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

\AtBeginDocument{{\noindent\small
\emph{Electronic Journal of Differential Equations},
Vol. 2013 (2013), No. 175, pp. 1--11.\newline
ISSN: 1072-6691. URL: http://ejde.math.txstate.edu or http://ejde.math.unt.edu
\newline ftp ejde.math.txstate.edu}
\thanks{\copyright 2013 Texas State University - San Marcos.}
\vspace{9mm}}

\begin{document}
\title[\hfilneg EJDE-2013/175\hfil Existence of positive bounded solutions]
{Existence of positive bounded solutions
for nonlinear elliptic systems}

\author[F. Toumi \hfil EJDE-2013/175\hfilneg]
{Faten Toumi}  % in alphabetical order

\address{Faten Toumi \newline
 D\'{e}partement de Math\'{e}matiques, Facult\'{e} des
Sciences de Tunis, Campus Universitaire, 2092 Tunis, Tunisia}
\email{Faten.Toumi@fsb.rnu.tn}

\thanks{Submitted August 15, 2012. Published July 29, 2013.}
\subjclass[2000]{34B15, 34B27, 35J66}
\keywords{Green function; Kato class; nonlinear elliptic systems; 
\hfill\break\indent positive solution; Schauder fixed point theorem}

\begin{abstract}
 In this article, we study a class of nonlinear elliptic systems
 in regular domains of $\mathbb{R}^n(n\geq 3)$ with compact boundary.
 More precisely, we prove the existence of bounded positive continuous
 solutions to the system $\Delta u=\lambda f(.,u,v)$,
 $\Delta v=\mu g(.,u,v)$, subject to some Dirichlet conditions.
 Our approach is essentially based on properties of functions in
 a Kato class $K^{\infty }(D)$ and the Schauder fixed point theorem.
\end{abstract}

\maketitle
\numberwithin{equation}{section}
\newtheorem{theorem}{Theorem}[section]
\newtheorem{proposition}[theorem]{Proposition}
\newtheorem{corollary}[theorem]{Corollary}
\newtheorem{definition}[theorem]{Definition}
\newtheorem{example}[theorem]{Example}
\newtheorem{remark}[theorem]{Remark}
\allowdisplaybreaks


\section{Introduction}

 The study of elliptic equations has strong motivations. In fact,
such equations model many phenomena in biology, ecology, combustion theory 
\cite{b2,f1}, chemical reactions, population genetics \cite{f2}
 etc. For instance, many steady state problems arise in the
description of physics phenomena such as fluid dynamics \cite{a2},
wave phenomena, nonlinear field theory \cite{b3} etc. As
consequence, the study of the existence of positive solutions and their
asymtotic behaviour of such problems are of interest. A typical model
example of these is the nonlinear eigenvalue problem
\[
\Delta u=\lambda f(u)\quad \text{in }D,
\]
where $\lambda $ is a positive parameter. For an extensive review on the
existence results of positive solutions of the above problem we refer the
reader to the work of Lions \cite{l2}.

 Recently, many researchers extended the study of nonlinear
elliptic scalar equations to\ nonlinear elliptic systems. For some recent
results, we give a short account.

 Lair and Wood \cite{l1} studied the existence of entire
nonnegative solutions for the semilinear elliptic system
\begin{gather*}
\Delta u=p(| x| )v^{r}, \\
\Delta v=q(| x| )u^{s},
\end{gather*}
in  $\mathbb{R}^n$, where $r>0$ and $s>0$. The authors proved the
existence of entire bounded solutions and large ones in the sublinear
 and superlinear cases, provided that the potentials $p$ and $q$ satisfy either
\[
\int_0^{\infty }tp(t)dt<\infty \quad \text{and}\quad
\int_0^{\infty }tq(t)dt<\infty
\]
or
\[
\int_0^{\infty }tp(t)dt=\infty\quad \text{and}\quad
\int_0^{\infty }tq(t)dt=\infty .
\]

