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

\AtBeginDocument{{\noindent\small
\emph{Electronic Journal of Differential Equations},
Vol. 2011 (2011), No. 79, pp. 1--9.\newline
ISSN: 1072-6691. URL: http://ejde.math.txstate.edu or http://ejde.math.unt.edu
\newline ftp ejde.math.txstate.edu}
\thanks{\copyright 2011 Texas State University - San Marcos.}
\vspace{9mm}}

\begin{document}
\title[\hfilneg EJDE-2011/79\hfil Existence and asymptotic behaviour]
{Existence and asymptotic behaviour of positive solutions for semilinear
 elliptic systems in the Euclidean plane}

\author[A. Ghanmi, F. Toumi\hfil EJDE-2011/79\hfilneg]
{Abdeljabbar Ghanmi, Faten Toumi}  % in alphabetical order

\address{Abdeljabbar Ghanmi\newline
 D\'{e}partement de Math\'{e}matiques, 
 Facult\'{e} des Sciences de Tunis, 
 Campus Universitaire, 2092 Tunis, Tunisia}
\email{ghanmisl@yahoo.fr}

\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 March 31, 2011. Published June 20, 2011.}
\subjclass[2000]{34B27, 35J45, 45M20}
\keywords{Green function; semilinear elliptic systems;
positive solution}

\begin{abstract}
 We study the semilinear elliptic system
 $$
 \Delta u=\lambda p(x)f(v),\Delta v=\lambda q(x)g(u),
 $$
 in an unbounded domain $D$ in $ \mathbb{R}^2$ with compact
 boundary subject to some Dirichlet conditions. We give existence
 results according to the monotonicity of the nonnegative
 continuous functions $f$ and $g$. The potentials $p$ and $q$ are
 nonnegative and required to satisfy some hypotheses related on a
 Kato class.
\end{abstract}

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


\section{Introduction}

Semilinear elliptic systems of the form
\begin{equation} \label{e1}
\begin{gathered}
\Delta u=F(u,v), \\
\Delta v=G(u,v),
\end{gathered}
\end{equation}
in $\mathbb{R}^{n}$ have been extensively treated recently.
Lair and Wood \cite{l1}  studied  the  semiliniar elliptic system
\begin{equation} \label{e2}
\begin{gathered}
\Delta u=p(|x|)v^{\alpha }, \\
\Delta v=q(|x|)u^{\beta },
\end{gathered}
\end{equation}
in $\mathbb{R}^{n}$ ($n\geq 3$).
They showed the existence of entire positive radial solutions. More
precisely, for the sublinear case where
$\alpha ,\beta \in (0,1)$, they proved the existence
of bounded solutions of \eqref{e2} if $p$ and $q$
satisfy the decay conditions
\begin{equation} \label{e3}
\int_0^{\infty }tp(t)dt<\infty ,\quad
\int_0^{\infty }tq(t)dt<\infty ,
\end{equation}
and the existence of large solutions  if
\begin{equation} \label{e4}
\int_0^{\infty }tp(t)dt=\infty ,\quad
\int_0^{\infty }tq(t)dt=\infty .
\end{equation}
For the superlinear case, where $\alpha ,\beta \in ( 1,+\infty ) $.
The authors proved the existence of an entire large positive solution
of problem \eqref{e2}, provided that the functions $p$ and $q$
satisfy \eqref{e3}.

Peng and Song \cite{p1} considered the semilinear
elliptic system
\begin{equation} \label{e5}
\begin{gathered}
\Delta u=p(|x|)f(v), \\
\Delta v=q(|x|)g(u),
\end{gathered}
\end{equation}
in $\mathbb{R}^{n}$ ($n\geq 3$),
under the assumptions:
\begin{itemize}
\item[(A1)] The functions $p$ and $q$ satisfy condition \eqref{e3}.

\item[(A2)] The functions $f$ and $g$ are positive
nondecreasing, satisfying the Keller-Osserman condition
\cite{k1,o1}
\begin{equation} \label{e}
\int_1^{\infty}\frac{1}{\sqrt{\int_0^{s}f(t)dt}}ds<\infty, \quad
\int_1^{\infty}\frac{1}{\sqrt{\int_0^{s}g(t)dt}}ds<\infty .
\end{equation}

\item[(A3)] The functions $f$ and $g$ are convex on $[0,+\infty )$.

\end{itemize}
The authors proved the existence of an entire large positive
solution of problem \eqref{e5}. We remark that Peng and Song
extended their results to the superlinear case in \cite{l1}.

