\documentclass[reqno]{amsart} \usepackage{hyperref} \AtBeginDocument{{\noindent\small \emph{Electronic Journal of Differential Equations}, Vol. 2012 (2012), No. 239, pp. 1--10.\newline ISSN: 1072-6691. URL: http://ejde.math.txstate.edu or http://ejde.math.unt.edu \newline ftp ejde.math.txstate.edu} \thanks{\copyright 2012 Texas State University - San Marcos.} \vspace{8mm}} \begin{document} \title[\hfilneg EJDE-2012/239\hfil Positive solutions] {Positive solutions for nonlinear elliptic systems} \author[A. Ben Dekhil\hfil EJDE-2012/239\hfilneg] {Adel Ben Dekhil} % in alphabetical order \address{Adel Ben Dekhil \newline D\'epartement de Math\'ematiques\\ Facult\'e des Sciences de Tunis \\ Universit\'e de Tunis El Manar, Campus Universitaire, 2092 Tunis, Tunisia} \email{Adel.Bendekhil@ipein.rnu.tn} \thanks{Submitted October 14, 2012. Published December 28, 2012.} \subjclass[2000]{35B08, 35B09, 35J47} \keywords{Semilinear elliptic systems; positive large solution; positive bounded solution} \begin{abstract} In this article, we study the existence of positive solutions for the system \begin{gather*} \Delta u=H(x,u,v),\\ \Delta v=K(x,u,v),\text{in }\mathbb{R}^n\; (n\geq 3), \end{gather*} where $H,K: \mathbb{R}^n\times[0,\infty)\times[0,\infty)\to[0,\infty)$ are continuous functions satisfying $H(x,u,v)\leq p_1(|x|)F(u+v)$ and $K(x,u,v)\leq q_1(|x|)G(u+v)$. In terms of the growth of the variable potential functions $p_1,q_1$ and the nonlinearities $F$ and $G$, we establish some sufficient conditions for the existence of positive continuous solutions for this system and we discuss whether these solutions are bounded or blow up at infinity. \end{abstract} \maketitle \numberwithin{equation}{section} \newtheorem{theorem}{Theorem}[section] \newtheorem{corollary}[theorem]{Corollary} \newtheorem{example}[theorem]{Example} \newtheorem{remark}[theorem]{Remark} \allowdisplaybreaks \section{Introduction} Semilinear elliptic systems of the form $$\label{S1} \begin{gathered} \Delta u=H(x,u,v),\quad \text{in } \mathbb{R}^n\;(n\geq 3) , \\ \Delta v=K(x,u,v),\quad \text{in } \mathbb{R}^n, \end{gathered}$$ have been studied intensively in the previous few years and various results concerning the existence and nonexistence of positive entire large or bounded solutions have been obtained. We refer the reader to \cite{AR, CR1, DZ, GMRZ, GR, AL1, LW2, LZZ, PS, Qi, WW, ZL, ZZ, ZSX} and their references for recent results concerning the existence and qualitative analysis of solutions of \eqref{S1}. The interest in systems of nonlinear stationary equations is motivated by applications to theory of Newtonian fluids and nonlinear optics. More precisely, coupled nonlinear stationary systems arise in the description of several physical phenomena such as the propagation of pulses in birefringent optical fibers and Kerr-like photorefractive media, see \cite{AA, Men}. When $H(x,u,v)=p(|x|)v^{\alpha}$, $K(x,u,v)=q(|x|)u^{\beta}$, $0< \alpha \leq \beta$, Lair and Wood in \cite{LW2} considered the existence and nonexistence of entire positive radial solutions to \eqref{S1} under the conditions of integrability