\documentclass[reqno]{amsart} \AtBeginDocument{{\noindent\small {\em Electronic Journal of Differential Equations}, Vol. 2003(2003), No. 94, pp. 1--22.\newline ISSN: 1072-6691. URL: http://ejde.math.swt.edu or http://ejde.math.unt.edu \newline ftp ejde.math.swt.edu (login: ftp)} \thanks{\copyright 2003 Southwest Texas State University.} \vspace{9mm}} \begin{document} \title[\hfilneg EJDE--2003/94\hfil Entire solutions of second-order] {On nonnegative entire solutions of second-order semilinear elliptic systems} \author[Tomomitsu Teramoto \hfil EJDE--2003/94\hfilneg] {Tomomitsu Teramoto} % in alphabetical order \address{Tomomitsu Teramoto \newline Faculty of Economics, Management \& Information Science\\ Onomichi University\\ 1600 Hisayamada, Onomichi Hiroshima\\ 722-8506, Japan} \email{teramoto@onomichi-u.ac.jp} \date{} \thanks{Submitted February 6, 2003. Published September 9, 2003.} \subjclass[2000]{35J60, 35B05} \keywords{Elliptic system, nonnegative entire solutions} \begin{abstract} We consider the second-order semilinear elliptic system $$\Delta u_i=P_i(x)u_{i+1}^{\alpha_i}\quad\mbox{in }\mathbb{R}^N, \quad i=1,2,\dots,m$$ with nonnegative continuous functions $P_i$. We establish nonexistence criteria of nonnegative nontrivial entire solutions for this system. We also proved a Liouville type theorem for nonnegative entire solutions. \end{abstract} \maketitle \numberwithin{equation}{section} \newtheorem{theorem}{Theorem}[section] \newtheorem{lemma}[theorem]{Lemma} \newtheorem{remark}[theorem]{Remark} \numberwithin{equation}{section} \section{Introduction} This paper concerns the second-order semilinear elliptic system $$\label{problem} \begin{gathered} {\Delta u_1=P_1(x)u_2^{\alpha_1}},\\ {\Delta u_2=P_2(x)u_3^{\alpha_2}},\\ \vdots\\ {\Delta u_m=P_m(x)u_{m+1}^{\alpha_m}},\quad u_{m+1}=u_1, \end{gathered}$$ where $x\in\mathbb{R}^N$, $N\geq 1$, $m\geq 2$, and $\alpha_i>0$, $i=1,2,\dots,m$ are constants satisfying $\alpha_1\alpha_2\cdots\alpha_m>1$, and the functions $P_i(x)$ are nonnegative continuous functions on $\mathbb{R}^N$. We are concerned with the problem of existence and nonexistence of nonnegative nontrivial entire solutions of \eqref{problem}. By an entire solution of \eqref{problem} we mean a vector function $(u_1,u_2,\dots,u_m)\in (C^2(\mathbb{R}^N))^m$ which satisfies \eqref{problem} at every point of $\mathbb{R}^N$. The problem of existence and nonexistence of nonnegative entire solutions for the scalar equation $${\Delta u=f(x,u),\quad x\in\mathbb{R}^N}$$ has been investigated by many authors, and numerous results have been obtained (see e.g. \cite{Cheng-Lin,Kawano,Kawano-Kusano-Naito,Ni} and references therein). In particular, when $f$ has the form $f(x,u)=P(x)u^{\alpha}$ with $\alpha>0$ and nonnegative function $P$, critical decay rate of $P$ to admit nonnegative entire solutions has been characterized. On the other hand, very little is known about this problem for elliptic system \eqref{problem} except for the case $m=2$. For $m=2$ we refer to \cite{Deng, Kawano, Kawano-Kusano, Lair-Wood, Teramoto, Teramoto-Usami, Yarur}. In \cite{Deng, Teramoto, Yarur}, the system \eqref{problem} with $m=2$ has been considered under the conditions $\alpha_i\geq 1$, $i=1,2$, and nonexistence criteria of nonnegative nontrivial entire solutions have been obtained. The result is described roughly as follows: \begin{theorem}\label{previous result 1} Let $N\geq 3,~m=2$ and $\alpha_i\geq 1,i=1,2$. Suppose that $P_i,~i=1,2$, satisfy $$\label{pre ass} {P_i(x)\geq \frac{C_i}{|x|^{\lambda_i}},\quad |x|\geq r_0>0,\quad i=1,2,}$$ where $C_i>0$ and $\lambda_i,~i=1,2$, are constants. If $(\lambda_1,\lambda_2)$ satisfies $$\label{pre condi} \lambda_1-2+\alpha_1(\lambda_2-2)\leq 0\quad \mbox{or}\quad \lambda_2-2+\alpha_2(\lambda_1-2)\leq 0,$$ then the system \eqref{problem} does not possess any nonnegative nontrivial entire solutions. \end{theorem} However, if $\alpha_1$ or $\alpha_2$ is less than 1, Theorem \ref{previous result 1} cannot derive any information about the nonnegative nontrivial entire solutions. Recently, Teramoto and Usami \cite{Teramoto-Usami} have proved a Liouville type theorem for nonnegative entire solutions of \eqref{problem} with $m=2$ under the condition $\alpha_1\alpha_2>1$. The result is described as follows: \begin{theorem}\label{previous result 2} Let $N\geq 3,m=2,\alpha_1\alpha_2>1,0<\alpha_1<1$. Suppose that $P_i,~i=1,2$, satisfy \eqref{pre ass} for some constants $\lambda_i,~i=1,2$. If $(\lambda_1,\lambda_2)$ satisfies $$\lambda_1-2+\alpha_1(\lambda_2-2)\leq 0,$$ then the system \eqref{problem} does not possess nonnegative nontrivial entire solutions satisfying $${u_1(x)=O(\exp |x|^{\rho})\quad\mbox{as}\quad |x|\to\infty\quad \mbox{for some }\rho>0.}$$ \end{theorem} The aim of this paper is to extend Theorems \ref{previous result 1} and \ref{previous result 2} to the system \eqref{problem} with $m\geq 3$. Let us introduce some notation used throughout this paper. For any sequence $\{s_1,s_2,\dots,s_m\}$, we assume that $s_{m+j}=s_j$, $j=1,2,\dots$; that is, the suffixes should be taken in the sense of $\mathbb{Z}/m\mathbb{Z}$. Denote $$A=\alpha_1\alpha_2\cdots \alpha_m.$$ For real constants $\lambda_1,~\lambda_2,\dots,\lambda_m$, we put \begin{aligned} \Lambda_i & = \lambda_i-2+(\lambda_{i+1}-2)\alpha_i+(\lambda_{i+2}-2)\alpha_i\alpha_{i+1}+\dots \\ & \quad +(\lambda_{i+m-1}-2)\alpha_i\alpha_{i+1}\alpha_{i+2}\dots\alpha_{i+m-2} \\ & = {\lambda_i-2+\sum_{j=1}^{m-1} \Big\{(\lambda_{i+j}-2)\prod_{k=0}^{j-1}\alpha_{i+k}\Big\},\quad i=1,2,\dots,m,} \label{constant Lambda} \end{aligned} and $$\label{constant beta} {\beta_i=\frac{\Lambda_i}{A-1},\quad i=1,2,\dots,m.}$$ Since our assumptions imposed on $P_i,~1\leq i\leq m$, essentially take the forms $$\liminf_{|x|\to\infty}|x|^{\lambda_i}P_i(x)>0 \quad\mbox{or}\quad \limsup_{|x|\to\infty}|x|^{\lambda_i}P_i(x)<\infty,$$ all our results are formulated by means of the numbers $\lambda_i$, $\Lambda_i$, $\beta_i$, $1\leq i\leq m$. This paper is organized as follows. In Section 2, we give nonexistence criteria of nonnegative nontrivial entire solutions of \eqref{problem}. In Section 3, to show the sharpness of our nonexistence