Cirstea and Radulescu \cite{c2} gave existence
results for  system \eqref{e5}. They adopted the
assumptions (A1)-(A2) and the assumption
\begin{itemize}
\item[(A3')] $f,g\in C^{1}[0,+\infty )$,
$f(0)=g(0)=0$, $\lim_{t\to +\infty }\inf \frac{f(t)}{g(t)}>0$,
\end{itemize}
to prove the existence of entire large positive solutions.


Recently, Ghanmi et al \cite{g1} considered the
 semilinear elliptic system
\begin{gather*}
\Delta u=\lambda p(x)f(v), \\
\Delta v=\mu q(x)g(u),
\end{gather*}
in a domain $D$ of $\mathbb{R}^{n}$ ($n\geq 3$) with compact
boundary subject to some Dirichlet conditions.
They assumed that the functions $f$, $g$ are nonnegative continuous
monotone on $(0,\infty )$, the nonnegative
potentials $p$ and $q$ are required to satisfy some hypotheses
related to a Kato class \cite{b1,m1}. In particular,
in the case where $f$ and $g$ are nondecreasing and for given positive
constants $\lambda _0$, $\mu _0$, they showed that for each
$\lambda \in [ 0,\lambda _0)$ and $\mu \in [ 0,\mu _0)$, there exists a
positive bounded solution $( u,v) $ satisfying the boundary
conditions
\[
u\big|_{\partial ^{\infty }D}
=\varphi \mathbf{1}_{\partial D}+a \mathbf{1}_{\{\infty \}},\quad
v\big|_{\partial ^{\infty }D}=\psi \mathbf{1}_{\partial D}
+b \mathbf{1}_{\{\infty\}}
\]
where $\varphi $ and $\psi $ are
nontrivial nonnegative continuous functions on $\partial D$.

In this article, we consider an unbounded domain $D$ in
$\mathbb{R}^2$ with compact non\-empty boundary $\partial D$
consisting of finitely many Jordan curves. We are concerned
with the  semilinear elliptic system
\begin{equation} \label{Pab}
\begin{gathered}
\Delta  u=\lambda p(x)f(v),\quad\text{in }D \\
\Delta  v=\mu q(x)g(u),\quad\text{in }D \\
u\big|_{\partial D}=a\varphi ,\quad
v\big|_{\partial D}=b\psi ,\\
\lim_{|x|\to +\infty }\frac{u(x)}{\ln |x| }=\alpha ,\quad
\lim_{|x|\to +\infty }\frac{v(x)}{\ln | x| }=\beta ,
\end{gathered}
\end{equation}
where $a,b,\alpha $ and $\beta $ are nonnegative constants such
that $ a+\alpha >0$, $b+\beta >0$. The functions $\varphi $ and
$\psi $ are nontrivial nonnegative and continuous on $\partial D$.
We will give two existence results according to the monotoniciy
of the functions $f$  and $g$.

Throughout this paper, we denote by $H_D\varphi $ the bounded
continuous solution of the Dirichlet problem
\begin{equation} \label{e7}
\begin{gathered}
\Delta w=0\quad \text{in }D, \\
w\big|_{\partial D}=\varphi ,\quad
\lim_{| x| \to +\infty }\frac{w( x) }{\ln | x| }=0,
\end{gathered}
\end{equation}
where $\varphi $ is a nonnegative continuous function on
$\partial D$.

 We remark that the solution $H_D\varphi$  of \eqref{e7} belongs
to $\mathcal{C}(\overline{D}\cup \{ \infty\} )$ and satisfies
$\lim_{| x| \to +\infty }H_D\varphi ( x) =C>0$
(See \cite[p. 427]{d1}).

 For the sake of simplicity we denote
\begin{equation} \label{e8}
\widetilde{\varphi }:=aH_D\varphi +\alpha h,\quad
\widetilde{\psi } :=bH_D\psi +\beta h,
\end{equation}
where $h$ is the harmonic function defined  by \eqref{e10}, below.

The outline of this paper is as follows. In section 2, we will give
some notions related to the Green function $G_D$ of the domain $D$
associated to the Laplace operator $\Delta $ and properties of the
functions belonging to a some Kato class $K(D)$ (See \cite{m1,t1}).
In section 3, we will first give an example and
then we give the proof of the existence result for the problem
\eqref{Pab}. More precisely, we adopt in section 3 the following
hypotheses
\begin{itemize}
\item[(H1)]  The functions $f$, $g:[0,\infty )\to[ 0,\infty )$
are nondecreasing and continuous.

