{p-1}$ and $g:=cn_1\Phi^{\alpha +1}$
are nonnegative and $(-\Phi,0)$ is a solution of \eqref{a1},
which contradicts the (MP).

Similarly, if $\lambda_1(n,q)\leq d$, then the functions
$f:=bm_1\Psi^{\beta +1}$ and $g:=[d-\lambda_1(n,q)]n\Psi^{q-1}$
are nonnegative and $(0,-\Psi)$ is a solution of \eqref{a1},
a contradiction.

 The proof of (C4) can be  adapted from \cite{sera} as follow.
We assume that $\lambda_1(m,p)>a$ and $\lambda_1(n,q)>d$.
If one of the coefficients $\Theta, b$ or $c$ vanishes,
then (C4) is satisfied. We will then assume that $\Theta\neq  0$,
$b\neq  0$, $c\neq 0$ and that (C4) does not hold, i.e.
\begin{equation}\label{a5}
[\lambda_1(m,p)-a]^{\frac{\alpha +1}{p}}
[\lambda_1(n,q)-d]^{\frac{\beta +1}{q}}
\leq\Theta b^{\frac{\alpha +1}{p}}c^{\frac{\beta +1}{q}}
\end{equation}
Set $A=\big(\frac{\lambda_1(m,p)-a}{b}\big)^{\frac{\alpha +1}{p}}$
and $B=\big(\frac{\lambda_1(n,q)-d}{c}\big)^{\frac{\beta +1}{q}}$.
Then, by \eqref{a5}, one has $AB\leq\Theta$, which clear implies
that $Aa_{2\infty}\leq \frac{1}{B}a_{1\infty}$. One deduces that
there exists $\xi\in\mathbb{R}_+^*$ such that
$$
Aa_{2\infty}\leq \xi\leq\frac{1}{B}a_{1\infty}.
$$
Since the function $a_1(r)$ (respectively $a_2(r)$) is
decreasing (respectively increasing) on $\mathbb{R}_+^*$, one has
$$
Aa_2(r)\leq Aa_{2\infty}\leq\xi
\leq \frac{1}{B}a_{1\infty}\leq \frac{1}{B}a_1(r),\quad
\text{for all }r>0.
$$
But for any $x\in\mathbb{R}^N$, there exists $r>0$ such that
$$
Ak_2(x)\leq Aa_2(r)\quad \text{and}\quad \frac{1}{B}a_1(r)
\leq \frac{1}{B}k_1(x).
$$
Consequently we set
$$
Ak_2(x)\leq Aa_2(r)\leq \xi\leq \frac{1}{B}a_1(r)
\leq \frac{1}{B}k_1(x)
$$
for all $x\in\mathbb{R}^N$, i.e.,
\begin{gather}
 Ak_2(x)\leq \xi\quad \forall x\in\mathbb{R}^N \label{ei}\\
\frac{B}{k_1(x)}\leq\frac{1}{\xi}\quad \forall x\in\mathbb{R}^N\,.
\label{eii}
\end{gather}
Next let we set
$\xi=(\frac{c_1^q}{c_2^p})^{\frac{\alpha +1}{p}\,\frac{\beta +1}{q}}$,
where $c_1$ and $c_2$ are positive constants.

From \eqref{ei} and (H4), one easily gets,
$$
-[\lambda_1(m,p)-a]m(x)[c_2\Phi(x)]^{p-1}+bm_1(x)[c_1\Psi(x)]^{\beta +1}
\geq 0\quad \text{for all } x \in\mathbb{R}^N.
$$
Similarly, using \eqref{eii} and (H4), one has
$$
-[\lambda_1(n,q)-d]n(x)[c_1\Psi(x)]^{q-1}+cn_1(x)[c_2\Phi(x)]^{\alpha +1}
\geq 0\quad \text{for all }x \in\mathbb{R}^N.
$$
Hence
$$
f:=-[\lambda_1(m,p)-a]m(x)[c_2\Phi(x)]^{p-1}
+bm_1(x)[c_1\Psi(x)]^{\beta +1}\geq 0\quad \text{for all }
x \in\mathbb{R}^N
$$
and
$$
g:=-[\lambda_1(n,q)-d]n(x)[c_1\Psi(x)]^{q-1}
 +cn_1(x)[c_2\Phi(x)]^{\alpha +1}\geq 0\quad \text{for all }
x\in\mathbb{R}^N
$$
are nonnegative functions and $(-c_2\Phi,-c_1\Psi)$ is a solution
of \eqref{a1}. This is a contradiction with the (MP).

The condition is sufficient.
A detailed proof of this part can be found in \cite{bouch,ser}.
We give a sketch here.
Assume that the conditions (C1), (C2) and (C3) are satisfied.
Let $(u,v)$ be a solution of \eqref{a1} for $f,g\geq 0$.
Moreover, suppose that $u^-\not\equiv 0$ and $v^-\not\equiv 0$
and taking those functions as test function in \eqref{a1},
we find by H\"{o}lder inequality that
\begin{align*}
&[(\lambda_1(m,p)-a)^{\frac{\alpha +1}{p}}(\lambda_1(n,q)-d)
^{\frac{\beta +1}{q}}-b^{\frac{\alpha +1}{p}}c^{\frac{\beta +1}{q}}]\\
&\times \Big[\Big(\int_{\mathbb{R}^N}m|u^-|^p\Big)
\Big(\int_{\mathbb{R}^N}n|v^-|^q\Big)\Big]
^{\frac{\alpha +1}{p}\,\frac{\beta +1}{q}}\leq 0,
\end{align*}
which contradicts  assumption (C4). By applying regularity results
of \cite{ser, tolk} and the maximum principle of \cite{vaz},
one has in fact $u> 0$ and $v> 0$ a.e in $\mathbb{R}^N$.
\end{proof}