 Cirstea and Radulescu  \cite{c2} studied the
semilinear elliptic system
\begin{gather*}
\Delta u=p(x)f_1(v), \\
\Delta v=q(x)f_2(u),
\end{gather*}
in $\mathbb{R}^n$ $(n\geq 3)$, where the functions $f_1$ and $f_2$ are
nonincreasing on $(0,\infty )$ and $p$ and $q$ are radially
symmetric functions in $\mathbb{R}^n$. In particular, the authors
established the existence of positive solutions provided that the function
 $x\to f(cg(x))$ is  sublinear at infinity and superliner at $0$, for each $c>0$.
Moreover, the authors gave the behavior of solutions, that is, bounded
solutions or blow-up ones depending upon some additional conditions related
essentially to the potentials $p$ and $q$. Motivated by this work \cite{c2},
 Ghanmi et al \cite{g3} considered  the system
\begin{equation}
\begin{gathered}
\Delta u=\lambda p(x)f_1(v)\quad \text{in }D, \\
\Delta v=\mu q(x)f_2(u)\quad \text{in }D, \\
u\big|_{\partial D}=a\varphi ,\quad v\big|_{\partial D}=b\psi , \\
\lim_{| x| \to +\infty }u(x)=\alpha ,\quad
\lim_{| x| \to +\infty }v(x)=\beta \quad\text{(if $D$ is unbounded)},
\end{gathered}  \label{eP}
\end{equation}
where the potentials $p$ and $q$ belong to the Kato class
$K^{\infty }(D)$ defined below (See Definition \ref{def1}), the functions $f_1$ and
$f_2 $ are monotone. Indeed, the authors established two existence results
for the problem \eqref{eP} as $f_1$ and $f_2$ are nondecreasing
or nonincreasing. They used a variant of monotone iteration and the
properties of the Green function and potentials belonging to $K^{\infty
}(D)$. We note that the authors extended the results of Toumi
and Zeddini \cite{t2} and Ahtreya \cite{a4} to systems of
equations.  Garc\'{\i}a-Meli\'{a}n and Rossi \cite{g1}
considered the elliptic system
\begin{gather*}
\Delta u=u^{p}v^{q} \quad \text{in }\Omega\\
\Delta v=u^{r}v^{s} \quad \text{in }\Omega
\end{gather*}
where $p,s>1,q,r>0$ and $\Omega $ $\subset \mathbb{R}^n$ is a smooth
bounded domain, subject to different types of Dirichlet boundary conditions:
\begin{itemize}
\item[(C1)] $u=\alpha$, $v=\beta$,
\item[(C2)] $u=v=+\infty $ and
\item[(C3)] $u=+\infty$, $v=\alpha $ on $\partial D$, where $\alpha ,\beta >0$.
\end{itemize}
Under several
hypotheses on the parameters $p,q,r,s$, they showed the existence and
nonexistence, uniqueness and nonuniqueness of positive solutions. We mention
that the proofs in \cite{g1} were based on the method of sub and
super- solutions and the maximum principle. We remark that numerous works
treating nonlinear elliptic systems adopted many techniques employed in the
study of scalar equations, namely, the method of sub and super- solutions,
variational method, topology degree, fixed point index theory, see
\cite{a1,d1,d2,g2,g3}  for more details and references therein.

 In the present article, we consider a $C^{1,1}$-domain $D$ in
$\mathbb{R}^n(n\geq 3)$ with compact boundary $\partial D$. We fix two
nontrivial nonnegative continuous functions $\varphi $ and $\psi $ on
$\partial D$ and we will deal with the existence and the asymptotic behaviour
of bounded solutions (in the sense of distributions) to the nonlinear
elliptic system
\begin{equation}
\begin{gathered}
\Delta u=\lambda f(.,u,v)\quad \text{in }D, \\
\Delta v=\mu g(.,u,v)\quad \text{in }D, \\
u\big|_{\partial D}=a\varphi ,\quad v\big|_{\partial D}=b\psi , \\
\lim_{| x| \to +\infty }u(x)=\alpha , \quad
\lim_{| x| \to +\infty}v(x)=\beta \quad\text{(if $D$ is unbounded)},
\end{gathered}  \label{ePab}
\end{equation}
where the nonnegative constants $a,b,\alpha $ and $\beta $ are such that
$a+\alpha >0$, $b+\beta >0$.

 For this aim, we will use a fixed point argument to give two
existence results for problem \eqref{ePab}. We are essentially
inspired by the work \cite{g3}.

  Hereinafter, we denote by $H_{D}\varphi $ the bounded
continuous solution of the Dirichlet problem
\begin{equation} \label{e1.1}
\begin{gathered}
\Delta u=0 \quad \text{in }D, \\
u=\varphi \quad\text{on }\partial D, \\
\lim_{| x| \to +\infty }u(x)=0,\quad\text{if $D$ is unbounded},
\end{gathered}
\end{equation}
where $\varphi $ is a nontrivial nonnegative continuous function on
 $\partial D$. Morerover, we denote
\begin{equation} \label{e1.2}
h=1-H_{D}1
\end{equation}
and we remark that $h=0$ when $D$ is bounded.

 For a nonnegative measurable function $f$, we denote by $Vf$  the
potential function defined in $D$ by
\[
Vf(x)=\int_{D}G_{D}(x,y)f(y)dy,
\]
 where $G_{D}$ is the Green function of the Laplace operator
$\Delta $ in $D$ with Dirichlet conditions.

 Throughout this article, we fix a nontrivial nonnegative continuous
function $\Phi $ on $\partial D$ and we will use combinations of the
following hypotheses
\begin{itemize}
\item[(H1)]  $f$ and $g$ are nonnegative measurable functions on
$D\times (0,\infty )\times (0,\infty )$ such that
for each $x\in D$ the function
$(u,v)\mapsto (f(x,u,v),g(x,u,v))$ is continuous on
$(0,\infty )\times (0,\infty )$.

\item[(H2)] For all $0<u\leq u_1,0<v\leq v_1$ and $x\in D$,
\[
f(x,u,v)\leq f(x,u_1,v_1), \quad g(x,u,v)\leq g(x,u_1,v_1).
\]

\item[(H3)]  For all $c_1,c_2>0$, the
functions $f(.,c_1,c_2)$ and $g(.,c_1,c_2)$
are in $K^{\infty }(D)$.

\item[(H4)] For $\omega :=aH_{D}\varphi +\alpha h$ and
$\theta :=bH_{D}\psi +\beta h$, we have
\begin{gather}
\lambda _0 =\inf_{x\in D}\frac{\omega (x)}{Vf(.,\omega ,\theta )(x)}>0, 
\label{e1.3}\\
\mu _0 =\inf_{x\in D}\frac{\theta (x)}{Vg(.,\omega ,\theta )(x)}>0.
\label{e1.4}
\end{gather}

\item[(H5)] For all $0\leq u\leq u_1,0\leq v\leq v_1$ and $x\in D$,
\[
f(x,u_1,v_1)\leq f(x,u,v)\quad \text{and}\quad
g(x,u_1,v_1)\leq g(x,u,v).
\]

\item[(H6)] For $h_0=H_{D}\Phi $. The functions
$x\mapsto \widetilde{p}(x):=\frac{f(x,h_0(x),h_0(x))}{h_0(x)}$ and
$x\mapsto \widetilde{q}(x):=\frac{g(x,h_0(x),h_0(x))}{h_0(x)}$ belong to
$K^{\infty}(D)$.