or non integrability of the functions $r \to rp(r)$ and $r\to r q(r)$ on $(0,\infty)$. Their results were extended by C\^{i}rstea and R\u{a}dulescu \cite{CR1}, Wang and Wood \cite{WW}, Ghergu and R\u adulescu \cite{GR}, Peng and Song \cite{PS}, Ghanmi et al \cite{GMRZ}, Li et al \cite{LZZ}, Zhang \cite{ZZ}. In \cite{ZSX}, the authors considered the case where $H(x,u,v)=p(x)f(v)$, $K(x,u,v)=q(x)g(u)$ with $p,q$ nontrivial nonnegative continuous on $[0,\infty)$ and $f, g$ continuous , nondecreasing on $[0,\infty)$ that are positive on $(0, \infty)$. In the case where $p, q$ are radial and under the condition \begin{eqnarray} \int_1^{\infty}\Big(\int_0^t(f(s)+g(s))ds\Big)^{-1/2}\,dt=\infty\,, \end{eqnarray} they proved that \eqref{S1} has one positive solution $(u,v)$. Moreover, if $$\int_0^{\infty}sp(s)\,ds=\int_0^{\infty}sq(s)\,ds=\infty\,,$$ then every positive radial entire solution $(u,v)$ of \eqref{S1} is large (i.e. $\lim_{x \to \infty} u(x)=\lim_{x \to \infty}v(x)=\infty$). And if $$\int_0^{\infty}sp(s)\,ds<\infty \text{ and } \int_0^{\infty}sq(s)\,ds< \infty\,,$$ then every positive radial entire solution $(u,v)$ of $\eqref{S1}$ is bounded. By using the sub-supersolution method, they establish some conditions on $p$, $q$ in order to prove the existence of positive bounded solutions in the case where $p, q$ are non radial. Their results extend partially those of Zhang \cite{ZZ}, where the existence of entire positive radial large solutions or bounded ones for \eqref{S1} was studied under the condition $\int_1^{\infty}\frac{ds}{f(s)+g(s)}= \infty$. Their results do not cover the cases where \begin{gather*} H(x,u,v)=a_1(x)v^{\alpha_1}+b_1(x)u^{\gamma_1}+c_1(x)(u+v)^{\beta_1} +d_1(x)u^{\delta_1} v^{\lambda_1},\\ K(x,u,v)=a_2(x)v^{\alpha_2}+b_2(x)u^{\gamma_2}+c_2(x)(u+v)^{\beta_2} +d_2(x)u^{\delta_2} v^{\lambda_2}\,, \end{gather*} where $\alpha_i$, $\beta_i$, $\gamma_i$, $\delta_i$, $\lambda_i \in(0,\infty)$. Our aim in this paper is to extend the results in \cite{DZ} for the particular case of the Dirichlet laplacian and to extend those in \cite{ZSX} to a wider class of functions $H(x,u,v)$ and $K(x,u,v)$. More precisely, our results apply in particular to the previous examples of functions $H$, $K$ and to the case where $H(x,u,v)=p(x)f(u,v)$ , $K(x,u,v)=q(x)g(u,v)$ with $f$ and $g$ are nondecreasing with respect to first and the second variables. To this aim, we assume that $H$ and $K$ satisfy the following hypotheses: \begin{itemize} \item[(H1)] $H$, $K:\mathbb{R}^n\times[0,\infty)\times [0,\infty)\to[0,\infty)$ are continuous. \item[(H2)] There exist nonnegative functions $p_i$, $q_i$, $f_i$, $g_i$, $1\leq i\leq 2$, $F$ and $G$ satisfying for each $x\in\mathbb{R}^n$ and $(u,v)\in[0,\infty)\times[0,\infty)$, \begin{gather*} p_2(|x|)g_1(v)g_2(u)\leq H(x,u,v)\leq p_1(|x|)F(u+v),\\ q_2(|x|)f_1(u)f_2(v)\leq K(x,u,v)\leq q_1(|x|)G(u+v), \end{gather*} with $f_i$, $g_i$, $1\leq i\leq 2$, $F$, $G:[0,\infty)\to[0,\infty)$ are nondecreasing continuous, positive on $(0,\infty)$ and $p_i$, $q_i:[0,\infty)\to[0,\infty)$ are continuous. \item[(H3)] There exist $c>0$ such that $$\int_0^t\sqrt{\Lambda(s)}ds0, where for \alpha>0, $$\label{E2} \begin{gathered} L_\alpha(t)=\int_\alpha^t\frac{ds}{\sqrt{2(\mathcal{F}(s)+\mathcal{G}(s))}},\quad t \geq \alpha, \\ \Lambda(r)=\max_{s\in[0,r]}(p_1(s)+q_1(s)),\quad r\geq0\,,\\ \mathcal{F}(r)=\int_0^rF(s)ds\,, \quad \mathcal{G}(r)=\int_0^rG(s)ds\,,\quad r\geq 0. \end{gathered}$$ \end{itemize} We note that L_\alpha has an inverse function L_\alpha^{-1} from [\alpha, \infty) to [0,L_{\alpha}(\infty)), where L_{\alpha}(\infty)=\lim_{r\to \infty}L_{\alpha}(r) \in (0,\infty]. \begin{remark} \label{rmk1.1} \rm If L_\alpha(\infty)=\infty, then$$ \int_{\alpha}^\infty\frac{ds}{\sqrt{\mathcal{F}(s)}}= \int_{\alpha}^\infty\frac{ds}{\sqrt{\mathcal{G}(s)}}=\infty. $$\end{remark} \begin{remark} \label{rmk1.2} \rm By \cite{AL1}, we see that if \int_1^\infty\frac{ds}{\sqrt{\mathcal{F}(s)}}<\infty, then \int_1^\infty\frac{ds}{F(s)}<\infty. In other words, if \int_1^\infty\frac{ds}{F(s)}=\infty, then \int_1^\infty\frac{ds}{\sqrt{\mathcal{F}(s)}}=\infty. Conversely, if \int_1^\infty\frac{ds}{\sqrt{\mathcal{F}(s)}}=\infty, then \int_1^\infty\frac{ds}{F(s)}=\infty does not hold. For example, for \beta>0 and F(t)=2(1+t)(\ln(t+1))^{2\beta-1}(\ln(t+1)+\beta), we have \mathcal{F}(t)=(t+1)^2(\ln(t+1))^{2\beta}. So we can see that \int_1^\infty\frac{ds}{F(s)}=\infty if and only if 0<\beta\leq\frac{1}{2} and \int_1^\infty\frac{ds}{\sqrt{\mathcal{F}(s)}}=\infty if and only if \;0<\beta\leq1. \end{remark} To discuss the existence of positive radial solutions to these nonlinear systems, we study the system of nonlinear differential equations $$\label{M} \begin{gathered} \frac{1}{A}(A y')'=g(t,y,z), \quad \text{in } (0,\infty), \\ \frac{1}{B}(B z')'=f(t,y,z),\quad \text{ in } (0,\infty),\\ \lim_{t\to0^+}A(t)y'(t)=\lim_{t\to0^+}B(t)z'(t)=0,\\ y(0)=a>0,\quad z(0)=b>0, \end{gathered}$$ where the continuous functions A, B:[0,\infty) \to [0,\infty) are nondecreasing, differentiable and positive on (0,\infty). We assume that f and g satisfy the following hypotheses. \begin{itemize} \item[(A1)] f, g:(0,\infty)\times[0,\infty)\times[0,\infty)\to[0,\infty) are continuous, nondecreasing with respect to the second and the third variables. \item[(A2)] There exist nonnegative functions h_i, k_i, \xi_i, \omega_i, 1\leq i\leq 2, \phi and \psi satisfying for each (t,u,v)\in(0,\infty)\times[0,\infty)\times[0,\infty), \begin{gather*} h_2(t)\omega_1(v)\omega_2(u)\leq g(t,u,v)\leq h_1(t)\phi(u+v), \\ k_2(t)\xi_1(u)\xi_2(v)\leq f(t,u,v)\leq k_1(t)\psi(u+v), \end{gather*} with \xi_i, \omega_i, 1\leq i\leq 2, \phi, \psi:[0,\infty)\to[0,\infty) are nondecreasing continuous, positive on (0,\infty) and h_i, k_i:(0,\infty)\to[0,\infty) continuous and satisfying$$ \int_0^1\frac{1}{A(t)}\Big(\int_0^tA(s)\,h_i(s)\,ds\Big)dt<\infty,\quad \int_0^1\frac{1}{B(t)}\Big(\int_0^tB(s)\,k_i(s)\,ds\Big)dt<\infty. $$\item[(A3)] For all  r>0, we suppose that$$\sqrt{2} \int_0^r \sqrt{\widetilde{\Lambda}(s)}ds0$, \begin{gather*} M_\alpha(t)=\int_\alpha^t\frac{ds}{\sqrt{2(\Phi(s)+\Psi(s))}}\,,\quad t\geq \alpha\\ \Phi(r)=\int_0^r\phi(s)ds,\quad \Psi(r)=\int_0^r\psi(s)ds\,,\quad r\geq 