\section{Existence of positive solutions}

In this section, we prove the existence of positive solutions
for \eqref{a1} under conditions (C1), (C2) and (C3), by an
approximation method used in \cite{boc,bouch}.
For $\epsilon\in ]0,1[$, we define the following expression
\begin{gather*}
X_k:=\frac{|u_k|^{p-2}u_k}{1+|\epsilon^{1/p}u_k|^{p-1}},\quad
X:=\frac{|u|^{p-2}u}{1+|\epsilon^{1/p}u|^{p-1}},\\
Y_k:=\frac{|u_k|^{\alpha}u_k}{1+|\epsilon^{1/p}u_k|^{\alpha+1}},\quad
Y:=\frac{|u|^{\alpha}u}{1+|\epsilon^{1/p}u|^{\alpha+1}},\\
X_k':=\frac{|v_k|^{q-2}v_k}{1+|\epsilon^{1/q}v_k|^{q-1}},\quad
X':=\frac{|v|^{q-2}v}{1+|\epsilon^{1/q}v|^{q-1}},\\
Y_k':=\frac{|v_k|^{\beta }v_k}{1+|\epsilon^{1/q}v_k|^{\beta +1}},\quad
Y':=\frac{|v|^{\beta}v}{1+|\epsilon^{1/q}v|^{\beta +1}}.
\end{gather*}
On has the following result which will be useful later.

\begin{lemma}\label{lem1}
 If $(u_k,v_k)$ converges to $(u,v)$ in
$L^{p^*}(\mathbb{R}^N)\times L^{q^*}(\mathbb{R}^N)$ then
\begin{itemize}
 \item[(i)]  $X_k\to X$ in $L^{\frac{p^*}{p-1}}(\mathbb{R}^N)$,
$Y_k\to Y$ in $L^{\frac{p^*}{\alpha+1}}(\mathbb{R}^N)$ and
in $L^{q'}(m,\mathbb{R}^N)$.
\item[(ii)]$X_k'\to X'$ in $L^{\frac{q^*}{q-1}}(\mathbb{R}^N)$,
$Y_k'\to Y'$ in $L^{\frac{q^*}{\beta+1}}(\mathbb{R}^N)$ and in
$L^{p'}(n,\mathbb{R}^N)$.
\end{itemize}
\end{lemma}

\begin{proof}
 We give the proof for (i) and indicate that the same arguments
hold for (ii).
If $u_k\to u$ in $L^{p^*}(\mathbb{R}^N)$, then there exists a
subsequence denoted $(u_k)$ such that $u_k\to u$ almost every
where in $\mathbb{R}^N$ and $|u_k(x)|\leq l_1(x)$ for some
$l_1\in L^{p^*}(\mathbb{R}^N)$. Hence
\begin{gather*}
X_k(x)\to X(x)\quad\text{a.e. in }\mathbb{R}^N,\\
|X_k(x)|\leq |u_k(x)|^{p-1}\leq |l_1(x)|^{p-1}\quad
\text{in }L^{\frac{p^*}{p-1}},
\end{gather*}
which implies, by dominated convergence Theorem, that
$X_k\to X$ in $L^{\frac{p^*}{p-1}}$.

Similarly, on deduces from the convergence of $Y_k$ to $Y$ in
$L^{\frac{p^*}{\alpha+1}}$ that
\begin{gather*}
Y_k(x)\to Y(x)\quad\text{a.e in }\mathbb{R}^N,\\
|Y_k(x)|\leq |u_k(x)|^{\alpha +1}\leq |l_2(x)|^{\alpha+1}\quad
\text{in }L^{\frac{p^*}{\alpha+1}},
\end{gather*}
and the conclusion follows. Moreover, using H\"older inequality,
we have
$$
\|Y_k-Y\|_{L^{q'}(m,\mathbb{R}^N)}^{q'}
=\int_{\mathbb{R}^N}m|Y_k-Y|^{q'}
\leq \|m\|_{L^{N/p}(\mathbb{R}^N)}
\|Y_k-Y\|_{L^{\frac{p^*}{\alpha+1}}}^{q'}.
$$
\end{proof}

We are now in position to give the main result of this section.

\begin{theorem}\label{theo2}
 Assume that {\rm (H1), (H2), (H3), (C1), (C2), (C3)} are satisfied.
 Then for all $f\in L^{(p^*)'}(\mathbb{R}^N)$ and
$g\in L^{(q^*)'}(\mathbb{R}^N)$, the system \eqref{a1} has at
least one solution
$(u,v)\in D^{1,p}(\mathbb{R}^N)\times D^{1,q}(\mathbb{R}^N)$.
\end{theorem}

The proof is partly adapted from \cite{boc,bouch}.
We choose $r>0$ such that $a+r>0$ and $d+r>0$. The system \eqref{a1}
is then equivalent to
\begin{equation}\label{a9}
\begin{gathered}
 -\Delta_p u+rm|u|^{p-2}u=(a+r)m|u|^{p-2}u + bm_1|v|^\beta v
+f\quad\text{in }\mathbb{R}^N\\
-\Delta_q v+rn|v|^{q-2}v=cn_1|u|^{\alpha}u + (d+r)n|v|^{q-2}v
+g\quad\text{in }\mathbb{R}^N\\
u(x)\to 0, \quad v(x)\to 0\quad\text{as } |x|\to +\infty
\end{gathered}
\end{equation}
For $\epsilon\in ]0,1[$, let us introduce the system
\begin{equation}\label{Sepsilon}
\begin{gathered}
 -\Delta_p u_\epsilon +rm|u_\epsilon|^{p-2}u_\epsilon
 =m h(u_\epsilon)+m_1h_1(v_\epsilon) +f\quad\text{in }\mathbb{R}^N\\
-\Delta_q v_\epsilon+rn|v_\epsilon|^{q-2}v_\epsilon
 =n_1k_1(u_\epsilon) + nk(v_\epsilon) +g\quad\text{in }\mathbb{R}^N\\
u_\epsilon(x)\to 0, \quad v_\epsilon(x)\to 0\quad\text{as } |x|\to +\infty
\end{gathered}
\end{equation}
where
\begin{gather*}
h(u):=(a+r)\frac{|u|^{p-2}u}{1+|\epsilon^{1/p}u|^{p-1}},\quad
h_1(v):=b\frac{|v|^{\beta}v}{1+|\epsilon^{1/q}v|^{\beta+1}}, \\
k_1(u):=c\frac{|u|^{\alpha}u}{1+|\epsilon^{1/p}u|^{\alpha+1}},\quad
k(v):=(d+r)\frac{|v|^{q-2}v}{1+|\epsilon^{1/q}v|^{q-1}}.
\end{gather*}