criteria we give existence theorems of positive entire solutions for \eqref{problem} under the assumption that $P_i$ have radial symmetry. In the final section (Section 4), we prove a Liouville type theorem for nonnegative entire solutions. \section{A priori estimate and nonexistence results} \subsection{Growth estimate of nonnegative entire solutions} In this subsection, we study the estimate for nonnegative entire solutions of \eqref{problem} which will play an important role to prove nonexistence theorems for nonnegative nontrivial entire solutions. For a nonnegative function $v$ defined on $\mathbb{R}^N$, we denote its spherical mean over the sphere $|x|=r,~r>0$, $\bar v(r)$ by $${\bar v(r)=\frac{1}{\omega_N r^{N-1}}\int_{|x|=r}v(x)\,dS},$$ where $dS$ denotes the volume element in the surface integral, $\omega_N$ is the surface area of the unit sphere in $\mathbb{R}^N$. Moreover we introduce the function $\hat P(r),r\geq 0$, by $$\label{mean p} \hat P(r)=\begin{cases} \Big(\frac{1}{\omega_N r^{N-1}}\int_{|x|=r}P(x) ^{-\frac{\alpha'}{\alpha}}dS\Big)^{-\alpha/\alpha'}, & \alpha>1,\\ \min_{|x|=r}P(x), & \alpha=1, \end{cases}$$ where $1/\alpha+1/\alpha'=1$. We set $\hat{P}(r)=0$ if $\int_{|x|=r}P(x)^{-\alpha'/\alpha}dS=\infty$. We note that $\hat P=P$ when $P$ has radial symmetry. We have the following well-known result (see \cite[p.654]{Cheng-Lin}, \cite[p.508]{Ni} and \cite[p.70]{Noussair Swanson}). \begin{lemma}\label{lemma-1} Let $\alpha_i\geq 1$, $i=1,2,\dots,m$, and $(u_1,u_2,\dots,u_m)$ be a nonnegative entire solution of \eqref{problem}. Then its spherical mean $(\bar u_1,\bar u_2,\dots,\bar u_m)$ satisfies system of ordinary differential inequalities $$\label{system-ode} \begin{gathered} {(r^{N-1}\bar u_i'(r))'\geq r^{N-1}\hat P_i(r)\bar u_{i+1}(r)^{\alpha_i},\quad r>0,}\\ \bar u_i'(0)=0, \end{gathered}$$ where $i=1,2,\dots,m$. \end{lemma} Our main result is as follows. \begin{theorem}\label{estimate} Let $N\geq 3$, $\alpha_i\geq 1$, $i=1,2,\dots,m$, and $A>1$. Suppose that $P_i$, $i=1,2,\dots,m$, satisfy $$\label{condi P} {\liminf_{|x|\to\infty}|x|^{\lambda_i}P_i(x)>0,}$$ where $\lambda_i,i=1,2,\dots,m$, are constants. Let $(u_1,u_2,\dots,u_m)$ be a nonnegative entire solution of \eqref{problem}. Then $u_i$, $i=1,2,\dots,m$, satisfy $$u_i(x)\leq C_i|x|^{\beta_i}\quad \mbox{at }\infty\,,$$ where $C_i>0$ are constants and $\beta_i$ are defined by \eqref{constant beta}. \end{theorem} Assume that \eqref{condi P} holds. Then there are constants $C_i>0$, $i=1,2,\dots,m$, and $R_0>0$ such that $${P_i(x)\geq\frac{C_i}{|x|^{\lambda_i}},\quad |x|\geq R_0,\quad i=1,2,\dots,m}.$$ So we can see that $\hat P_i$, $i=1,2,\dots,m$, defined by \eqref{mean p} satisfy $$\label{condi hat P} {\hat P_i(r)\geq\frac{C_i}{r^{\lambda_i}},\quad r\geq R_0.}$$ \begin{proof}[Proof of Theorem \ref{estimate}] Let $(u_1,u_2,\dots,u_m)$ be a nonnegative entire solution of \eqref{problem}. We may assume that $(u_1,u_2,\dots,u_m)\not\equiv (0,0,\dots,0)$. Then, by Lemma \ref{lemma-1}, its spherical mean $(\bar u_1,\bar u_2,\dots,\bar u_m)$ satisfies the system of ordinary differential inequalities \eqref{system-ode}. Integrating \eqref{system-ode} over $[0,r]$, we have $${\bar u_i'(r)\geq r^{1-N}\int_0^r s^{N-1}\hat P_i(s)\bar u_{i+1}(s)^{\alpha_i}ds, \quad i=1,2,\dots,m.