\end{itemize}

\begin{remark} \label{rmk1} \rm
Let $\tau (x):=\delta (x)$ if $D$ is bounded and
$\tau (x):=\frac{\delta (x)}{(1+|x| )^{n-1}}$ if $D$ is unbounded.
Note that under hypothesis (H5) the condition:
``For all $c_1,c_2>0$,
\[
\frac{f(x,c_1\tau (x),c_2\tau (x))}{\tau (x)}\quad\text{and}\quad
\frac{g(x,c_1\tau (x),c_2\tau (x))}{\tau (x)}
\]
belong to $K^{\infty }(D)$''
implies  (H6). Indeed, from \cite{a3,z1}, there exists $c>0$ such
that for each $x\in D,h_0(x)\geq c\tau (x)$. Using (H5), we
obtain that $\frac{f(x,h_0(x),h_0(x))}{h_0(x)}\leq \frac{f(x,c\tau (x)
,c\tau (x))}{c\tau (x)}\in K^{\infty }(D)$.
Similarly, we obtain that
$\frac{g(x,h_0(x),h_0(x))}{h_0(x)}\in K^{\infty }(D)$ and so
(H6) is satisfied.
\end{remark}

 Our paper is organized as follows. In Section 2, we give the first
existence result concerning problem \eqref{ePab}. More precisely
we prove the following result.

\begin{theorem} \label{thm1}
Assume that {\rm (H1)--(H4)} are satisfied. Then
for each $\lambda \in [0,\lambda _0)$ and
$\mu \in [ 0,\mu _0)$,  problem \eqref{ePab} has a positive
continuous bounded solution $(u,v)$ satisfying on $D$
\begin{gather*}
(1-\frac{\lambda }{\lambda _0})\omega (x)\leq
u(x)\leq \omega (x)\\
(1-\frac{\mu }{\mu _0})\theta (x)\leq v(
x)\leq \theta (x).
\end{gather*}
\end{theorem}

As a consequence of Theorem \ref{thm1}, we will prove the following result.

\begin{corollary} \label{coro1}
Let $\xi _1,\xi _2:(0,+\infty )\to (0,+\infty)$ be two continuous functions.
 Assume that {\rm (H1)--(H4)} hold. Then for each
$\lambda \in [0,\lambda _0)$ and $\mu \in [0,\mu _0)$, the  problem
\begin{equation} \label{eQab}
\begin{gathered}
\Delta u+\xi _1(u)| \nabla u| ^{2}=\lambda f(.,u,v)\quad \text{in }D, \\
\Delta v+\xi _2(v)| \nabla v| ^{2}=\mu g(.,u,v)\quad \text{in }D, \\
u\big|_{\partial D}=a\varphi ,\quad v\big|_{\partial D}=b\psi , \\
\lim_{| x| \to +\infty }u(x)=\alpha , \quad
\lim_{| x| \to +\infty }v(x)=\beta \quad \text{(if $D$ is unbounded)}.
\end{gathered}
\end{equation}
has a positive continuous bounded solution $(u,v)$.
\end{corollary}

 Section 3 is dedicated to the second existence result for
system \eqref{ePab} for $a=b=1$ and $\lambda =\mu =1$. So for a
fixed nontrivial nonnegative continuous function $\Phi $ on $\partial D$, we
prove the second result of this work.

\begin{theorem} \label{thm2}
Assume {\rm (H1), (H5), (H6)} are satisfied. Then there exists a constant
 $c>1$ such that if $\varphi \geq c\Phi $ and
$\psi \geq c\Phi $ on $\partial D$, problem \eqref{ePab}, with $a=1$ and $b=1$,
has a positive continuous solution $(u,v)$. Moreover, for each $x\in D$,
$(u,v)$ satisfies
\begin{gather*}
\alpha h(x)+H_{D}\Phi (x)\leq u(x)
\leq \alpha h(x)+H_{D}\varphi \\
\beta h(x)+H_{D}\Phi (x)\leq v(x)\leq
\beta h(x)+H_{D}\psi .
\end{gather*}
\end{theorem}

 In the remainder of this section we will recall some notation and
results needed in the rest of this paper.

 $\mathcal{B}(D)$ is the set of Borel measurable functions in $D$ and
$\mathcal{C}_0(D)$ is the set of continuous ones vanishing
continuously on $\partial D\cup \{ \infty\} $. The exponent $+$
means that only the nonnegative functions are considered.

  We note that $\mathcal{C}(\overline{D}\cup \{\infty \} )$ and
$\mathcal{C}(\overline{D}\cup \{ \infty \})\times
\mathcal{C}(\overline{D}\cup \{ \infty \} )$  are two
Banach spaces endowed with uniform norm $\| u\| _{\infty}
=\sup_{x\in \overline{D}\cup \{ \infty \} }| u(x)| $ and
$\| (u,v)\|_{\infty }=\max (\| u\| _{\infty },\|v\| _{\infty })$,
respectively.