\begin{lemma} \label{lem7}
 Under hypothesis of Theorem \ref{theo2}, system \eqref{Sepsilon}
 admits at least a couple of solution $(u,v)$ in
$D^{1,p}(\mathbb{R}^N)\times D^{1,q}(\mathbb{R}^N)$.
\end{lemma}

\begin{proof}
 We give the proof in several steps.

\textbf{Step 1.}
Construction of sub-super solution for \eqref{Sepsilon}:
Since the functions $h,h_1,k$ and $k_1$ are bounded, there exists
a constant $M>0$ such that
$$
|h(u)|\leq M,\quad |h_1(v)|\leq M, \quad |k_1(u)|\leq M,
\quad |k(v)|\leq M
$$
for all $(u,v)\in D^{1,p}(\mathbb{R}^N)\times D^{1,q}(\mathbb{R}^N)$.
Let $\xi^0\in D^{1,p}(\mathbb{R}^N)$
(respectively $\eta^0\in D^{1,q}(\mathbb{R}^N) $) be a solution of
$$
-\Delta_pu+rm|u|^{p-2}u=(m+m_1)M+f
$$
(respectively $-\Delta_qv+rm|v|^{q-2}v=(n+n_1)M+g)$,
and let $\xi_0 \in D^{1,p}(\mathbb{R}^N)$
(respectively $\eta_0\in D^{1,q}(\mathbb{R}^N)$) be solution of
$$
-\Delta_pu+rm|u|^{p-2}u=-(m+m_1)M+f
$$
(respectively
$-\Delta_qv+rm|v|^{q-2}v=-(n+n_1)M+g$).
Then $(\xi^0,\eta^0)$ (respectively $(\xi_0,\eta_0)$) is a super
 solution (respectively sub solution) of system \eqref{Sepsilon} since
\begin{gather*}
\begin{aligned}
&-\Delta_p\xi^0+rm|\xi^0|^{p-2}\xi^0-mh(\xi^0)-m_1h_1(\eta)-f\\
&\geq  -\Delta_p\xi^0+rm|\xi^0|^{p-2}\xi^0-(m+m_1)M-f=0\quad
\forall \eta\in [\eta_0,\eta^0],
\end{aligned}\\
\begin{aligned}
 &-\Delta_q\eta^0+rn|\eta^0|^{q-2}\eta^0-n_1k_1(\xi)-nk(\eta^0)-g\\
&\geq  -\Delta_q\eta^0+rn|\eta^0|^{q-2}\eta^0-(n+n_1)M-g=0\quad
 \forall \eta\in [\xi_0,\xi^0],
\end{aligned}\\
\begin{aligned}
 &-\Delta_p\xi_0+rm|\xi_0|^{p-2}\xi_0-mh(\xi_0)-m_1h_1(\eta)-f\\
&\leq -\Delta_p\xi_0+rm|\xi_0|^{p-2}\xi_0-(m+m_1)M-f=0\quad
 \forall \eta\in [\eta_0,\eta^0],
\end{aligned}\\
\begin{aligned}
 &-\Delta_q\eta_0+rn|\eta_0|^{q-2}\eta_0-n_1k_1(\xi)-nk(\eta_0)-g\\
&\leq -\Delta_q\eta_0+rn|\eta_0|^{q-2}\eta_0-(n+n_1)M-g=0\quad
\forall \eta\in [\xi_0,\xi^0]\,.
\end{aligned}
\end{gather*}

\textbf{Step 2.} Definition of operator $T$.
Denote by $K=[\xi_0,\xi^0]\times[\eta_0,\eta^0]$ and define
the operator $T:(u,v)\to (w,z)$ such that
\begin{equation}\label{a10}
\begin{gathered}
 -\Delta_pw+rm|w|^{p-2}w=mh(u)+m_1h_1(v)+f\quad\text{in }\mathbb{R}^N\\
-\Delta_qz+rm|z|^{q-2}z=n_1k_1(u)+nk(v)+g\quad\text{in }\mathbb{R}^N\\
w(x)\to 0,\quad  z(x)\to 0\quad\text{as } |x|\to +\infty
\end{gathered}
\end{equation}

\textbf{Step 3.} Let us prove that $T(K)\subset K$.
If $(u,v)\in K$ then we have
\begin{equation} \label{a10'}
\begin{aligned}
&-(\Delta_pw-\Delta_p\xi^0)+rm(|w|^{p-2}w-|\xi^0|^{p-2}\xi^0 )\\
&= m[h(u)-M]+m_1[h_1(v)-M])
\end{aligned}
\end{equation}
Taking $(w-\xi^0)^+$ as test function in \eqref{a10'}, we have
\begin{align*}
&\int_{\mathbb{R}^N}(|\nabla w|^{p-2}\nabla w
 -|\nabla \xi^0|^{p-2}\nabla \xi^0)\nabla (w-\xi^0)^+\\
&+r\int_{\mathbb{R}^N}m(|w|^{p-2}w-|\xi^0|^{p-2}\xi^0)(w-\xi^0)^+\\
&=\int_{\mathbb{R}^N}[m(h(u)-M)+m_1(h_1(v)-M)](w-\xi^0)^+\leq 0.
\end{align*}
Hence by the monotonicity of the function $x\mapsto \|x\|^{p-2}x$
and by the monotonicity of the p-Laplacian, we deduce that
$(w-\xi^0)^+=0$ and then $w\leq \xi^0$. Similarly we get
$\xi_0\leq w$ by taking $(w-\xi_0)^-$ as test function
in \eqref{a10'}. So we have $\xi_0\leq w\leq \xi^0$ and
$\eta_0\leq z\leq \eta^0$ and the step is complete.