0\\ \widetilde{\Lambda}(r)=\max_{s\in[0,r]}(h_1(s)+k_1(s)),\quad r\geq0. \end{gather*} \end{itemize} For any nonnegative measurable functions$\varphi$in$(0,\infty)$, we define \begin{gather*} S_A\varphi(t)=\int_0^t\frac{1}{A(r)}\Big(\int_0^rA(s)\varphi(s)ds\Big)dr\,,\quad S_B\varphi(t)=\int_0^t\frac{1}{B(r)}\Big(\int_0^rB(s)\varphi(s)ds\Big)dr. \end{gather*} Now, we are ready to give our existence result for \eqref{M}. \begin{theorem} \label{AX} Under the hypotheses {\rm (A1)--(A3)}, system \eqref{M} has a positive solution$(y,z)\in\left(C([0,\infty))\cap C^1((0,\infty))\right)^2$satisfying for each$t\in[0,\infty)$\begin{gather*} a+ \omega_1(b)\omega_2(a)S_A(h_2)(t)\leq y(t) \leq M_{a+b}^{-1}\Big(\sqrt{2}\int_0^t \sqrt{\widetilde{\Lambda}(s)}ds\Big), \\ b+\xi_1(a)\xi_2(b)S_B(k_2)(t)\leq z(t) \leq M_{a+b}^{-1}\Big(\sqrt{2}\int_0^t\sqrt{\widetilde{\Lambda}(s)}ds\Big)\,, \end{gather*} where$M_{a+b}^{-1}$is the inverse function of$M_{a+b}$which is defined from$[0,M_{a+b}(\infty))$to$[\alpha,\infty)$. \end{theorem} \begin{remark} \label{rmk1.4} \rm In the case$A(t)=B(t)$, the solution$(y,z)$of system \eqref{M} satisfies \begin{gather*} a+\omega_1(b)\omega_2(a)S_A(h_2)(t)\leq y(t) \leq M_{a+b}^{-1}\Big(\int_0^t \sqrt{\widetilde{\Lambda}(s)}ds\Big),\\ b+\xi_1(a)\xi_2(b)S_A(k_2)(t)\leq z(t) \leq M_{a+b}^{-1}\Big(\int_0^t \sqrt{\widetilde{\Lambda}(s)}ds\Big) \,,\quad \text{for } t\geq 0. \end{gather*} \end{remark} Now we give the existence result for system \eqref{S1} under the following hypotheses. \begin{itemize} \item[(H4)] The function$r\to r^{2(n-1)}\,(p_1(r)+q_1(r))$is nondecreasing for large$r$. \item[(H5)] There exists a positive constant$\varepsilon$such that $$\int_0^\infty r^{1+\varepsilon}(p_1(r)+q_1(r))dr<\infty,$$ \end{itemize} \begin{theorem} \label{AA} Under assumptions {\rm (H1)--(H5)}, system \eqref{S1} has a positive entire bounded continuous solution. \end{theorem} \begin{example} \label{examp1.1} \rm Let$\alpha_i$,$\beta_i$,$\gamma_i$,$\delta_i$,$\lambda_i \in (0,\infty)$satisfying$\max(\alpha_i,\beta_i, \gamma_i, \delta_i+\lambda_i)<1$for$1\leq i\leq 2$and$2<\sigma<2(n-1)$. Put \begin{gather*} H(x,u,v)=a_1(x)v^{\alpha_1}+b_1(x)u^{\gamma_1}+c_1(x)(u+v)^{\beta_1} +d_1(x)u^{\delta_1} v^{\lambda_1}, \\ K(x,u,v)=a_2(x)v^{\alpha_2}+b_2(x)u^{\gamma_2}+c_2(x)(u+v)^{\beta_2} +d_2(x)u^{\delta_2} v^{\lambda_2}\,, \end{gather*} where and$a_i$,$b_i$,$c_i$,$d_i$are nontrivial nonnegative continuous function in$\mathbb{R}^n$satisfying $$\sum_{i=1}^2\Big(\max_{|y|\leq s}a_i(y)+\max_{|y|\leq s}b_i(y) +\max_{|y|\leq s}c_i(y)+\max_{|y|\leq s}d_i(y)\Big) \leq \frac{1}{1+s^{\sigma}}.$$ Then problem \eqref{S1} has a positive continuous bounded solution$(u,v)$. Indeed, the hypotheses (H1), (H2), (H4) and (H5) are clearly satisfied. Since$\max(\alpha_i,\beta_i, \gamma_i, \delta_i+\lambda_i)<1$for$1\leq i\leq 2$and \begin{gather*} H(x,u,v)\leq \frac{1}{1+|x|^{\sigma}} [(u+v)^{\alpha_1}+(u+v)^{\beta_1}+(u+v)^{\gamma_1}+(u+v)^{\delta_1 +\lambda_1}],\\ K(x,u,v)\leq \frac{1}{1+|x|^{\sigma}} [(u+v)^{\alpha_2}+(u+v)^{\beta_2}+(u+v)^{\gamma_2}+(u+v) ^{\delta_2+\lambda_2}] \end{gather*} we deduce that$L_1(\infty)=\infty$and so hypothesis (H3) is satisfied. \end{example} \begin{example} \label{examp1.2} \rm Let$2<\sigma<2(n-1)$, and let$\beta_1, \beta_2>0$be such that$\beta_0=\max(\beta_1,\beta_2)>1$and $$\label{cond-ex2} 1+\frac{2}{\sigma-2}<\frac{1}{2(\beta_0-1)(Log(3))^{\beta_0}}.