 If $f\in L_{\rm loc}^{1}(D)$ and $Vf\in L_{\rm loc}^{1}(D) $,
then we have $\Delta (Vf)=-f$  in $D$ (in the sense of distributions)
see \cite{c1}.

\begin{definition}[\cite{b1,m1}] \label{def1} \rm
A Borel measurable function $p$ in $D$ belongs to the class $K^{\infty
}(D)$ if $p$ satisfies
\begin{equation} \label{e1.5}
\lim_{\alpha \to 0}\Big(\sup_{x\in D}\int_{D\cap
B(x,\alpha )}\dfrac{\rho (y)}{\rho (x)}
G_{D}(x,y)|p(y)|dy\Big)=0,
\end{equation}
and
\begin{equation} \label{e1.6}
\lim_{M\to +\infty }\Big(\sup_{x\in D}\int_{D\cap
(| y| \geq M)}\dfrac{\rho (y)}{\rho (
x)}G_{D}(x,y)|p(y)|dy\Big)=0\quad \text{(if $D$ is unbounded),}
\end{equation}
where $\rho (x)=\min (1,\delta (x))$ and
$\delta (x)$ is the Euclidean distance between $x$ and $\partial D$.
\end{definition}

\begin{proposition} \label{prop1}
Let $p$ be a nonnegative function in $K^{\infty }(D)$, then
\begin{itemize}
\item[(i)] The function $x\mapsto \frac{\rho (x)}{1+| x| ^{n-1}}p(x)\in L^{1}(D)$.

\item[(ii)] $\alpha _{p}=\sup_{x,y\in D}\int_{D}
\frac{G_{D}(x,z)G_{D}(z,y)}{G_{D}(x,y)}p(z)dz<\infty$.

\item[(iii)] For any nonnegative superharmonic function $h$
in $D$ we have
\begin{equation} \label{e1.7}
\int_{D}G_{D}(x,y)h(y)p(y)dy\leq
\alpha _{p}h(x),\forall x\in D.
\end{equation}

\item[(iv)] The potential $Vp$ $\in \mathcal{C}_0(D)$.


\item[(v)] If $h_0$ is a positive harmonic function in $D$,
continuous and bounded in $\overline{D}$, then the family of functions
\[
\mathfrak{F}_{p}=\Big\{ \int_{D}G_{D}(.,y)h_0(y)
v(y)dy:| v| \leq p\Big\}
\]
is relatively compact in $\mathcal{C}_0(D)$.
\end{itemize}
\end{proposition}

\begin{proof}
These properties were proved in \cite{m1} for $\mathcal{C}^{1,1}$-bounded
domains in $\mathbb{R}^n$ and in \cite{b1,t2} for $\mathcal{C}^{1,1}$-unbounded
domains with compact boundary.
\end{proof}


\section{Proof of Theorem \ref{thm1}}

 In this section, we are concerned with the first existence result
for the system \eqref{ePab}. More precisely, we will give proofs
of Theorem \ref{thm1} and Corollary \ref{coro1}. Moreover, we will give some examples to
illustrate Theorem \ref{thm1}.

\begin{proof}[Proof of Theorem \ref{thm1}]
We shall use a fixed point argument. Let $\lambda _0,\mu _0$ be the
constants given by \eqref{e1.3} and \eqref{e1.4}.
Let $\lambda \in [0,\lambda _0)$ and $\mu \in [0,\mu_0)$. Recall that
$\omega =aH_{D}\varphi +\alpha h$ and $\theta =bH_{D}\psi +\beta h$.
Consider the non-empty closed convex set $\Lambda $ given by
\[
\Lambda =\Big\{ (u,v)\in \mathcal{C}(\overline{D}\cup
\{ \infty \} )\times \mathcal{C}(\overline{D}\cup
\{ \infty \} ):(1-\frac{\lambda }{\lambda _0}
)\omega \leq u\leq \omega ,\;
(1-\frac{\mu }{\mu _0}) \theta \leq v\leq \theta \Big\} .
\]
Let $T$ be the integral operator defined on $\Lambda $ by
\begin{align*}
T(u,v)
&=(\omega -\lambda \int_{D}G_{D}(.,y)
f(y,u(y),v(y))dy,\theta -\mu \int_{D}G_{D}(.,y)g(y,u(y),v(y))dy)\\
&=(T_1(u,v),T_2(u,v)).
\end{align*}
 We shall prove that the family $T(\Lambda )$ is
relatively compact in
$\mathcal{C}(\overline{D}\cup \{ \infty \} )\times
\mathcal{C}(\overline{D}\cup \{ \infty\} )$.
Let $(u,v)\in \Lambda $. It is obvious to
see that $T_1(u,v)\leq \omega $ and $T_2(u,v)\leq \theta $.
Then for each $x\in \overline{D}\cup \{ \infty \}$,
\begin{gather*}
\| T_1(u,v)\| _{\infty }\leq \|
\omega \| _{\infty }\leq \alpha +a\| \varphi \|_{\infty }:=c_1,\\
\| T_2(u,v)\| _{\infty }\leq \|
\theta \| _{\infty }\leq \beta +b\| \psi \|
_{\infty }:=c_2.
\end{gather*}
So
\[
\| T(u,v)\| _{\infty }\leq \max (c_1,c_2).
\]
Hence $T(\Lambda )$ is uniformly bounded.