\textbf{Step 4.} $T$ is completely continuous:

$\bullet$  We will first prove that $T$ is continuous. Indeed
let $(u_k,v_k)\to (u,v) \in D^{1,p}(\mathbb{R}^N)
\times D^{1,q}(\mathbb{R}^N)$, we will prove that
$(w_k,z_k)=T(u_k,v_k)$ converges to $(w,z)=T(u,v)$.
\begin{equation}\label{a11}
\begin{aligned}
&(-\Delta_pw_k+rm|w_k|^{p-2}w_k)-(-\Delta_pw+rm|w|^{p-2}w)\\
&= m[h(u_k)-h(u)]+m_1[h_1(v_k)-h_1(v)]\\
&= (a+r)m(X_k-X)+bm_1(Y_k'-Y'),
\end{aligned}
\end{equation}
where $X_k,X,Y_k'$ and $Y'$ are previously define in Lemma \ref{lem1}.
Then taking $(w_k-w)$ as test function in \eqref{a11}, we get
\begin{align*}
&\int_{\mathbb{R}^N}(|\nabla w_k|^{p-2}\nabla w_k-|\nabla w|^{p-2}w)\nabla (w_k-w)\\
&\leq \int_{\mathbb{R}^N}(|\nabla w_k|^{p-2}\nabla w_k-|\nabla w|^{p-2}w)\nabla (w_k-w)\\
&\quad+ r \int_{\mathbb{R}^N}m(| w_k|^{p-2} w_k-|w|^{p-2}w)(w_k-w)\\
&= (a+r)\int_{\mathbb{R}^N} m(X_k-X)(w_k-w)+b\int_{\mathbb{R}^N} m_1(Y_k'-Y')(w_k-w).
\end{align*}
Using H\"older inequality, we obtain
$$
\int_{\mathbb{R}^N} m(X_k-X)(w_k-w)\leq \|m\|_{L^{N/p}(\mathbb{R}^N)}
\|X_k-X\|_{L^{p^*/(p-1)}(\mathbb{R}^N)}.
\|w_k-w\|_{L^{p^*}(\mathbb{R}^N)}
$$
and
\begin{align*}
\int_{\mathbb{R}^N} m_1(Y_k'-Y')(w_k-w)
&\leq \int_{\mathbb{R}^N} [m^{1/p}(w_k-w)][n^{(\beta +1)/q}(Y_k'-Y')]\\
&\leq  \|w_k-w\|_{L^{p}(m,\mathbb{R}^N)}.\|Y_k'-Y'\|_{L^{p'}
(n,\mathbb{R}^N)},
\end{align*}
since $\frac{\beta +1}{q}=\frac{1}{p'}$. Consequently
\begin{align*}
&0\leq \int_{\mathbb{R}^N}(|\nabla w_k|^{p-2}\nabla w_k
 -|\nabla w|^{p-2}w)\nabla (w_k-w)\\
&\leq (a+r)\|m\|_{L^{N/p}(\mathbb{R}^N)}\|X_k-X\|_{L^{p^*/(p-1)}
 (\mathbb{R}^N)}\|w_k-w\|_{L^{p^*}(\mathbb{R}^N)}\\
&\quad +b\|Y_k'-Y'\|_{L^{p'}(n,\mathbb{R}^N)}.\|w_k-w\|_{L^{p}
 (m,\mathbb{R}^N)}
\end{align*}
Using then the inequality
\begin{equation}\label{a12}
\|x-y\|^p\leq c[(\|x\|^{p-2}x-\|y\|^{p-2}y)(x-y)]^{s/2}
[\|x\|^p+\|y\|^p]^{1-s/2},
\end{equation}
where $x,y\in\mathbb{R}^N$, $c=c(p)>0$ and $s=2$ if $p\geq 2$,
$s=p$ if $1<p<2$ (cf. e.g. \cite{lind}), one easily obtains that
$w_k\to w$ in $D^{1,p}(\mathbb{R}^N)$. Similarly, we have
$z_k\to z$ in $D^{1,q}(\mathbb{R}^N)$.


