\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 $$\begin{gathered} \Delta u=\lambda p(x)f_1(v)\quad \text{in }D, \\ \Delta v=\mu q(x)f_2(u)\quad \text{in }D, \\ u\big|_{\partial D}=a\varphi ,\quad v\big|_{\partial D}=b\psi , \\ \lim_{| x| \to +\infty }u(x)=\alpha ,\quad \lim_{| x| \to +\infty }v(x)=\beta \quad\text{(if D is unbounded)}, \end{gathered} \label{eP}$$ 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 $$\begin{gathered} \Delta u=\lambda f(.,u,v)\quad \text{in }D, \\ \Delta v=\mu g(.,u,v)\quad \text{in }D, \\ u\big|_{\partial D}=a\varphi ,\quad v\big|_{\partial D}=b\psi , \\ \lim_{| x| \to +\infty }u(x)=\alpha , \quad \lim_{| x| \to +\infty}v(x)=\beta \quad\text{(if D is unbounded)}, \end{gathered} \label{ePab}$$ 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 $$\label{e1.1} \begin{gathered} \Delta u=0 \quad \text{in }D, \\ u=\varphi \quad\text{on }\partial D, \\ \lim_{| x| \to +\infty }u(x)=0,\quad\text{if D is unbounded}, \end{gathered}$$ where $\varphi$ is a nontrivial nonnegative continuous function on $\partial D$. Morerover, we denote $$\label{e1.2} h=1-H_{D}1$$ 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 $00$, 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 $$\label{eQab} \begin{gathered} \Delta u+\xi _1(u)| \nabla u| ^{2}=\lambda f(.,u,v)\quad \text{in }D, \\ \Delta v+\xi _2(v)| \nabla v| ^{2}=\mu g(.,u,v)\quad \text{in }D, \\ u\big|_{\partial D}=a\varphi ,\quad v\big|_{\partial D}=b\psi , \\ \lim_{| x| \to +\infty }u(x)=\alpha , \quad \lim_{| x| \to +\infty }v(x)=\beta \quad \text{(if D is unbounded)}. \end{gathered}$$ 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 $$\label{e1.5} \lim_{\alpha \to 0}\Big(\sup_{x\in D}\int_{D\cap B(x,\alpha )}\dfrac{\rho (y)}{\rho (x)} G_{D}(x,y)|p(y)|dy\Big)=0,$$ and $$\label{e1.6} \lim_{M\to +\infty }\Big(\sup_{x\in D}\int_{D\cap (| y| \geq M)}\dfrac{\rho (y)}{\rho ( x)}G_{D}(x,y)|p(y)|dy\Big)=0\quad \text{(if D is unbounded),}$$ 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 $$\label{e1.7} \int_{D}G_{D}(x,y)h(y)p(y)dy\leq \alpha _{p}h(x),\forall x\in D.$$ \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 $$\label{e2.3} T_1(u,v)(x)\geq (1-\frac{\lambda }{\lambda _0})\omega (x).$$ Similarly, we have $$\label{e2.4} T_2(u,v)(x)\geq (1-\frac{\mu }{\mu _0})\theta (x).$$ 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 $$\label{e2.7} \begin{gathered} \Delta u_1=\lambda \rho _1'(\rho _1^{-1}(u_1))f(.,\rho _1^{-1}(u_1),\rho _2^{-1}(v_1))\quad \text{in }D, \\ \Delta v_1=\mu \rho _2'(\rho _2^{-1}(v_1))g(.,\rho _1^{-1}(u_1),\rho _2^{-1}( v_1))\quad \text{in } D, \\ u_{_1/_{\partial D}}=\rho _1(a\varphi ),\quad v_{_1/_{\partial D}}=\rho _2(b\psi )\\ \lim_{|x| \to +\infty }u_1(x)=\rho _1(\alpha ),\quad \lim_{|x| \to +\infty }v_1(x)=\rho _2(\beta )\quad \text{(if D is unbounded)}. \end{gathered}$$ 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 $$\label{eP11} \begin{gathered} \Delta u=f(.,u,v)\quad \text{in }D, \\ \Delta v=g(.,u,v)\quad \text{in }D, \\ u\big|_{\partial D}=\varphi ,\quad v\big|_{\partial D}=\psi , \\ \lim_{| x| \to +\infty }u(x)=\alpha ,\quad \lim_{| x| \to +\infty }v(x)=\beta \quad\text{(if D is unbounded)}. \end{gathered}$$ 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 $$\label{e3.1} H_{D}\varphi (x)\geq ch_0(x),\quad H_{D}\psi (x)\geq ch_0(x),\forall x\in D.$$ 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 $$\label{e3.4} Vf(.,w+\alpha h,z+\beta h)(x)\leq \alpha _{ \widetilde{p}}h_0(x).$$ So, by \eqref{e3.1} and \eqref{e3.4}, we obtain $$\label{e3.5} L_1(w,z)(x)\geq (1+\alpha _{\widetilde{q}})h_0(x) \geq h_0(x)>0.$$ Similarly, we prove that $$\label{e3.6} L_2(w,z)(x)\geq h_0(x)>0.$$ 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. \begin{thebibliography}{99} \bibitem{a1} G.\ A.\ Afrouzi, E.\ Graily; \emph{Uniqueness results for a class of semilinear elliptic systems}, International Journal of Nonlinear Science, Vol.9, No.3 (2010) 259-264. \bibitem{a2} D.\ Aronson, M. G. Crandall, L. A. Peletier; \emph{Stabilization of soltuions of a degenerate nonlinear diffusion problem}, Nonlinear.\ Anal. 6 (1982) 1001-1022. \bibitem{a3} D. Armitage, S. 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