\item[(H2)] The functions $\widetilde{p}:=pf(\widetilde{\psi})$ and
$\widetilde{q}:=qg(\widetilde{\varphi })$ belong to the Kato class
$K(D)$.

\item[(H3)] $\lambda _0:=\inf_{x\in D}
\frac{\widetilde{\varphi }(x)}{V(\widetilde{p})(x)}>0$ and
$\mu_0:=\inf_{x\in D}\frac{\widetilde{\psi }(x)}{V(\widetilde{q})(x)}
>0 $, where $V$ is the Green kernel defined by \eqref{e9} below.

\end{itemize}

 We prove the following result.

\begin{theorem} \label{thm1}
Assume  {\rm (H1)--(H3)},
then for each $\lambda \in [ 0,\lambda _0)$ and
$\mu \in [ 0,\mu _0)$,  problem \eqref{Pab} has a positive
continuous solution $(u,v)$ satisfying, on $D$,
\begin{gather*}
(1-\frac{\lambda }{\lambda _0})[aH_D\varphi +\alpha h]\leq u\leq
aH_D\varphi +\alpha h, \\
(1-\frac{\mu }{\mu _0})[bH_D\psi +\beta h]\leq v\leq bH_D\psi +\beta
h.
\end{gather*}

\end{theorem}

In the last section, we fix $\lambda =\mu =1$ and a nontrivial
nonnegative continuous function $\Phi $ on $\partial D$ and we
note $h_0=H_D\Phi $.
Then we give an existence result for problem \eqref{Pab} with
$a=1$ and $b=1$,
under the following hypotheses:
\begin{itemize}
\item[(H4)]  The functions $f, g:[0,\infty )\to [ 0,\infty )$
are nonincreasing and continuous.

\item[(H5)]  The functions $p_0:=p\frac{f(h_0)}{h_0}$
and $q_0:=q\frac{g(h_0)}{h_0}$ belong to the Kato class $K(D)$.

\end{itemize}
More precisely, we obtain the following result.

\begin{theorem} \label{thm2}
Assume {\rm (H4)--(H5)}, then there exists a constant $c>1$ such
that if $\varphi \geq c\Phi$ and $\psi \geq c\Phi$ on $\partial D$,
then  problem \eqref{Pab} with $a=1$ and $b=1$ has a positive
continuous solution $(u,v) $ satisfying, on $D$,
\begin{gather*}
h_0 +\alpha h \leq u \leq H_D\varphi  +\alpha h, \\
h_0 +\beta h \leq v \leq H_D\psi  +\beta h.
\end{gather*}
\end{theorem}

Note that this result generalizes those by Athreya \cite{a2}
 and Toumi and Zeddini \cite{t1}, stated for semilinear
elliptic equations.

\section{Preliminaries}

In the reminder of this paper, we will adopt the following notation.

$\mathcal{C}(\overline{D}\cup \{ \infty\} )
=\{f\in \mathcal{C}(\overline{D}):\lim_{|x| \to +\infty }f(x)
 \text{ exists}\}$.
 We note that $\mathcal{C}(\overline{D}\cup \{
\infty \} )$ is a Banach space endowed with the uniform norm
$\Vert f\Vert _{\infty }=\sup_{x\in D}|f(x)|$.

 For $x\in D$, we denote by $\delta _D(x)$ the
distance from $x$ to $\partial D$, by
$\rho _D(x):=min(1,\delta _D(x))$
and by $\lambda _D( x) :=\delta _D( x) (1+\delta _D( x) )$.

Let $f$ and $g$ be two positive functions on a
set $S$. We denote $f\sim g$ if there exists a constant $c>0$ such that
\[
\frac{1}{c}g(x)\leq f(x)\leq cg(x)\quad \text{for all }x\in S.
\]

 For a Borel measurable and nonnegative function $f $ on $D$,
we denote by $Vf$  the Green kernel of $f$ defined on $D$ by
\begin{equation} \label{e9}
Vf(x)=\int_DG_D(x,y)f(y)dy.
\end{equation}
We recall that 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 distributional sense
(See \cite[p 52]{c1}).

