\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 $\Delta_p u:= \mathop{\rm div}(|\nabla u|^{p-2}\nabla u)$, $1
0$, 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 $1
0; m\in L_{\rm loc}^\infty(\mathbb{R}^N)\cap L^{N/p}
(\mathbb{R}^N)$ and
$n\in L_{\rm loc}^\infty(\mathbb{R}^N)\cap L^{N/q}(\mathbb{R}^N)$
\item[(H2)] $ 0 \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 $1 a$;
\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
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