 Next, by  hypotheses (H2) and (H3), it follows that for each
$(u,v)\in \Lambda $,
\begin{gather} \label{e2.1}
f(.,u,v)\leq f(.,c_1,c_2)=:q_1\in K^{\infty }(D),\\
\label{e2.2}
g(.,u,v)\leq g(.,c_1,c_2)=:q_2\in K^{\infty }(D).
\end{gather}
Therefore,
\begin{gather*}
\mathcal{A}_1:=\Big\{ \int_{D}G_{D}(.,y)f(y,u(y)
,v(y))dy:(u,v)\in \Lambda \Big\} \subseteq
\mathfrak{F}_{q_{_1}},
\\
\mathcal{A}_2:=\Big\{ \int_{D}G_{D}(.,y)g(y,u(y)
,v(y))dy:(u,v)\in \Lambda \Big\} \subseteq
\mathfrak{F}_{q_{_2}}.
\end{gather*}
Now, by Proposition \ref{prop1} (v), the families $\mathfrak{F}_{q_{_1}}$ and
$\mathfrak{F}_{q_{_2}}$ are relatively compact in $\mathcal{C}_0(D)$.
Therefore $\mathcal{A}_1$ and $\mathcal{A}_2$ are equicontinuous in
$\overline{D}\cup \{ \infty \} $. Now, since the functions
$\omega $ and $\theta $ belong to
$\mathcal{C}(\overline{D}\cup \{ \infty \} )$, we deduce that
$T_1(\Lambda )$ and $T_2(\Lambda )$ are equicontinuous in
$\overline{D}\cup \{ \infty \} $. Hence, $T(\Lambda )$ is equicontinuous in
$\overline{D}\cup \{ \infty \} $. Using Arzela-Ascoli theorem, we
obtain that $T(\Lambda )$ is relatively compact in
$\mathcal{C} (\overline{D}\cup \{ \infty \} )\times \mathcal{C}
(\overline{D}\cup \{ \infty \} )$.

 Now, we claim that the operator $T$ maps $\Lambda $ to itself.
Indeed, since $T(\Lambda )$ is equicontinuous on
$\overline{D} \cup \{ \infty \} $, it follows that for each
$(u,v) \in \Lambda ,T(u,v)\in $ $\mathcal{C}(\overline{D}\cup
\{ \infty \} )\times \mathcal{C}(\overline{D}\cup
\{ \infty \} )$. On the other hand, using hypothesis
(H2), we conclude that for each $x\in D$,
\[
T_1(u,v)(x)\geq \omega (x)-\lambda \int_{D}G_{D}(x.,y)f(y,\omega (y),\theta
(y))dy.
\]
So by \eqref{e1.3}, it follows that
\begin{equation} \label{e2.3}
T_1(u,v)(x)\geq (1-\frac{\lambda }{\lambda _0})\omega (x).
\end{equation}
Similarly, we have
\begin{equation} \label{e2.4}
T_2(u,v)(x)\geq (1-\frac{\mu }{\mu _0})\theta (x).
\end{equation}
Then, by \eqref{e2.3} and \eqref{e2.4}, we deduce by that
$T(\Lambda)\subset \Lambda $.

 Next, let us prove that $T$ is a continuous mapping in the
supremum norm. Let $\{ (u_{k},v_{k})\} _{k}$ be a
sequence in $\Lambda $ which converges uniformly to a function
$(u,v)$ in $\Lambda $. Then, for each $x\in D$, we have
\[
| T_1(u_{k},v_{k})(x)-T_1(u,v)(x)|
\leq \int_{D}G_{D}(x,y) | f(y,u_{k}(y),v_{k}(y))-f(y,u(y),v(y))| dy.
\]
On the other hand, by  (H2), we have
\[
| f(y,u_{k}(y),v_{k}(y))-f(y,u(y),v(y))| \leq 2f(y,c_1,c_2)
=2q_1(y)\in K^{\infty }(D).
\]
Since, by Proposition \ref{prop1} (iv), the function $Vq_1$ is bounded, we deduce
by  (H1) and the dominated convergence theorem
that for all $x\in D$,
\[
T_1(u_{k},v_{k})(x)\to T_1(u,v)(x)\quad \text{as }k\to +\infty .
\]
Similarly,
\[
T_2(u_{k},v_{k})(x)\to T_2(
u,v)(x)\quad \text{as }k\to +\infty .
\]
Therefore,
\[
T(u_{k},v_{k})(x)\to T(u,v)
(x)\quad \text{as }k\to +\infty .
\]
As $T(\Lambda )$ is relatively compact in $\mathcal{C}(
\overline{D}\cup \{ \infty \} )\times \mathcal{C}(
\overline{D}\cup \{ \infty \} )$, we conclude that the
pointwise convergence implies the uniform convergence; that is,
\[
\| T(u_{k},v_{k})-T(u,v)\|_{u}\to 0\quad \text{as }k\to +\infty .
\]
Hence $T$ is a compact mapping on from $\Lambda $ to itself.
By the Schauder fixed point theorem, there exists $(u,v)\in \Lambda $ such that
$T(u,v)=(u,v)$. That is,
\begin{gather} \label{e2.5}
u(x)=w(x)-\lambda \int_{D}G_{D}(x,y)f(y,u(y),v(y))dy,\\
 \label{e2.6}
v(x)=\theta (x)-\mu \int_{D}G_{D}(x,y)f(
y,u(y),v(y))dy.
\end{gather}
Now, let us prove that $(u,v)$ is a solution of the problem
\eqref{ePab}. Since $q_1,q_2\in K^{\infty }(D)$,
it follows by Proposition \ref{prop1} $(i)$, that
$q_1,q_2\in L_{\rm loc}^{1}(D)$. Using \eqref{e2.1} and \eqref{e2.2},
we deduce that $f(.,u,v),g(.,u,v)\in L_{\rm loc}^{1}(D)$ and
$Vf(.,u,v),Vg(.,u,v)\in \mathcal{C}_0(D)$. Thus applying
$\Delta $ on both sides of \eqref{e2.5} and \eqref{e2.6} respectively, we
obtain that $(u,v)$ satisfies the  elliptic system (in
the sense of distributions)
\begin{gather*}
\Delta u = \lambda f(.,u,v)\quad \text{in }D, \\
\Delta v = \mu g(.,u,v)\quad \text{in }D.
\end{gather*}
Moreover, since the functions $Vf(.,u,v)$ and $Vg(.,u,v)$ are in
 $\mathcal{C}_0(D)$, we conclude that
\begin{gather*}
\lim_{x\to z\in \partial D}u(x)=a\varphi (z), \quad
\lim_{| x| \to \infty }u(x)=\alpha ,\\
\lim_{x\to z\in \partial D}v(x)=b\psi (z),\quad
\lim_{| x| \to \infty }v(x)=\beta .
\end{gather*}
This completes the proof.
\end{proof}