 We note that the Green function satisfies
\[
G_D( x,y) \sim \ln (1+\frac{\lambda _D(
x) \lambda _D( y) }{| x-y|^2})
\]
on $D^2$ (See \cite{m2}).


\begin{definition} \label{def1} \rm
A Borel measurable function $q$ in $D$ belongs to the Kato class
$K(D) $ if $q$ satisfies
\[
\lim_{\alpha \to 0}\Big( \sup_{x\in D}\int_{D\cap
B(x,\alpha )}\frac{\rho _D( y) }{\rho _D( x) }
G_D(x,y)|q(y)|dy\Big) =0,
\]
and
\[
\lim_{M\to +\infty }\Big( \sup_{x\in D}\int_{D\cap
(| y| \geq M)}\frac{\rho _D( y) }{\rho
_D( x) }G_D(x,y)|q(y)|dy\Big) =0.
\]
\end{definition}

\begin{example} \label{exmp1} \rm
Let $p>1$ and $\gamma ,\theta \in \mathbb{R}$ such that
$\gamma <2-\frac{2}{p}<\theta $. Then using the H\"{o}lder
inequality and the same arguments as
in  \cite[Proposition 3.4]{t1}  it follows that for each
$f\in L^{p}( D) $, the function defined in $D$ by
$\frac{f(x) }{(1+\vert x| )^{\theta -\gamma }( \delta
_D( x) ) ^{\gamma }}$ belongs to $K(D) $.
\end{example}

 Throughout this article, $h$ will be the function defined
on $D$ by
\begin{equation} \label{e10}
h( x) =2\pi \lim_{| y| \to +\infty }G_D( x,y)
\end{equation}

\begin{proposition}[\cite{u1}] \label{prop3}
The function $h$ defined by \eqref{e10} is harmonic positive in $D$
and satisfies
\[
\lim_{x\to z\in \partial D}h( x) =0, \quad
\lim_{| x| \to +\infty }\frac{h(x) }{\ln | x| }=1.
\]
\end{proposition}

 In the sequel, we use the notation
\begin{gather} \label{e11}
\| q\| _D=\sup_{x\in D}\int_D\frac{
\rho _D( y) }{\rho _D( x) }G_D(x,y)|q(y)|dy,\\
\label{e12}
\alpha _{q}=\sup_{x,y\in D}\int_D\frac{
G_D(x,z)G_D(z,y)}{G_D(x,y)}|q(z)|dz.
\end{gather}
It is shown in \cite{t1}, that
if $q\in K(D)$, then
$\|q\| _D<\infty$, and $\alpha _{q}\sim \| q\| _D$. %(13)
For stating our results we need the following result.

\begin{proposition}[\cite{t1}] \label{prop4}
 Let $q$ in $K( D) $, then the following assertions hold

\begin{itemize}
\item[(i)] For any nonnegative superharmonic function $w$ in $D$,
we have
\begin{equation} \label{e14}
V(wq)(x)=\int_DG_D(x,y)w( y) |q(y)|dy\leq \alpha
_{q}w(x),\forall x\in D.
\end{equation}

\item[(ii)] The potential
$Vq \in \mathcal{C}(\overline{D}\cup{{\infty}})$ and
$\lim_{x \to z\in \partial D}Vq(x)=0$.

\item[(iii)] Let $\Lambda _{q}=\{ p\in K( D) :|p| \leq q\}$.
Then the family of functions
\[
\mathfrak{F}_{q}=\{ \int_DG_D( .,y)
h_0( y) p( y) dy:p\in \Lambda _{q}\}
\]
is uniformly bounded and equicontinuous in $\overline{D}\cup
\{ \infty \} $. Consequently, it is relatively compact
in $\mathcal{C}( \overline{D}\cup \{ \infty \}) $.
\end{itemize}
\end{proposition}

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

Before stating the proof, we give an
example where (H2)  and (H3) are satisfied.

\begin{example} \label{exmp2} \rm
Let $D=\overline{B(0,1)}^{c}$ be the exterior of the unit closed disk.
Let $ \alpha =b=1$ and $\beta =a=0$. Assume that $\psi \geq c_1>0$
on $\partial D$. Let $p_1,\widetilde{q}$ be nonnegative functions
in $K(D)$ such that the function $p:=p_1h$ is in $K(D)$.
Then using the fact that the function $f$ is
continuous and $H_D\psi $ is bounded on $D$ we obtain that
$\widetilde{p}:=pf(H_D\psi )\in K(D)$ and so the hypothesis
(H2)  is satisfied. Now, since
$\widetilde{p_1}:=p_1f(H_D\psi )\in K(D)$ then by
Proposition \ref{prop4} (i) we obtain
\[
V(\widetilde{p})\leq \alpha_{\widetilde{p_1}}h.
\]
Therefore, for each $x\in D$
\[
\frac{h(x)}{V(\widetilde{p})( x) }\geq \frac{1}{\alpha _{p_1}}
>0;
\]
that is, $\lambda _0>0$.  On the other hand we have
\[
\frac{H_D\psi (x)}{V(\widetilde{q})( x) }\geq \frac{c_1}{
\alpha _{\widetilde{q}}}>0,
\]
which yields $\mu _0>0$. Thus the hypothesis (H3) is satisfied.
\end{example}

