\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 $$\label{e1} \begin{gathered} \Delta u=F(u,v), \\ \Delta v=G(u,v), \end{gathered}$$ in $\mathbb{R}^{n}$ have been extensively treated recently. Lair and Wood \cite{l1} studied the semiliniar elliptic system $$\label{e2} \begin{gathered} \Delta u=p(|x|)v^{\alpha }, \\ \Delta v=q(|x|)u^{\beta }, \end{gathered}$$ 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 $$\label{e3} \int_0^{\infty }tp(t)dt<\infty ,\quad \int_0^{\infty }tq(t)dt<\infty ,$$ and the existence of large solutions if $$\label{e4} \int_0^{\infty }tp(t)dt=\infty ,\quad \int_0^{\infty }tq(t)dt=\infty .$$ 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 $$\label{e5} \begin{gathered} \Delta u=p(|x|)f(v), \\ \Delta v=q(|x|)g(u), \end{gathered}$$ 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} $$\label{e} \int_1^{\infty}\frac{1}{\sqrt{\int_0^{s}f(t)dt}}ds<\infty, \quad \int_1^{\infty}\frac{1}{\sqrt{\int_0^{s}g(t)dt}}ds<\infty .$$ \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 $$\label{Pab} \begin{gathered} \Delta u=\lambda p(x)f(v),\quad\text{in }D \\ \Delta v=\mu q(x)g(u),\quad\text{in }D \\ u\big|_{\partial D}=a\varphi ,\quad v\big|_{\partial D}=b\psi ,\\ \lim_{|x|\to +\infty }\frac{u(x)}{\ln |x| }=\alpha ,\quad \lim_{|x|\to +\infty }\frac{v(x)}{\ln | x| }=\beta , \end{gathered}$$ 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 $$\label{e7} \begin{gathered} \Delta w=0\quad \text{in }D, \\ w\big|_{\partial D}=\varphi ,\quad \lim_{| x| \to +\infty }\frac{w( x) }{\ln | x| }=0, \end{gathered}$$ 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 $$\label{e8} \widetilde{\varphi }:=aH_D\varphi +\alpha h,\quad \widetilde{\psi } :=bH_D\psi +\beta h,$$ 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 $$\label{e9} Vf(x)=\int_DG_D(x,y)f(y)dy.$$ 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 $$\label{e10} h( x) =2\pi \lim_{| y| \to +\infty }G_D( x,y)$$ \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 $$\label{e14} V(wq)(x)=\int_DG_D(x,y)w( y) |q(y)|dy\leq \alpha _{q}w(x),\forall x\in D.$$ \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 $$\label{e15} f( v_{k}) \leq f( \widetilde{\psi }), \quad g( u_{k}) \leq g( \widetilde{\varphi }) .$$ 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 $$\label{e16} \begin{gathered} u=\widetilde{\varphi }-\lambda V(pf( v) ) \\ v=\widetilde{\psi }-\mu V(qg( u) ). \end{gathered}$$ 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 $$\label{e19} V(qg(w+\alpha h))\leq V(qg(h_0)).$$ 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 $$\label{e20} v\geq h_0>0.$$ So, $Tw\leq H_D\varphi$. On the other hand, by \eqref{e20}, the monotonicity of $f$ and Proposition \ref{prop4} (i), we obtain $$\label{e21} V(pf(v))\leq V(pf(h_0))=V(p_0h_0)\leq \alpha _{p_0}h_0.$$ 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 $p_0:=p\frac{( H_D\Phi ( x) ) ^{-\theta }}{H_D\Phi ( x) }\in K(D)$. 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