\documentclass[reqno]{amsart} \usepackage{hyperref} \AtBeginDocument{{\noindent\small \emph{Electronic Journal of Differential Equations}, Vol. 2010(2010), No. 16, pp. 1--5.\newline ISSN: 1072-6691. URL: http://ejde.math.txstate.edu or http://ejde.math.unt.edu \newline ftp ejde.math.txstate.edu} \thanks{\copyright 2010 Texas State University - San Marcos.} \vspace{9mm}} \begin{document} \title[\hfilneg EJDE-2010/16\hfil Existence of entire positive solutions] {Existence of entire positive solutions for a class of semilinear elliptic systems} \author[Z. Zhang\hfil EJDE-2010/16\hfilneg] {Zhijun Zhang} \address{Zhijun Zhang \newline School of Mathematics and Information Science, Yantai University, Yantai, Shandong, 264005, China} \email{zhangzj@ytu.edu.cn} \thanks{Submitted October 22, 2009. Published January 27, 2010.} \thanks{Supported by grants 10671169 from NNSF of China, and 2009ZRB01795 from NNSF of \hfill\break\indent Shandong Province} \subjclass[2000]{35J55, 35J60, 35J65} \keywords{Semilinear elliptic systems; entire solutions; existence} \begin{abstract} Under simple conditions on $f_i$ and $g_i$, we show the existence of entire positive radial solutions for the semilinear elliptic system \begin{gather*} \Delta u =p(|x|)f_1(v)f_2(u)\\ \Delta v =q(|x|)g_1(v)g_2(u), \end{gather*} where $x\in \mathbb{R}^N$, $N\geq 3$, and $p,q$ are continuous functions. \end{abstract} \maketitle \numberwithin{equation}{section} \newtheorem{theorem}{Theorem}[section] \newtheorem{remark}[theorem]{Remark} \section{Introduction} The purpose of this paper is to investigate the existence of entire positive radial solutions to the semilinear elliptic system \begin{equation} \begin{gathered} \Delta u=p(|x|)f_1(v)f_2(u),\quad x \in R^N , \\ \Delta v=q(|x|)g_1(v)g_2(u),\quad x \in R^N, \end{gathered} \label{e1.1} \end{equation} where $N\geq 3$. We assume that $p,q,f_i,g_i$ ($i=1, 2$) satisfy the following hypotheses. \begin{itemize} \item[(H1)] The functions $p,q,f_i,g_i:[0,\infty)\to [0,\infty)$ are continuous; \item[(H2)] the functions $f_i$ and $g_i$ are increasing on $[0,\infty)$. \end{itemize} Denote \begin{gather*} P(\infty):=\lim_{r\to \infty}P(r),\quad P(r)=\int_{0}^{r}t^{1-N} \Big(\int_{0}^{t} s^{N-1}p(s) ds\Big)dt,\quad r\geq 0,\\ Q(\infty):=\lim_{r\to \infty}Q(r),\quad Q(r)=\int_{0}^{r}t^{1-N} \Big(\int_{0}^{t} s^{N-1}q(s)ds\Big)dt,\quad r\geq 0,\\ F(\infty):=\lim_{r\to \infty}F(r),\quad F(r)=\int_a^r\frac {ds}{f_1(s)f_2(s)+g_1(s)g_2(s)},\quad r\geq a>0. \end{gather*} We see that $F'(r)=\frac {1}{f_1(r)f_2(r)+g_1(r)g_2(r)}>0$, for $r>a$ and $F$ has the inverse function $F^{-1}$ on $[a, \infty)$. This problem arises in many branches of mathematics and physics and has been discussed by many authors; see, for instance, \cite {CR}-\cite{LW1}, \cite {LZZ,PS,WW} and the references therein. When $f_2=g_1\equiv 1$, $f_1(v)=v^\alpha$, $g_2(u)=u^\beta$, $0<\alpha\leq \beta$, Lair and Wood \cite {LW1} considered the existence and nonexistence of entire positive radial solutions to \eqref{e1.1}. Their results were extended by C\^{\i}rstea and R\u adulescu \cite{CR}, Wang and Wood \cite{WW}, Ghergu and R\u adulescu \cite {GR}, Peng and Song \cite{PS}, Ghanmi, M\^{a}agli, R\u{a}dulescu and Zeddini \cite {GMRZ}, and the authors of this article in \cite {LZZ}. When $f_1(v)=v^{\alpha_1}$, $f_2(u)=u^{\alpha_2}$, $g_1(v)=v^{\beta_1}$, $g_2(u)=u^{\beta_2}$, where $\alpha_1>0$, $\beta_2>0$, $\alpha_2>1$ and $\beta_1>1$, Garc\'ia-Meli\'an and Rossi \cite {GMR}, Garc\'ia-Meli\'an \cite {GM} have studied the existence, uniqueness and exact blow-up rate near the boundary of positive solutions to system \eqref{e1.1} on a bounded domain. In this paper, we give simple conditions on $f_i$ and $g_i$ to show the existence of entire positive radial solutions to \eqref{e1.1}. Our main results are as the following. \begin{theorem} \label{thm1.1} Under hypotheses {\rm (H1)--(H2)} and \begin{itemize} \item[(H3)] $F(\infty)=\infty$, \end{itemize} system \eqref{e1.1} has one positive radial solution $(u,v) \in C^2([0,\infty))$. Moreover, when $P(\infty)<\infty$ and $Q(\infty)<\infty$, $u$ and $v$ are bounded; when $P(\infty)=\infty =Q(\infty)$, $\lim _{r\to \infty}u(r)=\lim _{r\to \infty}v(r)=\infty$. \end{theorem} \begin{theorem} \label{thm1.2} Under hypotheses {\rm (H1)--(H2)} and \begin{itemize} \item[(H4)] $F(\infty)<\infty$; \item[(H5)] $P(\infty)<\infty$, $Q(\infty)<\infty$; \item[(H6)] there exist $b>a$ and $c>a$ such that $P(\infty)+Q(\infty)1$ or $\beta_1+\beta_2>1$. \end{remark} \begin{remark} \label{rmk1.3} \rm By \cite {LW2}, we see that $P(\infty)=\infty$ if and only if $\int_0^\infty sp(s)ds=\infty$. \end{remark} \section{Proof of Theorems \ref{thm1.1} and \ref{thm1.2}} Note that radial solutions of \eqref{e1.1} are solutions of the ordinary differential equation system \begin{gather*} u''+\frac {N-1}{r}u'=p(r)f_1(v)f_2(u),\\ v''+\frac {N-1}{r}v'=q(r)g_1(v)g_2(u). \end{gather*} Thus solutions of \eqref{e1.1} are simply solutions of \begin{gather*} u(r)=b+\int_{0}^{r}t^{1-N} \Big(\int_{0}^{t} s^{N-1}p(s)f_1(v(s))f_2(u(s)) ds\Big)dt,\quad r\geq 0, \\ v(r)=c+\int_{0}^{r}t^{1-N} \Big(\int_{0}^{t} s^{N-1}q(s)g_1(v(s))g_2(u(s)) ds\Big)dt,\quad r\geq 0. \\ \end{gather*} Let $\{u_{m}\}_{m\geq 0}$ and $\{v_{m}\}_{m\geq 0}$ be the sequences of positive continuous functions defined on $[0,\infty)$ by \begin{gather*} u_{0}(r)\equiv b, \quad v_{0}(r)\equiv c,\\ u_{m+1}(r)=b+\int_{0}^{r} t^{1-N} \Big(\int_{0}^{t} s^{N-1}p(s)f_1(v_{m}(s))f_2(u_{m}(s)) ds\Big)dt,\quad r\geq 0,\\ v_{m+1}(r)=c+\int_{0}^{r} t^{1-N} \Big(\int_{0}^{t} s^{N-1}q(s)g_1(v_{m}(s))g_2(u_{m}(s)) ds\Big)dt,\quad r\geq 0. \end{gather*} Obviously, for all $ r\geq 0$ and $m\in {\mathbb{N}}$, $u_{m}(r)\geq b$, $v_{m}(r)\geq c$ and $$ v_0\leq v_1,\quad u_0\leq u_1, \quad \forall r\geq 0. $$ Hypothesis (H2) yields $$ u_1(r)\leq u_2(r),\quad v_1(r)\leq v_2(r), \quad \forall r\geq 0. $$ Continuing this line of reasoning, we obtain that the sequences $\{u_m\}$ and $\{v_m\}$ are increasing on $[0, \infty)$. Moreover, we obtain by (H1) and (H2) that, for each $r>0$, \begin{align*} u_{m+1}'(r) &=r^{1-N} \int_{0}^{r} s^{N-1}p(s)f_1(v_{m}(s))f_2(u_{m}(s)) ds\\ &\leq f_1(v_{m}(r))f_2(u_{m}(r))P'(r)\\ &\leq f_1\big(v_{m+1}(r)+u_{m+1}(r)\big)f_2 \big(v_{m+1}(r)+u_{m+1}(r)\big)P'(r)\\ &\leq \Big[ f_1\big(v_{m+1}(r)+u_{m+1}(r)\big)f_2 \big(v_{m+1}(r)+u_{m+1}(r)\big)\\ &\quad +g_1\big(v_{m+1}(r)+u_{m+1}(r)\big)g_2 \big(v_{m+1}(r)+u_{m+1}(r)\big)\Big]P'(r)\,, \end{align*} \begin{align*} v_{m+1}'(r) &=r^{1-N} \int_{0}^{r} s^{N-1}q(s)g_1(v_{m}(s))g_2(u_{m}(s)) ds\\ &\leq g_1\big(v_{m}(r))g_2(u_{m}(r)\big)Q'(r)\\ &\leq g_1\big(v_{m+1}(r)+u_{m+1}(r)\big)g_2 \big(v_{m+1}(r)+u_{m+1}(r)\big)Q'(r)\\ &\leq \Big[f_1\big(v_{m+1}(r)+u_{m+1}(r)\big)f_2 \big(v_{m+1}(r)+u_{m+1}(r)\big)\\ &\quad +g_1\big(v_{m+1}(r)+u_{m+1}(r)\big) g_2\big(v_{m+1}(r)+u_{m+1}(r)\big)\Big]Q'(r) \end{align*} and \[ \int_{b+c}^{v_{m+1}(r)+u_{m+1}(r)}\frac {d\tau}{f_1(\tau) f_2(\tau)+g_1(\tau) g_2(\tau)} \leq Q(r)+P(r). \] Consequently, \begin{equation} F\big(u_m(r)+v_m(r)\big)-F(b+c)\leq P(r)+Q(r), \quad \forall r\geq 0. \label{e2.1} \end{equation} Since $F^{-1}$ is increasing on $[0, \infty)$, we have \begin{equation} u_m(r)+v_m(r)\leq F^{-1}\big (F(b+c)+P(r)+Q(r)\big),\quad \forall r\geq 0. \label{e2.2} \end{equation} (i) When (H3) holds, we see that \begin{equation} F^{-1}(\infty)=\infty.\label{e2.3} \end{equation} It follows that the sequences $\{u_m\}$ and $\{v_m\}$ are bounded and equicontinuous on $[0,c_0]$ for arbitrary $c_0>0$. It follows by Arzela-Ascoli theorem that $\{u_m\}$ and $\{v_m\}$ have subsequences converging uniformly to $u$ and $v$ on $[0, c_0]$. By the arbitrariness of $c_0>0$, we see that $(u, v)$ are positive entire solutions of \eqref{e1.1}. Moreover, when $P(\infty)<\infty$ and $Q(\infty)<\infty$, we see by \eqref{e2.2} that $$ u(r)+v(r)\leq F^{-1}\big(F(b+c)+P(\infty)+Q(\infty)\big),\quad \forall r\geq 0; $$ and,when $P(\infty)=\infty =Q(\infty)$, by (H2) and the monotones of $\{u_m\}$ and $\{v_m\}$, $$ u(r)\geq b +f_1(c)f_2(b)P(r),\quad v(r)\geq c +g_1(c)g_2(b)Q(r),\quad \forall r\geq0. $$ Thus $\lim _{r\to \infty} u(r)=\lim_{r\to \infty}v(r)=\infty$. \noindent(ii) When (H4)--(H6) hold, we see by \eqref{e2.1} that \begin{equation} F(u_m(r)+v_m(r))\leq F(b+c)+P(\infty)+Q(\infty)