\begin{proof}[Proof of Theorem \ref{thm1}]
Let $\lambda \in [0,\lambda _0) $ and
$\mu \in [0,\mu _0) $. We intend to prove that the problem
\eqref{Pab} has a positive continuous solution.
To this aim we define the
sequences $( u_{k}) _{k\in\mathbb{N}}$ and
$( v_{k}) _{k\in\mathbb{N}}$ as follows:
\begin{gather*}
v_0=\widetilde{\psi }, \\
u_{k}=\widetilde{\varphi }-\lambda V( pf(v_{k})) , \\
v_{k+1}=\widetilde{\psi }-\mu V( qg(u_{k})) ,
\end{gather*}
where $\widetilde{\varphi }$ and $\widetilde{\psi }$ are defined
by \eqref{e8}. We shall prove by induction that for each
$k\in \mathbb{N}$,
\begin{gather*}
0<\big( 1-\frac{\lambda }{\lambda _0}\big) \widetilde{\varphi }\leq
u_{k}\leq u_{k+1}\leq \widetilde{\varphi }, \\
0<\big( 1-\frac{\mu }{\mu _0}\big) \widetilde{\psi }\leq v_{k+1}\leq
v_{k}\leq \widetilde{\psi }.
\end{gather*}
First, using hypothesis (H3) we obtain, on $D$,
\[
\lambda _0V( pf(\widetilde{\psi })) \leq \widetilde{\varphi }.
\]
Then by the monotonicity of  $f$, it follows that
\[
\widetilde{\varphi }\geq u_0=\widetilde{\varphi }-\lambda V( pf(
\widetilde{\psi })) \geq \big(1-\frac{\lambda }{\lambda _0}\big) \widetilde{\varphi }>0.
\]
So
\[
v_1-v_0=-\mu V( qg(u_0) )\leq 0
\]
and consequently
\[
u_1-u_0=\lambda V( p[f(v_0) -f(v_1)])\geq 0.
\]
Moreover, the hypothesis (H3) yields
\[
\mu _0V( qg(\widetilde{\varphi })) \leq \widetilde{\psi }.
\]
Then using the fact that the function $g$ is nondecreasing we have
\[
v_1\geq \widetilde{\psi }-\mu V( qg(\widetilde{\varphi })) \geq
\big( 1-\frac{\mu }{\mu _0}\big) \widetilde{\psi }>0.
\]
In addition, we have $u_1\leq \widetilde{\varphi }$, then it follows that
\[
u_0\leq u_1\leq \widetilde{\varphi }\quad\text{and}\quad
v_1\leq v_0\leq \widetilde{\psi }.
\]
Suppose that
\[
u_{k}\leq u_{k+1}\leq \widetilde{\varphi }\quad\text{and}\quad
( 1-\frac{\mu }{\mu _0}) \widetilde{\psi }\leq v_{k+1}\leq v_{k}.
\]
Therefore,
\begin{gather*}
v_{k+2}-v_{k+1}=\mu V( q[g(u_{k}) -g(u_{k+1})])\leq 0,\\
u_{k+2}-u_{k+1}=\lambda V( p[f(v_{k+1}) -f(v_{k+2})])\geq 0.
\end{gather*}
Furthermore, since $u_{k+1}\leq \widetilde{\varphi }$ the
monotonicity of the function $g$ yields
\[
v_{k+2}\geq \widetilde{\psi }-\lambda V( qg(\widetilde{\varphi }
)) \geq ( 1-\frac{\mu }{\mu _0}) \widetilde{\psi }>0.
\]
Thus, we obtain
\[
u_{k+1}\leq u_{k+2}\leq \widetilde{\varphi }\quad\text{and}\quad
\big( 1-\frac{\mu }{\mu _0}\big) \widetilde{\psi }
\leq v_{k+2}\leq v_{k+1}.
\]
Hence, the sequences $( u_{k})$ and $( v_{k})$ converge respectively
to two functions $u$ and $v$ satisfying
\[
0<\big(1-\frac{\lambda }{\lambda _0}\big) \widetilde{\varphi }\leq
u\leq \widetilde{\varphi }, \quad
0<\big( 1-\frac{\mu }{\mu _0}\big) \widetilde{\psi }\leq v\leq
\widetilde{\psi }.
\]
Furthermore, for each $k\in \mathbb{N}$,
we have
\begin{equation} \label{e15}
f( v_{k}) \leq f( \widetilde{\psi }), \quad
g( u_{k}) \leq g( \widetilde{\varphi }) .
\end{equation}
Therefore, using hypothesis (H2) and Proposition \ref{prop4} (ii)
we deduce by Lebesgue's theorem that $V(pf( v_{k}) )$ and
$V(qg( u_{k}) )$ converge respectively to $V(pf( v) )$
and $V(qg( u) )$ as $k$ tends to infinity. Then, on $D$,
$(u,v)$ satisfies
\begin{equation} \label{e16}
\begin{gathered}
u=\widetilde{\varphi }-\lambda V(pf( v) ) \\
v=\widetilde{\psi }-\mu V(qg( u) ).
\end{gathered}
\end{equation}
Moreover, by \eqref{e16} and the monotonicity of the functions
$f$ and $g$ we obtain
$pf( v) \leq \widetilde{p}$ and $qg( u) \leq
\widetilde{q}$. So $pf( v) ,qg( u) \in K(D)$ and
consequently by Proposition \ref{prop4} (ii)  we have
$V( pf( v) ) ,V( qg( u) ) \in \mathcal{C}(\overline{D}\cup
\{ \infty \} ) $. Now using the fact that
the functions $\widetilde{\varphi }$ and $\widetilde{\psi }$
are continuous we conclude that $u$ and $v$ are continuous and
satisfy in the distributional sense $\Delta u=\lambda pf( v) $
and $\Delta v=\mu qg( u) $ in $D$. Now, since
$H_D\varphi =\varphi $ on $\partial
D,\lim_{x\to z\in \partial D}h(x)=0$, and
 $\lim_{x\to z\in \partial D}V( \widetilde{p}) (x)=0$,
we conclude that $\lim_{x\to z\in \partial D}u(x)=a\varphi
( z) $. By similar arguments we have
$\lim_{x\to z\in \partial D}v(x)=b\psi ( z) $. Furthermore,
by Proposition \ref{prop4} (ii) and Proposition \ref{prop3}, we have
$\lim_{|x| \to +\infty }\frac{1}{h( x) }V(pf(v) )=0$ and
$\lim_{| x| \to +\infty } \frac{1}{h( x) }V(qg( u) )=0$. Hence
$(u,v) $ is a continuous positive solution of the problem \eqref{Pab},
which completes the proof.
\end{proof}