$$ Let$p, q$be two nontrivial nonnegative continuous functions in$\mathbb{R}^n$such that $$\max_{|y|\leq s}p(y)+\max_{|y|\leq s}q(y)\leq \frac{1}{1+s^{\sigma}}.$$ Then the problem \begin{gather*} \Delta u=2p(x)(u+v+2)(Log(u+v+2))^{2\beta_1-1}(Log(u+v+2)+\beta_1)\\ \Delta v=2q(x)(u+v+2)(Log(u+v+2))^{2\beta_2-1}(Log(u+v+2)+\beta_2)\,, \end{gather*} in$\mathbb{R}^n$with$n\geq 3$, has a positive bounded continuous solution$(u,v)$. Indeed, the hypotheses (H1), (H2), (H4) and (H5) are clearly satisfied. Now, (H3) follows from \eqref{cond-ex2} and the fact that for each$t\geq1, $$\int_0^t\frac{ds}{\sqrt{1+s^{\sigma}}} \leq \int_0^1\frac{ds}{\sqrt{1+s^{\sigma}}}+ \int_1^t\frac{ds}{\sqrt{1+s^{\sigma}}} \leq 1+\int_1^t\frac{ds}{s^{\frac{\sigma}{2}}}\leq 1+\frac{2}{\sigma-2}$$ and \begin{align*} \int_1^{\infty}\frac{ds}{\sqrt{2(s+2)^2 ((\log(s+2))^{2\beta_1}+(\log(s+2))^{2\beta_2})}} &\geq \int_1^{\infty}\frac{ds}{2(s+2)(Log(s+2))^{\beta_0}}\\ &=\frac{1}{2(\beta_0-1)(Log(3))^{\beta_0-1}}. \end{align*} From Theorem \ref{AA}, we have the following corollaries in the case whereH(x,u,v)=H(|x|,u,v)$and$K(x,u,v)=K(|x|,u,v)$. \end{example} \begin{corollary} \label{coro1.6} Under hypotheses {\rm (H1)--(H3)}, problem \eqref{S1} has one positive solution. Under the additional hypothesis \begin{itemize} \item[(H6)]$\int_0^{\infty}sp_2(s)\,ds=\int_0^{\infty}sq_2(s)\,ds=\infty$, \end{itemize} every positive radial entire solution$(u,v)$of \eqref{S1} is large and satisfies $$u(0)+g_1(v(0))g_2(u(0))P_2(r)\leq u(r),\quad v(0)+f_1(u(0))f_2(v(0))Q_2(r)\leq v(r) \,,$$ for all$r\geq 0$, where $$P_2(r)=\int_0^rt^{1-n}\Big(\int_0^ts^{n-1}p_2(s)\,ds\Big)\,dt,\quad Q_2(r)=\int_0^rt^{1-n}\Big(\int_0^ts^{n-1}q_2(s)\,ds\Big)\,dt.$$ \end{corollary} For the next corollary, we use the assumption \begin{itemize} \item[(H7)]$L_\alpha(\infty)=\infty$. \end{itemize} \begin{corollary} \label{coro1.7} Assume that {\rm (H1), (H2), (H4), (H7)} are satisfied. If \eqref{S1} has a nonnegative radial entire large solution, then $$\int_0^\infty r^{1+\varepsilon}(p_1(r)+q_1(r))dr =\infty,\quad \forall\;\varepsilon>0.$$ \end{corollary} An other result for the radial case is given under the assumption \begin{itemize} \item[(H8)]$\sqrt{p_1+q_1}\in L^1(0,\infty)$. \end{itemize} \begin{theorem} \label{BB} Assume that {\rm (H1), (H2), (H4), (H8)} are satisfied. If$(u,v)$is a positive entire large radial solution of \eqref{S1}, then$F$and$G$satisfy the Keller-Osserman condition $$L_1(\infty)=\int_1^{\infty}\frac{ds}{\sqrt{2 (\mathcal{F}(s)+\mathcal{G}(s))}}<\infty.