$\bullet$  We now prove that operator $T$ is compact.
Let $(u_k,v_k)$ be a bounded sequence in
$D^{1,p}(\mathbb{R}^N)\times D^{1,q}(\mathbb{R}^N)$ and set
$(w_k,z_k)=T(u_k,v_k)$. We have
\begin{equation}\label{a13}
 -\Delta_pw_k+rm|w_k|^{p-2}w_k=mh(u_k)+m_1h_1(v_k)+f
\end{equation}
Taking $w_k$ as test function in \eqref{a13}, we get
\begin{align*}
&\int_{\mathbb{R}^N}|\nabla w_k |^p+r\int_{\mathbb{R}^N} m|w_k|^p\\
&= \int_{\mathbb{R}^N} mh(u_k)w_k
+\int_{\mathbb{R}^N} m_1h_1(v_k)w_k+\int_{\mathbb{R}^N} fw_k\\
&\leq (a+r)\int_{\mathbb{R}^N} m|u_k|^{p-1}|w_k|+b\int_{\mathbb{R}^N}m_1|v_k|^{\beta+1}|w_k|+\int_{\mathbb{R}^N}|f\|w_k|\\
&\leq \big[(a+r)\|u_k\|_{L^p(m,\mathbb{R}^N)}^{p-1}
 +b\|v_k\|_{L^q(n,\mathbb{R}^N)}^{\beta+1}\big].\|w_k\|_{L^p(m,\mathbb{R}^N)}\\
&\quad +\|f\|_{L^{(p^*)'}(\mathbb{R}^N)}\|w_k\|_{L^{p^*}(\mathbb{R}^N)}.
\end{align*}
Hence $w_k$ is bounded in $D^{1,p}(\mathbb{R}^N)$ and consequently,
up to a subsequence $w_k$ converges to $w$ weakly in
$D^{1,p}(\mathbb{R}^N)$ and strongly in $L^p(m,\mathbb{R}^N)$.
Now taking $(w_k-w_q)$ as test function in \eqref{a13}, we have
\begin{align*}
&\int_{\mathbb{R}^N}|\nabla w_k|^{p-2}\nabla w_k\nabla (w_k-w_q)
 +r \int_{\mathbb{R}^N} m|w_k|^{p-2}w_k(w_k-w_q)\\
&=\int_{\mathbb{R}^N}[mh(u_k)+m_1h_1(v_k)](w_k-w_q)
 +\int_{\mathbb{R}^N} f(w_k-w_q)
\end{align*}
and consequently
\begin{align*}
&\int_{\mathbb{R}^N}(|\nabla w_k|^{p-2}\nabla w_k-|\nabla w_q|^{p-2}\nabla w_q)\nabla (w_k-w_q)\\
&\leq \int_{\mathbb{R}^N}(|\nabla w_k|^{p-2}\nabla w_k
 -|\nabla w_q|^{p-2}\nabla w_q)\nabla (w_k-w_q)\\
&\quad +r \int_{\mathbb{R}^N} m(|w_k|^{p-2}w_k-|w_q|^{p-2}w_q)(w_k-w_q)\\
&= \int_{\mathbb{R}^N} m[h(u_k)-h(u_q)](w_k-w_q)+ \int_{\mathbb{R}^N} m_1[h_1(v_k)-h_1(v_q)](w_k-w_q)\\
&\leq \big[\|u_k\|_{L^p(m,\mathbb{R}^N)}^{p-1}
 +\|u_q\|_{L^p(m,\mathbb{R}^N)}^{p-1}
 +\|v_k\|_{L^q(n,\mathbb{R}^N)}^{\beta+1}\\
&\quad +\|v_q\|_{L^q(n,\mathbb{R}^N)}^{\beta+1}\big]
\|w_k-w_q\|_{L^p(m,\mathbb{R}^N)}.
\end{align*}
We then deduce that
$$
\int_{\mathbb{R}^N}(|\nabla w_k|^{p-2}\nabla w_k-|\nabla w_q|^{p-2}
\nabla w_q)\nabla (w_k-w_q)\to 0.
$$
 From \eqref{a12}, we conclude that $w_k$ converges to $w$
in $D^{1,p}(\mathbb{R}^N)$. Similarly, we prove that $z_k$
converges to $z$ in $D^{1,q}(\mathbb{R}^N)$.

Since the set $K$ is convex, bounded and closed in
$D^{1,q}(\mathbb{R}^N)\times D^{1,q}(\mathbb{R}^N)$, applying
Schauder's fixed point theorem, then there exists a fixed point
for $T$ which gives the existence of solution of system \eqref{Sepsilon}.
\end{proof}



\begin{proof}[Proof of Theorem \ref{theo2}]
The proof will be given by three steps.