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

In the sequel, we recall that $h_0=H_D\Phi$ is a fixed positive
harmonic function in $D$ and $h$ is the function defined by
\eqref{e10}.

\begin{proof}
Let $\alpha _{p_0}$ and $\alpha _{q_0}$ be the constants defined by
\eqref{e12} associated respectively to the functions $p_0$ and
$q_0$ given in the hypothesis (H5). Put
$c=1+\alpha _{p_0}+\alpha _{q_0}$ . Suppose that
\[
\varphi ( x) \geq c\Phi ( x) \quad\text{and}\quad
\psi (x) \geq c\Phi ( x),\quad \forall x\in \partial D.
\]
Then by the maximum principle it follows that for each $x\in D$
\begin{gather} \label{e17}
H_D\varphi ( x) \geq ch_0( x),\\
\label{e18}
H_D\psi ( x) \geq ch_0( x) .
\end{gather}
Consider the nonempty convex set $\Omega $ given by
\[
\Omega :=\{ w\in \mathcal{C}( \overline{D}\cup \{ \infty
\} ) :h_0\leq w\leq H_D\varphi \} .
\]
Let $T$ be the operator defined on $\Omega $ by
\[
Tw:=H_D\varphi -V(pf[\beta h+H_D\psi -V(qg(w+\alpha h))] ).
\]
We shall prove that the operator $T$ has a fixed point.
First, let us prove that the operator $T$ maps $\Omega $
to its self. Let $w\in \Omega $. Since
$w+\alpha h\geq h_0$, then from hypothesis (H4)
we deduce that
\begin{equation} \label{e19}
V(qg(w+\alpha h))\leq V(qg(h_0)).
\end{equation}
Therefore, using \eqref{e18} and \eqref{e19} we obtain
\begin{align*}
v&:=\beta h+H_D\psi -V(qg(w+\alpha h)\geq \beta h+H_D\psi
-V(q_0h_0)\\
&\geq \beta h+H_D\psi -\alpha _{q_0}h_0\geq \beta h+(
c-\alpha _{q_0}) h_0.
\end{align*}
This yields
\begin{equation} \label{e20}
v\geq h_0>0.
\end{equation}
So, $Tw\leq H_D\varphi$.
On the other hand, by \eqref{e20}, the monotonicity of $f$
and Proposition \ref{prop4} (i), we obtain
\begin{equation} \label{e21}
V(pf(v))\leq V(pf(h_0))=V(p_0h_0)\leq \alpha _{p_0}h_0.
\end{equation}
Then, by \eqref{e17} and \eqref{e21}, we have
\[
Tw\geq H_D\varphi -\alpha _{p_0}h_0\geq ( 1+\alpha
_{q_0}) h_0\geq h_0.
\]
Hence $T\Omega \subseteq \Omega $. Next, let us prove that the
set $T\Omega $ is relatively compact in
$\mathcal{C}( \overline{D}\cup \{ \infty\} ) $.
Let $w\in \Omega $, then by (H4), (H5) and using Proposition
\ref{prop4} (iii)
it follows that the family of functions
\[
\big\{ \int_DG( .,y) p(y)f[\beta h+H_D\psi
-V(qg(w+\alpha h))] ( y) dy:w\in \Omega \big\}
\]
is relatively compact in $\mathcal{C}( \overline{D}\cup
\{ \infty\} ) $. Since $H_D\varphi \in \mathcal{C}( \overline{D}
\cup \{ \infty\} ) $ we deduce that $T\Omega $ is
relatively compact in $\mathcal{C}( \overline{D}\cup \{ \infty\} ) $.