$$ \end{theorem} \section{Proof of main results} \begin{proof}[Proof of Theorem \ref{AX}] Let$(y_m)_{m\geq0}$and$(z_m)_{m\geq0}$be the sequences of positive continuous functions defined on$[0,\infty)$by \begin{gather*} y_0(t)=a,\;\;z_0(t)=b,\\ y_{m+1}(t)=a+\int_0^t\frac{1}{A(r)} \Big(\int_0^rA(s)g(s,y_m(s),z_m(s))ds\Big)dr,\\ z_{m+1}(t)=b+\int_0^t\frac{1}{B(r)}\Big(\int_0^rB(s)f(s,y_m(s),z_m(s))ds\Big)dr. \end{gather*} Clearly$y_m,z_m\in C([0,\infty))\cap C^1((0,\infty))$and are positive, so we deduce by (A1) that$(y_m)_{m\geq0}$and$(z_m)_{m\geq0}$are nondecreasing sequences and for each$m\in{\mathbb{N}}$, the functions$t\to y_m(t)$and$t\to z_m(t)$are nondecreasing. Hence, for each$t\in(0,\infty),$\begin{gather*} y'_m(t) = \frac{1}{A(t)}\int_0^tA(s)g(s,y_{m-1}(s),z_{m-1}(s))ds, \\ z'_m(t) = \frac{1}{B(t)}\int_0^tB(s)f(s,y_{m-1}(s),z_{m-1}(s))ds \end{gather*} Which implies $\left(A(t)y'_m(t)\right)'= A(t)g(t,y_{m-1}(t),z_{m-1}(t)),\quad \leq A(t)g(t,y_{m}(t),z_{m}(t)) .$ Multiplying this expression by$2A(t)y'_m(t)$and integrate on$[0,t], we obtain \begin{align*} \left(A(t)y'_m(t)\right)^2 &\leq 2\int_0^t A^2(s)h_1(s)\left(\phi(y_{m}(s)+z_{m}(s))\right)\,y'_m(s)\,ds \\ &\leq 2\widetilde{\Lambda}(t)A^2(t) \int_0^t\left(\phi(y_{m}(s)+z_{m}(s))\right)\,(y'_m(s)+z'_m(s))\,ds\\ &= 2\widetilde{\Lambda}(t)A^2(t)\int_{a+b}^{y_{m}(t)+z_{m}(t)}\phi(s)\,ds. \end{align*} Which implies $y'_m(t) \leq \sqrt{\widetilde{\Lambda}(t)}\sqrt{2\Phi({y_{m}(t)+z_{m}(t)})}.$ Similarly, we have $z'_m(t) \leq \sqrt{\widetilde{\Lambda}(t)}\sqrt{2\Psi({y_{m}(t)+z_{m}(t)})}.$ Then $y'_m(t)+z'_m(t)\leq \sqrt{2}\sqrt{\widetilde{\Lambda}(t)} \sqrt{2(\Phi(y_m(t)+z_m(t))+\Psi(y_m(t)+z_m(t)))}.$ Therefore, \begin{align*} M_{a+b}(y_m(t)+z_m(t)) &= \int_{a+b}^{y_m(t)+z_m(t)}\frac{ds}{\sqrt{2\left(\Phi(s)+\Psi(s)\right)}}\,ds\\ &= \int_0^t\frac{y'_m(s)+z'_m(s)}{\sqrt{2\left(\Phi(y_m(s)+z_m(s))+\Psi(y_m(s) +z_m(s))\right)}}\,ds \\ &\leq \sqrt{2}\int_0^r\sqrt{\widetilde{\Lambda}(s)}ds. \end{align*} SinceM_{a+b}^{-1}$is increasing on$[0,M_{a+b}(\infty))$, we obtain $y_m(t)+z_m(t)\leq M_{a+b}^{-1} \Big(\sqrt{2}\int_0^t\sqrt{\widetilde{\Lambda}(s)}ds\Big),\quad \forall\,r\geq0.$ Therefore, the sequences$(y_m)_{m\geq0}$and$(z_m)_{m\geq0}$converge to two functions$y$and$z$that, for each$t\in[0,\infty)$, satisfy \begin{gather*} y(t)=a+\int_0^t\frac{1}{A(r)}\Big(\int_0^rA(s)g(s,y(s),z(s))ds\Big)dr,\\ z(t)=b+\int_0^t\frac{1}{B(r)}\left(\int_0^rB(s)f(s,y(s),z(s))ds\right)dr. \end{gather*} Hence,$y$,$z\in C([0,\infty))\cap C^1((0,\infty))$and$(y,z)$is a solution of \eqref{M} satisfying \begin{gather*} a+g_1(b)g_2(a)S_A(h_2)(t)\leq y(t)\leq M_{a+b}^{-1} \Big(\sqrt{2}\int_0^t\sqrt{\widetilde{\Lambda}(s)}ds\Big), \\ b+f_1(a)f_2(b)S_B(k_2)(t)\leq z(t)\leq M_{a+b}^{-1} \Big(\sqrt{2}\int_0^t\sqrt{\widetilde{\Lambda}(s)}ds\Big). \end{gather*} In the case where$A(t)=B(t)$, we multiply the inequality $\left(A(t)(y'_m(t)+z'_m(t))\right)' \leq A(t)[g(t,y_{m}(t),z_{m}(t))+f(t,y_m(t),z_m(t))]$ by$2A(t)(y'_m(t)+z'_m(t))$, we integrate on$[0,t]and we proceed as below to obtain $\left(A(t)(y'_m(t)+z'_m(t))\right)^2 \leq (A(t))^2 \,2\widetilde{\Lambda}(t) \int_0^{y_{m}(t)+z_{m}(t)}\left(\phi(s)+\psi(s)\right)\,ds.