\textbf{Step 1.}
 We show that $(u_\epsilon,v_\epsilon)$ is bounded in
$D^{1,p}(\mathbb{R}^N)\times D^{1,q}(\mathbb{R}^N)$.
Indeed denoting by
$t_\epsilon =\max (\|u_\epsilon\|_{D^{1,p}(\mathbb{R}^N)}^p,
\|v_\epsilon\|_{D^{1,q}(\mathbb{R}^N)}^q)$,
$z_\epsilon=t_\epsilon^{-1/p}u_\epsilon$ and
$w_\epsilon=t_\epsilon^{-1/q}v_\epsilon$.
Since $(u_\epsilon,v_\epsilon)$ is solution of \eqref{Sepsilon},
 we have
\begin{gather*}
 -\Delta_pz_\epsilon+rm|z_\epsilon|^{p-2}z_\epsilon
=\frac{(a+r)m|z_\epsilon|^{p-2}z_\epsilon}{1+t_\epsilon^{1/p'}
|\epsilon^{1/p}z_\epsilon|^{p-1}}
+\frac{bm_1|w_\epsilon|^{\beta}w_\epsilon}{1+t_\epsilon^{1/p'}
|\epsilon^{1/q}w_\epsilon|^{\beta+1}}+t_\epsilon^{-1/p'}f\,,
\\
-\Delta_qw_\epsilon+rm|w_\epsilon|^{q-2}w_\epsilon
=\frac{cn_1|z_\epsilon|^{\alpha}z_\epsilon}{1+t_\epsilon^{1/q'}
|\epsilon^{1/p}z_\epsilon|^{\alpha+1}}+\frac{(d+r)n|w_\epsilon|^{q-2}
w_\epsilon}{1+t_\epsilon^{1/q'}|\epsilon^{1/q}w_\epsilon|^{q-1}}
+t_\epsilon^{-1/q'}g.
\end{gather*}
Hence taking $z_\epsilon$ as test function in the first equation, we get
\begin{align*}
&\int_{\mathbb{R}^N}|\nabla z_\epsilon|^p
+ r\int_{\mathbb{R}^N}m|z_\epsilon|^p \\
&\leq (a+r)\int_{\mathbb{R}^N}m|z_\epsilon|^p
 +b\int_{\mathbb{R}^N}m_1|w_\epsilon|^{\beta+1}|z_\epsilon|
 +t_\epsilon^{-1/p'}\int_{\mathbb{R}^N}|f\| z_\epsilon|,
\end{align*}
which implies, by H\"older inequality and (H2),
\begin{align*}
\int_{\mathbb{R}^N}|\nabla z_\epsilon|^p
&\leq a\int_{\mathbb{R}^N}m|z_\epsilon|^p+b
\Big(\int_{\mathbb{R}^N}m|z_\epsilon|^p\Big)^{1/p}
\Big(\int_{\mathbb{R}^N}n|w_\epsilon|^q\Big)^{(\beta+1)/q}\\
&\quad +t_\epsilon^{-1/p'}\|f\|_{L^{(p^*)'}(\mathbb{R}^n)}
\Big(\int_{\mathbb{R}^N}|z_\epsilon|^{p^*}\Big)^{1/{p^*}}
\end{align*}
By the imbedding of $D^{1,p}(\mathbb{R}^N)$ in $L^{p^*}(\mathbb{R}^N)$,
we deduce that
\begin{align*}
 \|z_\epsilon\|_{D^{1,p}}^p
&\leq \frac{a}{\lambda_1(m,p)}
\|z_\epsilon\|_{D^{1,p}}^p
+b\frac{\|z_\epsilon\|_{D^{1,p}}}{[\lambda_1(m,p)]^{1/p}}
\frac{\|w_\epsilon\|_{D^{1,q}}^{\beta+1}}{[\lambda_1(n,q)]^{(\beta+1)/q}}\\
&\quad + c_1t_\epsilon^{-1/p'}\|f\|_{L^{(p^*)'}}\|z_\epsilon\|_{D^{1,p}},
\end{align*}
where $c_1=c_1(p,N)$ is the constant of the imbedding of
$D^{1,p}(\mathbb{R}^N)$ into $L^{p^*}(\mathbb{R}^N)$, and consequently
\begin{equation}\label{a14}
\begin{aligned}
[\lambda_1(m,p)-a]\Big(\frac{\|z_\epsilon\|_{D^{1,p}}}
{[\lambda_1(m,p)]^{1/p}}\Big)^{p-1}
&\leq b \Big(\frac{\|w_\epsilon\|_{D^{1,q}}}{[\lambda_1(n,q)]^{1/q}}\Big)
^{\beta+1}\\
&\quad + c_1t_\epsilon^{-1/p'}[\lambda_1(m,p)]^{1/p}\|f\|_{L^{(p^*)'}}.
\end{aligned}
\end{equation}
Similarly, we obtain
\begin{equation}\label{a15}
\begin{aligned}
 [\lambda_1(n,q)-d]\Big(\frac{\|w_\epsilon\|_{D^{1,q}}}{[\lambda_1(n,q)]
^{1/q}}\Big)^{q-1}
&\leq c \Big(\frac{\|z_\epsilon\|_{D^{1,p}}}
{[\lambda_1(m,p)]^{1/p}}\Big)^{\alpha+1}\\
&\quad + c_2t_\epsilon^{-1/q'}[\lambda_1(n,q)]^{1/q}\|g\|_{L^{(q^*)'}}
\end{aligned}
\end{equation}
Now assume that $u_\epsilon$ (or $v_\epsilon$)is unbounded
in $D^{1,p}(\mathbb{R}^N)$ ( in $D^{1,q}(\mathbb{R}^N)$).
Then $t_\epsilon\to +\infty$ and it follows from \eqref{a14}
and \eqref{a15}, that
\begin{align*}
&[\lambda_1(m,p)-a]^{\frac{\alpha+1}{p}}
[\lambda_1(n,q)-d]^{\frac{\beta+1}{q}}
\Big(\frac{\|z_\epsilon\|_{D^{1,p}}}{[\lambda_1(m,p)]^{1/p}}\Big)
^{\frac{(\alpha+1)(\beta+1)}{q}}\\
&\times \Big(\frac{\|w_\epsilon\|_{D^{1,q}}}{[\lambda_1(n,q)]^{1/q}}\Big)
^{\frac{(\alpha+1)(\beta+1)}{p}}\\
&\leq b^{\frac{\alpha+1}{p}}c^{\frac{\beta+1}{q}}
\Big(\frac{\|z_\epsilon\|_{D^{1,p}}}{[\lambda_1(m,p)]^{1/p}}\Big)
^{\frac{(\alpha+1)(\beta+1)}{q}}
\Big(\frac{\|w_\epsilon\|_{D^{1,q}}}{[\lambda_1(n,q)]^{1/q}}\Big)
^{\frac{(\alpha+1)(\beta+1)}{p}},
\end{align*}
which implies
\begin{align*}
&[(\lambda_1(m,p)-a)^{\frac{\alpha+1}{p}}(\lambda_1(n,q)-d)
^{\frac{\beta+1}{q}}-b^{\frac{\alpha+1}{p}}c^{\frac{\beta+1}{q}}]\\
&\times \Big(\frac{\|z_\epsilon\|_{D^{1,p}}}{\lambda_1(m,p)^{1/p}}
\frac{\|w_\epsilon\|_{D^{1,q}}}{\lambda_1(n,q)^{1/q}}\Big)
^{\frac{(\alpha+1)(\beta+1)}{p}}\leq 0.
\end{align*}
But this is a contradiction since conditions (C1), (C2) and
(C3) hold.

\textbf{Step 2.}
 Using the same arguments as in \cite{sera}, we easily prove
that $(\epsilon^{\frac{1}{p}}u_\epsilon,\epsilon^{\frac{1}{q}}
v_\epsilon)\to (0,0)$ in $D^{1,p}(\mathbb{R}^N)\times D^{1,q}
(\mathbb{R}^N)$.