 Now we  prove the continuity of the operator $T$ in $
\Omega $ in the supremum norm. Let $( w_{k})$
 be a sequence in $\Omega $ which converges uniformly to a function
$w$ in $\Omega $. Then using  \eqref{e20} and the monotonocity of $f$ we
have, for each $x$ in $D$,
\begin{align*}
&p( x) | f( \beta h+H_D\psi -V(qg(w_{k}+\alpha
h))) ( x) -f( \beta h+H_D\psi -V(qg(w+\alpha
h))) ( x) |\\
& \leq 2f( h_0) p( x) \leq 2\|h_0\| _{\infty }p_0( x)
\end{align*}
Using the fact that $Vp_0$ is bounded, we conclude by the
continuity of $f$ and the dominated convergence theorem that
for all $x\in D$,
$Tw_{k}( x) \to Tw(x)$ as $k\to +\infty$.
Consequently, as $T\Omega $ is relatively compact in
$\mathcal{C}(\overline{D}\cup \{ \infty \} ) $, we deduce that the
pointwise convergence implies the uniform convergence; that is,
\[
\| Tw_{k}-Tw\| _{\infty }\quad\text{as } k\to +\infty
\]
Therefore, $T$ is a continuous mapping of $\Omega $ to itself.
So, since $ T\Omega $ is relatively compact in
$\mathcal{C}( \overline{D}\cup \{ \infty\} ) $,
it follows that $T$ is compact mapping on
$\Omega $. Thus, the Schauder fixed-point theorem yields the
existence of $ w\in \Omega $ such that
\[
w=H_D\varphi -V(pf[\beta h+H_D\psi -V(qg(w+\alpha h))] .
\]
Put $u(x)=w( x) +\alpha h( x) $ and $v(x)=\beta
h( x) +H_D\psi ( x) -V(qg(u))( x) $ for
$x\in D$. Then $( u,v) $ is a positive continuous solution of
\eqref{Pab} with $a=1, b=1$, for the same arguments as in the
proof of Theorem \ref{thm1}.
\end{proof}