\begin{proof}[Proof of Corollary \ref{coro1}]
Let $i\in \{ 1,2\} $ and 
$\rho _{i}(t) =\int_0^{t}\exp (\int_0^{s}\xi _{i}(r)dr)ds$. 
Then $\rho _{i}$ is a $\mathcal{C}^{2}-$ diffeomorphism from 
$(0,+\infty )$ to itself. Put $u_1=\rho _1(u)$ and 
$v_1=\rho _2(v)$. Then $(u_1,v_1)$ satisfies
\begin{equation} \label{e2.7}
\begin{gathered}
\Delta u_1=\lambda \rho _1'(\rho _1^{-1}(u_1))f(.,\rho _1^{-1}(u_1),\rho
_2^{-1}(v_1))\quad \text{in }D, \\
\Delta v_1=\mu \rho _2'(\rho _2^{-1}(v_1))g(.,\rho _1^{-1}(u_1),\rho _2^{-1}(
v_1))\quad \text{in } D, \\
u_{_1/_{\partial D}}=\rho _1(a\varphi ),\quad
v_{_1/_{\partial D}}=\rho _2(b\psi )\\
\lim_{|x| \to +\infty }u_1(x)=\rho _1(\alpha ),\quad
\lim_{|x| \to +\infty }v_1(x)=\rho _2(\beta )\quad \text{(if $D$ is unbounded)}.
\end{gathered}
\end{equation}
Put
\begin{gather*}
F(.,u_1,v_1):=\rho _1'(\rho_1^{-1}(u_1))f(.,\rho _1^{-1}(u_1),
 \rho _2^{-1}(v_1)),\\
G(.,u_1,v_1):=\rho _2'(\rho_2^{-1}(v_1))g(.,\rho _1^{-1}(u_1)
,\rho _2^{-1}(v_1)).
\end{gather*}
Then $F$ and $G$ satisfy  (H1)--(H4). Thus by Theorem \ref{thm1},
problem \eqref{e2.7} admits a positive bounded solution
$(u_1,v_1)$. So, it is easy to verify that
$(\rho_1^{-1}(u_1),\rho _2^{-1}(v_1))$ is
a positive bounded solution of the problem $(Q_{a,b})$. This
completes the proof.
\end{proof}

\begin{example} \label{examp1} \rm
Let $D$ be a $\mathcal{C}^{1,1}$-bounded domain in $\mathbb{R}^n(n\geq 3)$. 
Let $\varphi $ and $\psi $ be two nontrivial
nonnegative continuous functions on $\partial D$. Let $p,q$ be two
nonnegative functions in $L^{k}(D),k>\frac{n}{2}$ and suppose
that $m_1,m_2<1-\frac{n}{k}$. Let $r_1,r_2,s_1,s_2>0$. Then, the
 system
\begin{gather*}
\Delta u=\lambda \frac{p(x)}{(\delta (x))^{m_1}}
u^{r_1}v^{s_1}\text{\ in }D, \\
\Delta v=\mu \frac{q(x)}{(\delta (x))^{m_2}}
u^{r_2}v^{s_2}\text{ \ in }D, \quad
u\big|_{\partial D}=\varphi , \\
v\big|_{\partial D}=\psi .
\end{gather*}
has a positive bounded continuous solution. Indeed, from 
\cite[Proposition 2.3]{t1}, the functions 
$p_1(x):=p(x)/(\delta (x))^{m_1}$, and 
$q_1(x):=q(x)/(\delta (x))^{m_2}$ belong to $K^{\infty }(D)$ 
and so  (H3) is satisfied.
 From \cite[ Proposition 2.7(iii)]{t1}, there exists a constant 
$c>0$ such that we have for each $x\in D$
\[
Vp_1(x)\leq c\delta (x).
\]
So, for $f(x,u,v)=p_1(x)u^{r_1}v^{s_1}$, we have
\[
Vf(.,H_{D}\varphi ,H_{D}\psi )(x)\leq
c\| \varphi\| _{\infty }^{r_1}\| \psi \| _{\infty}^{s_1}\delta (x).
\]
In addition, since the function $\varphi $ is nontrivial nonnegative on
$\partial D$, then there exists a constant $c_1>0$ such that we have on $D$
\[
H_{D}\varphi (x)\geq c_1\delta (x).
\]
Thus,
\[
\lambda _0=\inf_{x\in D}\frac{H_{D}\varphi (x)
}{Vf(.,H_{D}\varphi ,H_{D}\psi )(x)}>\frac{c_1}{c\|
\varphi \| _{\infty }^{r_1}\| \psi \|
_{\infty }^{s_1}}>0.
\]
 Similarly, we prove that $\mu _0>0$ and so
assumption (H4) is satisfied.
\end{example}