\textbf{Step 3.}  Now we prove that $(u_\epsilon,v_\epsilon)$
converges strongly in $D^{1,p}(\mathbb{R}^N)\times D^{1,q}(\mathbb{R}^N)$
as $\epsilon \to 0$. Indeed from Step 1 and Step 2, we have
$(u_\epsilon,v_\epsilon)$ is bounded in
$D^{1,p}(\mathbb{R}^N)\times D^{1,q}(\mathbb{R}^N)$ and
$\epsilon^{\frac{1}{p}}u_\epsilon \to 0$ a.e in $\mathbb{R}^n$.
So up to a subsequence $(u_\epsilon,v_\epsilon)\to (u_0,v_0)$
in $L^p(m,\mathbb{R}^N)\times L^q(n,\mathbb{R}^N)$ and consequently
\begin{gather*}
\big|\frac{|u_\epsilon|^{p-2}u_\epsilon}{1+|\epsilon^{\frac{1}{p}}
u_\epsilon|^{p-1}}\big|
\leq |u_\epsilon|^{p-1}\leq l_1^{p-1}\quad\text{in }
L^{p'}(m,\mathbb{R}^N), \\
\frac{|u_\epsilon(x)|^{p-2}u_\epsilon(x)}{1+|\epsilon^{\frac{1}{p}}
u_\epsilon(x)|^{p-1}}\to |u_0(x)|^{p-2}u_0(x)\quad\text{a.e. in }
\mathbb{R}^N.
\end{gather*}
 From the dominated convergence theorem, we have
$h(u_\epsilon)\to h(u_0)$ in $L^{p'}(m,\mathbb{R}^N)$ as
$\epsilon \to 0$. Similarly,  we get $h_1(v_\epsilon)\to h_1(v_0)$
in $L^{q'}(n,\mathbb{R}^N)$, $k_1(u_\epsilon)\to k_1(u_0)$ in
$L^{p'}(m,\mathbb{R}^N)$ and $k(v_\epsilon)\to k(v_0)$ in
$L^{q'}(n,\mathbb{R}^N)$. We finally use \eqref{a12} to deduce
that $(u_\epsilon,v_\epsilon)\to (u_0,v_0)$ in
$D^{1,p}(\mathbb{R}^N)\times D^{1,q}(\mathbb{R}^N)$ as
$\epsilon \to 0$. Therefore, passing to the limit in \eqref{Sepsilon},
we obtain
\begin{gather*}
 -\Delta_pu_0=am|u_0|^{p-2}u_0+bm_1|v_0|^{\beta}v_0 +f\quad
 \text{in }\mathbb{R}^N,\\
 -\Delta_qv_0=cn_1|u_0|^{\alpha}u_0+dn|v_0|^{q-2}v_0 +g\quad
 \text{in }\mathbb{R}^N
\end{gather*}
which implies that $(u_0,v_0)$ is solution of \eqref{a1}.
\end{proof}

We remark that when $\alpha=\beta=0$ and $p=q=2$, we obtain
the results presented in \cite{fleck4,fleck5}.

 \section{Related results}

The tools used to establish the above results can be easily adapted
for the   problem
 \begin{equation}\label{a16}
   \begin{gathered}
   -\Delta_pu = a m(x)|u|^{p-2}u + b m_{1}(x)|u|^\alpha|v|^\beta v + f
\quad \text{in } \mathbb{R}^n \\
 -\Delta_qv = cn_1(x)|u|^\alpha u|v|^\beta + dn(x)|v|^{q-2}v + g
\quad \text{in } \mathbb{R}^n\\
   u(x)\to 0, \quad v(x)\to 0 \quad \text{as } |x|\to +\infty
\end{gathered}
\end{equation}
where we assume that the conditions (H1),(H2'),(H3) and (H4') hold,
 with
\begin{itemize}
\item[(H2')] $ 0<m_1(x)\leq m(x)^{\frac{\alpha +1}{p}}n(x)
^{\frac{\beta +1}{q}}$ and
$0<n_1(x)\leq m(x)^{\frac{\alpha +1}{p}}n(x)^{\frac{\beta +1}{q}}$
a.e. in  $\mathbb{R}^N$

\item[(H4')] $b,c\geq 0$; $\alpha,\beta\geq 0$;
$\frac{\alpha +1}{p}+\frac{\beta +1}{q}=1$.
\end{itemize}

Under these assumptions, one has the following results.


\begin{theorem}\label{theo3}
Assume that hypothesis {\rm (H1), (H2'), (H3), (H4')} are satisfied.
Then the (MP) holds for \eqref{a16} if
\begin{itemize}
 \item[(C1')]  $\lambda_1(m,p)>a$;
\item[(C2')] $\lambda_1(n,q)>d$;
\item[(C3')] $[\lambda_1(m,p)-a]^{\frac{\alpha +1}{p}}[\lambda_1(n,q)-d]
^{\frac{\beta +1}{q}}>b^{\frac{\alpha +1}{p}}c^{\frac{\beta +1}{q}}$;
\end{itemize}
Conversely, if the (MP) holds, then (C1'), (C2') and (C4') are satisfied,
 where
\begin{itemize}
\item[(C4')] $[\lambda_1(m,p)-a]^{\frac{\alpha +1}{p}}
 [\lambda_1(n,q)-d]^{\frac{\beta +1}{q}}
>\Theta b^{\frac{\alpha +1}{p}}c^{\frac{\beta +1}{q}}$.
\end{itemize}
\end{theorem}

\begin{theorem}\label{theo4}
 Assume that {\rm (H1), (H2'), (H3), (C1'), (C2'), (C3')}  hold.
 Furthermore assume that $m\in L^{(p^*)'}(\mathbb{R}^N)$ and
$m\in L^{(q^*)'}(\mathbb{R}^N)$. Then for all
$f\in L^{(p^*)'}(\mathbb{R}^N)$ and $g\in L^{(q^*)'}(\mathbb{R}^N)$,
the system \eqref{a16} has at least one solution
$(u,v)\in D^{1,p}(\mathbb{R}^N)\times D^{1,q}(\mathbb{R}^N)$.
\end{theorem}

The proofs of theorems \ref{theo3} and \ref{theo4} can be adapted
from  those of theorems \ref{theo1} and \ref{theo2} respectively.