\begin{example} \label{exmp3} \rm
 Let $D=\overline{B( 0,1) }^{c}$ be the exterior of the
unit closed disk, $0<\theta <1$ and $0<\gamma <1$. Let $p,q$ be two
nonnegative functions such that  the functions
$( \frac{| x| }{| x| -1}) ^{1+\theta }p(x)$ and
$( \frac{| x| }{| x|-1}) ^{1+\gamma }q(x)$ are in $K(D)$.
Suppose that the functions $\varphi$ and $\psi $ are nonnegative
continuous on $\partial D$. Then for a fixed nontrivial nonnegative
continuous function $\Phi$ in $\partial D$, there exists a constant
$c>1$ such that if $\varphi \geq c\Phi $ and $\psi \geq c\Phi $ on
$\partial D$, the  problem
\begin{gather*}
\Delta  u=p(x)v^{-\gamma },\quad\text{in }D \\
\Delta  v=q(x)u^{-\theta },\quad\text{in }D \\
u\big|_{\partial D}=\varphi ,\quad
v\big|_{\partial D}=\psi, \quad
\lim_{|x|\to +\infty }\frac{u(x)}{\ln \vert x \vert}=\alpha \geq
0,\quad
\lim_{|x|\to +\infty }\frac{v(x)}{\ln \vert x \vert}=\beta \geq 0,
\end{gather*}
has a positive continuous solution $(u,v)$ satisfying
\begin{gather*}
H_D\Phi ( x) +\alpha h( x) \leq u( x) \leq H_D\varphi ( x) +\alpha h( x) , \\
H_D\Phi ( x) +\beta h( x) \leq v( x) \leq H_D\psi ( x) +\beta h( x) ,
\end{gather*}
for each $ x\in D$.
Indeed, from \cite{a1} there exists $c_0>0$ such that for each
$x\in D$,
\[
c_0\frac{| x| -1}{| x| }\leq H_D\Phi ( x).
\]
It follows that $p_0:=p\frac{( H_D\Phi ( x)
) ^{-\theta }}{H_D\Phi ( x) }\in K(D)$. In a
similar way we have $q_0\in K(D)$. Thus the hypothesis
(H5) is satisfied.
\end{example}

\begin{thebibliography}{99}

\bibitem{a1} D. Armitage and S. Gardiner;
\emph{Classical Potential Theory},
Springer- Verlag 2001.

\bibitem{a2} S. Atherya;
\emph{On a singular Semilinear Elliptic Boundary Value
Problem and the Boundary Harnack Principle}, Potential Ana. 17, 293-301 $
( 2002) $.

\bibitem{b1} I. Bachar, H. M\^{a}agli and N. Zeddini;
\emph{Estimates on the Green Function and Existence of Positive
Solutions of Nonlinear Singular Elliptic
Equations}, Comm. Contemporary. Math, $Vol.5$, No.3 (2003) 401-434.

\bibitem{c1} K. L. Chung; Z. Zhao;
\emph{From Brownian Motion to Schr\"{o}dinger's
Equation}, Springer $(1995)$.

\bibitem{c2} F. C. Cirstea, V. D. Radulescu;
\emph{Entire solutions blowing up at
infinity for semilinear elliptic systems},
 J. Math. Pures. Appl. 81 (2002) 827-846.

\bibitem{d1} R. Dautray, J. L. Lions et al.;
\emph{Analyse math\'{e}matique et calcul
num\'{e}rique pour les sciences et les Thechniques, L'op\'{e}rateur de
Laplace}, Masson, 1987.

\bibitem{g1} A. Ghanmi, H. M\^{a}agli, S. Turki, N. Zeddini;
\emph{Existence of positive bounded solutions for some Nonlinear
elliptic systems}, J. Math. Annal . Appl. 352(2009), pp. 440-448.

\bibitem{k1} J. Keller;
\emph{On solutions to $\Delta u=f(u)$}, Comm. Pure and
Applied Math. 10, (1957), 503-510.

\bibitem{l1} A. V. Lair ; A. W. Wood;
\emph{Existence of entire large positive
solutions of semilinear elliptic systems},
J. Differential Equations. 164. No.2 (2000) 380-394.

\bibitem{m1} H. M\^{a}agli, M. Zribi;
\emph{On a new Kato class and singular
solutions of a nonlinear elliptic equation in bounded domains
of $\mathbb{R}^{n}$}. Positivity (2005)9:667--686.

\bibitem{m2} S. Masmoudi;
\emph{On the existence of positive solutions for some
nonlinear elliptic problems in unbounded domains in
$\mathbb{R}^2$}, Nonlinear Anal. 62 (3) (2005) 397-415.

\bibitem{o1} R. Osserman;
\emph{On the inequality $\Delta u\geq f(u)$}, Pacific J.
Math. 7, (1957), 1641-1647.

\bibitem{p1} Y. Peng, Y. Song;
\emph{Existence of entire large positive solutions of
a semilinear elliptic systems},
Appl. Math. Comput. 155 (2004), pp. 687-698.

\bibitem{t1} F. Toumi, N. Zeddini;
\emph{Existence of Positive Solutions for
Nonlinear Boundary Value Problems in Unbounded Domains}, Electron. J.
Differential Equations. Vol. 2005, No. 43, 1-14.

\bibitem{u1} U. Ufuktepe, Z. Zhao;
\emph{Positive Solutions of Non-linear Elliptic
Equations in the Euclidean Space}, Proc. Amer. Math. Soc. Vol.126, p.
3681-3692 (1998)

\end{thebibliography}

\end{document}