\begin{example} \label{examp2} \rm
Let $D=\overline{B(0,1)}^{c}$ be the exterior of the unit ball
in $\mathbb{R}^n$ $(n\geq 3)$. Suppose that $\gamma ,\sigma $ $>n$. 
Let $r_1,r_2,s_1,s_2>0$. Then, the  problem
\begin{gather*}
\Delta u=\lambda \frac{1}{| x| ^{\sigma -\gamma }(
| x| -1)^{\gamma }}u^{r_1}v^{s_1}\quad \text{in }D,\\
\Delta v=\mu \frac{1}{| x| ^{\sigma -\gamma }(
| x| -1)^{\gamma }}u^{r_2}v^{s_2}\quad \text{in }D, \\
u\big|_{\partial D}=\varphi ,\quad v\big|_{\partial D}=\psi , \\
\lim_{| x| \to +\infty }u(x)=\alpha ,\quad 
\lim_{| x| \to +\infty}v(x)=\beta.
\end{gather*}
has a positive continuous solution. In fact, from \cite{b1} the
functions $p(x):=\frac{1}{| x| ^{\sigma }}$
and $q(x):=\frac{1}{| x| ^{\gamma }}$ belong
to $K^{\infty }(D)$. Morerover, from  \cite[Proposition 3.5]{b1},
there exists a constant $c>0$ such that
\[
Vp(x)\leq c\frac{| x| -1}{| x| ^{n-1}}.
\]
So, for $f(x,u,v):=p(x)u^{r_1}v^{s_1}$, $\omega =H_{D}\varphi +\alpha h$
and $\theta =H_{D}\psi +\beta h$, there exists a constant $c_1>0$ such that
\[
Vf(.,\omega ,\theta )(x)\leq c_1\frac{| x|-1}{| x| ^{n-1}}.
\]
On the other hand, from \cite[page 258]{a3} there exists a constant $c_2>0$
such that  on $D$ we have
\[
\omega (x)\geq c_2\frac{| x| -1}{| x| ^{n-1}}.
\]
It follows that $\lambda _0=\inf_{x\in D}\frac{\omega (
x)}{Vf(.,\omega ,\theta )(x)}\geq \dfrac{c_2}{c_1}>0$.
Similarly, we prove that
$\mu _0=\inf_{x\in D}\frac{\theta (x)}{Vg(.,\omega ,\theta )(x)}>0$, for
$g(x,u,v):=q(x)u^{r_2}v^{s_2}$. Thus, the assumption (H4)
is satisfied.
\end{example}

\section{Proof of Theorem \ref{thm2}}

 In this section, we will be interested in \eqref{ePab} with
 $a=b=\lambda =\mu =1$; that is, we will study the problem
\begin{equation} \label{eP11}
\begin{gathered}
\Delta u=f(.,u,v)\quad \text{in }D, \\
\Delta v=g(.,u,v)\quad \text{in }D, \\
u\big|_{\partial D}=\varphi ,\quad v\big|_{\partial D}=\psi , \\
\lim_{| x| \to +\infty }u(x)=\alpha ,\quad
\lim_{| x| \to +\infty }v(x)=\beta \quad\text{(if $D$ is unbounded)}.
\end{gathered}
\end{equation}
where $\alpha ,\beta \geq 0$.  So, we recall that $\Phi $ is a
fixed nontrivial nonnegative continuous function on $\partial D$ and we put 
$h_0=H_{D}\Phi $.  First, we give the proof of Theorem \ref{thm2}. Then we give an
example of application to illustrate Theorem \ref{thm2}.

\begin{proof}[Proof of Theorem \ref{thm2}]
Let $\alpha _{\widetilde{p}}$ and $\alpha _{\widetilde{q}}$ be the constants
defined in Proposition \ref{prop1} (ii) associated to the functions 
$\widetilde{p}$ and $\widetilde{q}$ given in hypothesis (H6). 
Put $c=1+\alpha _{\widetilde{p}}+\alpha _{\widetilde{q}}$ and suppose that
\[
\varphi (x)\geq ch_0(x),\quad \psi (x)\geq ch_0(x),\forall x\in \partial D.
\]
Then, by the maximum principle, we have
\begin{equation} \label{e3.1}
H_{D}\varphi (x)\geq ch_0(x),\quad
H_{D}\psi (x)\geq ch_0(x),\forall x\in D.
\end{equation}
Now, let $\Gamma $ be the non-empty closed bounded convex set given by
\[
\Gamma =\{ (w,z)\in \mathcal{C}(\overline{D}\cup
\{ \infty \} )\times \mathcal{C}(\overline{D}\cup
\{ \infty \} ):h_0\leq w\leq H_{D}\varphi,\; h_0\leq z\leq H_{D}\psi \} .
\]
Consider the operator $L$ defined on $\Gamma $ by
\[
L(w,z)=(L_1(w,z),L_2(w,z)),
\]
where
\begin{gather*}
L_1(w,z)(x)=H_{D}\varphi (x)
-\int_{D}G_{D}(x,y)f(y,w(y)+\alpha h(
y),z(y)+\beta h(y))dy,
\\
L_2(w,z)(x)=H_{D}\psi (x)-\int_{D}G_{D}(x,y)g(y,w(y)
+\alpha h(y),z(y)+\beta h(y))dy.
\end{gather*}
We shall prove that the operator $L$ admits a fixed point in $\Gamma $.
Let $(w,z)\in \Gamma $. Then using  (H5) and (H6), it follows that
\begin{gather}  \label{e3.2}
f(.,w+\alpha h,z+\beta h)(x)\leq f(
.,h_0,h_0)(x)=h_0(x)\widetilde{p}(x),
\\  \label{e3.3}
g(.,w+\alpha h,z+\beta h)(x)\leq g(
.,h_0,h_0)(x)=h_0(x)\widetilde{q}(x).
\end{gather}
Now, using \eqref{e3.2}, \eqref{e3.3} and (H6), it follows that
\begin{gather*}
\mathcal{G}_1:=\Big\{ \int_{D}G_{D}(.,y)f(y,(w+\alpha
h)(y),(z+\beta h)(y))dy:(
w,z)\in \Gamma \Big\} \subseteq \mathfrak{F}_{\widetilde{p}},
\\
\mathcal{G}_2:=\Big\{ \int_{D}G_{D}(.,y)g(y,(w+\alpha
h)(y),(z+\beta h)(y))dy:(
w,z)\in \Gamma \Big\} \subseteq \mathfrak{F}_{\widetilde{q}}.
\end{gather*}
By Proposition \ref{prop1} (v), $\mathcal{G}_1$ and $\mathcal{G}_2$ are
equicontinuous in $\overline{D}\cup \{ \infty \} $. Thus, as in
the proof of Theorem \ref{thm1}, we conclude that $L(\Gamma )$ is
equicontinuous in $\overline{D}\cup \{ \infty \} $. Moreover,
$L(\Gamma )$ is uniformly bounded.