$ Which implies \begin{align*} y'_m(t)+z'_m(t) \leq \sqrt{\widetilde{\Lambda}(t)}\sqrt{2\Phi({y_{m}(t)+z_{m}(t)})}. \end{align*} So in this case(y,z)$satisfies \begin{gather*} a+\omega_1(b)\omega_2(a)S_A(h_2)(t)\leq y(t)\leq M_{a+b}^{-1}\Big(\int_0^t \sqrt{\widetilde{\Lambda}(s)}ds\Big), \\ b+\xi_1(a)\xi_2(b)S_A(k_2)(t)\leq z(t)\leq M_{a+b}^{-1}\left(\int_0^t \sqrt{\widetilde{\Lambda}(s)}ds\right) \,,\; \text{ for } t\geq 0. \end{gather*} This completes the proof. \end{proof} \begin{proof}[Proof of Theorem \ref{AA}] We will show that \eqref{S1} has a solution by finding a subsolution$(\underline{u},\underline{v})$and a supersolution$(\overline{u},\overline{v})$, such that$\underline{u}\leq\overline{u}$and$\underline{v}\leq\overline{v}$. To do this, we first prove the existence of a radial subsolution$(\underline{u},\underline{v})$to \eqref{S1} by considering the problem \begin{gather*} \triangle \underline{u}=p_1(|x|)F(\underline{u}+\underline{v}), \quad \text{in } \mathbb{R}^n\\ \triangle \underline{v}=q_1(|x|)G(\underline{u}+\underline{v}),\quad \text{in } \mathbb{R}^n, \end{gather*} with$n\geq 3$. That is,$(\underline{u},\underline{v})$satisfies $$\label{M2} \begin{gathered} \frac{1}{r^{n-1}} (r^{n-1}\underline{u}')' =p_1(r)F(\underline{u}+\underline{v}), \, r\in (0,\infty)\\ \frac{1}{r^{n-1}} (r^{n-1}\underline{v}')' =q_1(r)G(\underline{u}+\underline{v})\, r \in (0,\infty). \end{gathered}$$ By Theorem \ref{AX}, we conclude that system \eqref{M2} has a positive entire solution$(\underline{u},\underline{v})$. Now, since \begin{gather*} \triangle\underline{u} = p_1(|x|)F(\underline{u}+\underline{v}) \geq H(x,\underline{u},\underline{v}), \quad\text{in } \mathbb{R}^n, \\ \triangle\underline{v} = q_1(|x|)G(\underline{u}+\underline{v}) \geq K(x,\underline{u},\underline{v}),\quad\text{in } \mathbb{R}^n, \end{gather*} we deduce that$(\underline{u},\underline{v})$is a subsolution of system \eqref{S1}.\\ Next we prove that$(\underline{u},\underline{v})$is bounded. Since$(\underline{u},\underline{v})satisfies \begin{align*} (r^{n-1}\underline{u}'(r))'&=& r^{n-1}p_1(r)F(\underline{u}(r)+\underline{v}(r)),\\ (r^{n-1}\underline{v}'(r))'&=& r^{n-1}q_1(r)G(\underline{u}(r)+\underline{v}(r)), \end{align*} it follows that $$\label{F3} (r^{n-1}(\underline{u}'(r)+\underline{v}'(r)))' = r^{n-1}[p_1(r)F(\underline{u}(r)+\underline{v}(r)) +q_1(r)G(\underline{u}(r)+\underline{v}(r))].$$ ChooseR>0$so that$r^{2(n-1)}\,(p_1(r)+q_1(r))$is nondecreasing on$[R,\infty)$. Then, after multiplying$\eqref{F3}$by$r^{n-1}(\underline{u}'(r)+\underline{v}'(r))$and integrating from$R$to$r, we obtain \begin{align*} & \left(r^{n-1}(\underline{u}'(r)+\underline{v}'(r))\right)^2\\ &=C+2\Big(\int_R^rt^{2(n-1)} [p_1(t)F(\underline{u}(t)+\underline{v}(t))+q_1(r) \mathcal{G}(\underline{u}(r)+\underline{v}(r))]dt\Big) \\ &\leq C+2r^{2(n-1)}\big(p_1(r)+q_1(r)\big)\\ &\quad\times \Big(\int_R^r[\mathcal{F}'(\underline{u}(t) +\underline{v}(t))+\mathcal{G}'(\underline{u}(t)+\underline{v}(t))] (\underline{u}'(t)+\underline{v}'(t))\,dt\Big) \\ &\leq C+2r^{2(n-1)}(p_1(r)+q_1(r)) \big[\mathcal{F}(\underline{u}(r)+\underline{v}(r)) +\mathcal{G}(\underline{u}(r)+\underline{v}(r))\big], \end{align*} whereC=\left (R^{n-1}(\underline{u}'(R)+\underline{v}'(R))\right)^2. This yields \begin{align*} &\frac{\underline{u}'(r)+\underline{v}'(r)}{\sqrt{2}(\mathcal{F}(\underline{u}(r) +\underline{v}(r))+\mathcal{G}(\underline{u}(r) +\underline{v}(r)))^\frac{1}{2}}\\ &\leq\frac{\sqrt{C}r^{1-n}}{\sqrt{(\mathcal{F}(\underline{u}(r)+\underline{v}(r)) +\mathcal{G}(\underline{u}(r)+ \underline{v}(r)))}}+\sqrt{2}\,\sqrt{p_1(r)+q_1(r)}. \end{align*} Integrating the above inequality and using the fact that $$\mathcal{F}(\underline{u}(r)+\underline{v}(r)) +\mathcal{G}(\underline{u}(r)+\underline{v}(r))\geq\mathcal{F}(\underline{u}(R)+ \underline{v}(R))+\mathcal{G}(\underline{u}(R)+\underline{v}(R))=C_1,$$ for allr\geq R, and $\sqrt{p_1(r)+q_1(r)} \leq 2\sqrt{r^{1+\varepsilon}(p_1(r)+q_1(r))r^{-1-\varepsilon}} \leq r^{1+\varepsilon} (p_1(r)+q_1(r))+r^{-1-\varepsilon}\,,$ we obtain \begin{align*} &L_\alpha(\underline{u}(r)+\underline{v}(r))\\ &\leq \sqrt{2}\,\int_R^rs^{1+\varepsilon}(p_1(s)+q_1(s))ds +\sqrt{2}\, (\varepsilon R^\varepsilon)^{-1} +\sqrt{\frac{C}{C_1}}(2-n) (R^{2-n})^{-1}\,, \end{align*} where\alpha=\underline{u}(R)+\underline{v}(R)$. Letting$r\to\infty$, we deduce from hypothesis (H5) that$(\underline{u},\underline{v})$is bounded. Thus, since$(\underline{u},\underline{v})$is nondecreasing, we have $$\lim_{r\to\infty}\underline{u}(r)=M_1>0,\quad \lim_{r\to\infty}\underline{v}(r)=M_2>0.$$ Now, it is clear that$(\overline{u},\overline{v})=(M_1,M_2)$is a supersolution for \eqref{S1} and we have for$r\geq 0$, $$\overline{u}(r)\geq M_1\geq\underline{u}(r),\quad \overline{v}(r)\geq M_2\geq\underline{v}(r).$$ Hence the standard sub-supersolution method (see \cite{HER,SAT}) implies that \eqref{S1} has a bounded solution$(u,v)$such that$\underline{u}\leq u\leq\overline{u}$and$\underline{v}\leq v\leq\overline{v}$. This completes the proof. \end{proof} \begin{proof}[Proof of Theorem \ref{BB}] Let$(u,v)$be a positive entire large radial solution of \eqref{S1}. Then$(u,v)satisfies \begin{gather*} (r^{n-1}u'(r))'\leq r^{n-1}p_1(r)F(u(r)+v(r)),\\ (r^{n-1}v'(r))'\leq r^{n-1}q_1(r)G(u(r)+v(r)). \end{gather*} Adding these inequalities, we obtain \begin{align*} (r^{n-1}(u'(r)+v'(r)))' &\leq r^{n-1}p_1(r)F(u(r)+v(r))+ r^{n-1}q_1(r)G(u(r)+v(r))\\ &\leq r^{n-1}(p_1(r)+q_1(r))\left(F(u(r)+v(r))+ G(u(r)+v(r))\right). \end{align*} Multiplying the above inequality by2r^{n-1}(u'(r)+v'(r))$and integrating from$0$to$r, and using (H4), we obtain \begin{align*} \big(r^{n-1}(u'(r)+v'(r))\big)^2 & \leq 2\Big(\int_0^rs^{2(n-1)}(p_1(s)+q_1(s)) \Big(F(u(s)+v(s))\\ &\quad +G(u(s)+v(s))\Big)(u'(s)+v'(s))\,ds\Big) \\ & \leq 2r^{2(n-1)}(p_1(r)+q_1(r)) \int_0^r \Big(F(u(s)+v(s))\\ &\quad +G(u(s)+v(s))\Big)(u'(s)+v'(s))\,ds \\ & \leq 2r^{2(n-1)}(p_1(r)+q_1(r)) \left(\mathcal{F}(u(r)+v(r))+\mathcal{G}(u(r)+v(r))\right) \end{align*} which implies $$\frac{u'(r)+v'(r)}{\sqrt{2(\mathcal{F}(u(r)+v(r))+\mathcal{G}(u(r)+v(r)))}} \leq\sqrt{p_1(r)+q_1(r)}.$$ By integrating over[0,r]$, we obtain $$L_{u(0)+v(0)}(u(r)+(v(r))\leq\int_0^r\sqrt{p_1(s)+q_1(s)}ds.$$ Letting$r\to\infty$, we obtain that$F$and$G$satisfies the Keller-Osserman condition. \end{proof} \begin{thebibliography}{00} \bibitem{AA} N. 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