\subsection*{Acknowledgements}
The authors would like to express their gratitude to Professor
F. de Th\'elin for his constructive remarks and suggestions.
L. Leadi is grateful to the International Centre for Theoretical
Physics (ICTP) for financial support during his visit

\begin{thebibliography}{99}

\bibitem{all} W. Allegretto and Y. X. Huang;
 {\it Eigenvalues of the indefinite weight p-Laplacian in weighted
spaces}, Funck.Ekvac., 8 (1995), pp 233-242.

\bibitem{boc} L. Boccardo,  J. Fleckinger and F. de Th\'elin;
{\it Existence of solutions for some nonlinear cooperative systems}, Diff. and Int. Equations

\bibitem{bouch} M. Bouchekif, H. Serag and F. de Th\'elin;
 {\it On maximum principle and existence of solutions for some nonlinear elliptic systems}, Rev. Mat. Apl. 16, pp 1-16.

\bibitem{dra}P. Dr\'{a}bek, A.Kufner and . F. Nicolosi;
 {\it Quasilinear elliptic equations with degenerations and singularities}, De Gryter, 1997.

\bibitem{fig1}D.G.de Figueiredo and E.Mitidieri;
 {\it Maximum Principles for Linear
Elliptic Systems, Quaterno Matematico}, page 177 , 1988; Dip. Sc. Mat,
Univ Trieste.

\bibitem{fig2}D. G. de Figueiredo and E. Mitidieri;
 {\it Maximum Principles for Cooperative Elliptic Systems },
Comptes Rendus Acad.Sc.Paris, 310, (1990), 49-52 .

\bibitem{fig3}D. G. de Figueiredo and E. Mitidieri;
{\it A Maximum Princilpe for an Elliptic System and Applications
to Semilinear Problems }, SIAM J. Math. Anal., 17, (1986), 836-849.

\bibitem{fleck} J. Fleckinger, J. Hernandez and F. de Th\'elin;
 {\it  Principe du Maximum pour un Syst\`eme Elliptique non Lin\'eaire},
Comptes Rendus Acad. Sc. Paris, 314, (1992), 665-668.

\bibitem{fleck1} J. Fleckinger, J. Hernnadez and F. de Th\'elin;
 {\it On maximum principle and existence of postive solutions for some
cooperative elliptic systems}, J. Diff. and Int. Equa., vol8 (1995),
pp 69-85.

\bibitem{fleck2} J. Fleckinger, R. F. Man\'{a}sevich,
N. M. Stavrakakis and F. de Th\'elin;
 {\it Principal eigenvalues for some quasilinear elliptic equations
on $\mathbb{R}^N$}, Advances in Diff. Equa, vol2, No. 6, November
1997, pp 981-1003.

\bibitem{fleck3} J. Fleckinger, J. P. Gossez and F. de Th\'elin;
 {\it Antimaximum principle in $\mathbb{R}^N$: Local versus global},
J. Diff. Equa., 196 (2004), pp. 119-133.

\bibitem{fleck4} J. Fleckinger and H. Serag;
 {\it On maximum principle and existence of solutions for elliptic
systems on $\mathbb{R}^N$}, J. Egypt. Math. Soc., vol2 (1994), pp 45-51.

\bibitem{fleck5} J. Fleckinger and H. Serag;
 {\it Semilinear cooperative elliptic systems on $\mathbb{R}^N$},
Rend. di Mat., vol seri VII Roma (1995), pp. 89-108.

\bibitem{gil} D. Gilbarg and N. S. Trudinger;
 {\it Elliptic Partial Differential Equations of Second Order},
Springer-Verlag, Berlin 1983.

\bibitem{goss} J-P. Gossez and L. Leadi;
 {\it Asymmetric elliptic problems in
$\mathbb{R}^N$}, Elec. J. Diff. Equa., Conf 14 (2006), pp 207-222.

\bibitem{koz} H. Kozono and H. Sohr,
{\it New a priori estimates for the stokes equation in exterior domains},
 Indiana Univ. Math. J., 40 (1991), pp 1-27.

\bibitem{lind} P. Lindqvist;
 {\it On the equation $\mathop{\rm div}(|\nabla u|^{p-2}\nabla u)
+\lambda |u|^{p-2}u=0$}, Proc. Amer. Math. Soc., 109 (1990),
pp. 157-166. Addendum, Proc. Amer.Math.Soc., 116 (1992), pp 583-584.

\bibitem{pro} M. H. Protter and H. Weinberger;
 {\it Maximum Principles in Differential Equations}
{\it Prentice Hall, Englewood Cliffs,} 1967.

\bibitem{sera} H. M. Serag and E. A. El-Zahrani;
 {\it Maximum principle and existence of positive solutions for
nonlinear systems on $\mathbb{R}^N$}, Elec. J. Diff. Equa. (EJDE),
vol 2005(2005), No. 85, pp. 1-12.

\bibitem{ser} J. Serrin;
 {\it Local behavior of solutions of quasilinear
equations}, Acta Math., 111 (1964), 247-302.

\bibitem{tolk} P. Tolksdorf;
{\it Regularity for a more general class of quasilinear elliptic
equations}, J. Diff. Equat., 51 (1984), pp 126-150.

\bibitem{vaz} J. L. V\'{a}zquez;
{\it A strong maximum principle for some quasilinear elliptic
equations}, Appl. Math. and Optimization 12 (1984) pp 191-202.

\end{thebibliography}

\end{document}