 By Ascoli-Arzela theorem, we conclude that the family 
$L(\Gamma )$ is relatively compact in 
$\mathcal{C}(\overline{D}\cup \{ \infty \} )\times \mathcal{C}(\overline{D}
\cup \{ \infty \} )$. Next, let us prove that $L$ maps
 $\Gamma $ to itself. Let $(w,z)\in \Gamma $, since
 $L(\Gamma )$ is relatively compact in 
$\mathcal{C}(\overline{D} \cup \{ \infty \} )
\times \mathcal{C}(\overline{D} \cup \{ \infty \} )$, it follows that 
$L(w,z) \in \mathcal{C}(\overline{D}\cup \{ \infty \} )
\times \mathcal{C}(\overline{D}\cup \{ \infty \} )$.
On the other hand, by Proposition \ref{prop1} (iii) and \eqref{e3.2},
we obtain
\begin{equation} \label{e3.4}
Vf(.,w+\alpha h,z+\beta h)(x)\leq \alpha _{
\widetilde{p}}h_0(x).
\end{equation}
So, by \eqref{e3.1} and \eqref{e3.4}, we obtain
\begin{equation} \label{e3.5}
L_1(w,z)(x)\geq (1+\alpha _{\widetilde{q}})h_0(x)
\geq h_0(x)>0.
\end{equation}
Similarly, we prove that
\begin{equation} \label{e3.6}
L_2(w,z)(x)\geq h_0(x)>0.
\end{equation}
Thus, $L(\Gamma )\subset \Gamma$.

 Now, we proceed as in the proof of Theorem \ref{thm1} and using hypothesis 
(H5), we prove the continuity of the operator $L$ in the
supremum norm. Thus, we conclude that $L$ is a compact operator mapping from
$\Gamma $ to itself. Hence, the Schauder fixed point theorem ensures the
existence of $(w,z)\in \Gamma $ such that
\begin{gather*}
w(x)=H_{D}\varphi (x)-\int_{D}G_{D}(x,y)f(y,w(y)+\alpha h(y),z(y)
+\beta h(y))dy,
\\
z(x)=H_{D}\psi (x)-\int_{D}G_{D}(x,y)
g(y,w(y)+\alpha h(y),z(y)+\beta h(y))dy.
\end{gather*}
Put $u:=w+\alpha h$ and $v:=z+\alpha h$. It is easy to verify that
 $(u,v)$ is a positive continuous bounded solution of \eqref{eP11}.
\end{proof}

\begin{example} \label{examp3} \rm
Let $D$ be a $\mathcal{C}^{1,1}-$ domain in $\mathbb{R}^n(n\geq 3)$ 
with compact boundary. Let $h_0$ be a positive
harmonic bounded function in $D$ and $\tau $ be the function defined in
Remark 1 and $r_1,r_2,s_1,s_2>0$. Suppose that $p$ and $q$ are two
nonnegative functions such that 
$\widetilde{p}(x):=(\tau (x))^{-(r_1+s_1+1)}p(x)$
and $\widetilde{q}(x):=(\tau (x))^{-(r_2+s_2+1)}q(x)$ belong to
 $K^{\infty }(D)$. Then there exists a constant $c>1$ such that 
if $\varphi \geq ch_0$ and $\psi \geq ch_0$ on $\partial D$, 
the  system
\begin{gather*}
\Delta u=p(x)u^{-r_1}v^{-s_1}\quad \text{in }D, \\
\Delta v=q(x)u^{-r_2}v^{-s_2}\quad \text{in }D, \\
u\big|_{\partial D}=\varphi ,\quad v\big|_{\partial D}=\psi , \\
\lim_{| x| \to +\infty }u(x)=\alpha ,\quad 
\lim_{| x| \to +\infty }v(x)=\beta .
\end{gather*}
has a positive bounded continuous solution on $D$. 
\end{example}

\subsection*{Acknowledgements}
The author is greatly indebted to
Professor Habib Ma\^{a}gli for many helpful suggestions.

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\end{document}