}$$ Hence, we see that $\bar u_i'(r)\geq 0$ for $r\geq 0$. Integrating \eqref{system-ode} twice over $[R,r]$, $R\geq 0$ and $i=1,2,\dots,m$, we have $$\label{int R} {\bar u_i(r)\geq \bar u_i(R)+\frac{1}{N-2}\int_R^rs\big[1-(\frac{s}{r})^{N-2}\big] \hat P_i(s)\bar u_{i+1}(s)^{\alpha_i}ds}.$$ Since $(u_1,u_2,\dots,u_m)$ is nonnegative and nontrivial, there exists a point $x_*\in\mathbb{R}^N$ such that $u_{i_0}(x_*)>0$ for some $i_0\in\{1,2,\dots,m\}$; that is, $\bar u_{i_0}(r_*)>0,~r_*=|x_*|$. We may assume that $r_*\geq R_0$. Therefore, we see from \eqref{int R} with $R=r_*$ that $\bar u_i(r)>0$ for $r>r_*$, $i=1,2,\dots,m$. First, we will show that $$\label{mean sol estimate} {\bar u_i(r)=O(r^{\beta_i})\quad\mbox{as } r\to\infty,\quad i=1,2,\dots,m.}$$ Let us fix $R>r_*$ arbitrarily. Using \eqref{condi hat P} and the inequality $$s\big[1-(\frac{s}{r})^{N-2}\big]\geq \frac{N-2}{3^{N-2}}(r-s)\quad \mbox{for } R\leq r\leq 3R,$$ in \eqref{int R}, we have \begin{align*} \bar u_i(r) &\geq \bar u_i(R)+\frac{C_i}{3^{N-2}}\int_R^rs^{-\lambda_i}(r-s) \bar u_{i+1}(s)^{\alpha_i}ds \\ &\geq \hat C_iR^{-\lambda_i}\int_R^r(r-s)\bar u_{i+1}(s)^{\alpha_i}ds, \end{align*} where $R\leq r\leq 3R$ and $\hat C_i$ are some positive constants independent of $r$ and $R$. We put $$\label{function f} {f_i(r;R)=\hat C_iR^{-\lambda_i}\int_R^r(r-s)\bar u_{i+1}(s)^{\alpha_i}ds, \quad R\leq r\leq 3R.}$$ For simplicity of notation we write $f_i(r)=f_i(r;R)$ if there is no ambiguity. Clearly, $f_i(r)$, $i=1,2,\dots,m$, satisfy \begin{gather*} \bar u_i(r)\geq f_i(r),\quad f_i(R)=0,\\ f'_i(r)\geq 0,\quad f_i'(R)=0, \end{gather*} and $$f_i''(r) = {\hat C_i R^{-\lambda_i}\bar u_{i+1}(r)^{\alpha_i}} \geq {\hat C_i R^{-\lambda_i}f_{i+1}(r)^{\alpha_i},\quad R\leq r\leq 3R.}\label{double f}$$ From \eqref{function f} and the monotonicity of $\bar u_i$, we see that $$\label{estimate u} {f_i(r;R)\geq \frac{\hat C_i}{2}R^{-\lambda_i}\bar u_{i+1}(R)^{\alpha_i}(r-R)^2,\quad R\leq r\leq 3R.}$$ Let us fix $i\in\{1,2,\dots,m\}$. Multiplying \eqref{double f} by $f_{i+1}'(r)\geq 0$ and integrating by parts of the resulting inequality on $[R,r]$, we have $${f_{i+1}'(r)f_i'(r)\geq CR^{-\lambda_i}f_{i+1}(r)^{\alpha_i+1}, \quad R\leq r\leq 3R,}$$ where $C=\tilde C_i/(\alpha_i+1)$. For the rest of this article, $C$ denotes various positive constants independent of $r$ and $R$. Multiplying this inequality by $f_{i+1}'(r)\geq 0$ and integrating by parts, we obtain $${f_{i+1}'(r)^2f_i(r)\geq CR^{-\lambda_i}f_{i+1}(r)^{\alpha_i+2}, \quad R\leq r\leq 3R.}$$ From \eqref{double f}, we see that $${f_{i+1}'(r)^{2\alpha_{i-1}}f_{i-1}''(r) \geq CR^{-\lambda_i\alpha_{i-1}-\lambda_{i-1}}f_{i+1}(r)^{(\alpha_i+2) \alpha_{i-1}},\quad R\leq r\leq 3R.}$$ Again multiplying this relation by $f_{i+1}'(r)\geq 0$ and integrating by parts on $[R,r]$ twice, we have $${f_{i+1}'(r)^{2\alpha_{i-1}+2}f_{i-1}(r)\geq CR^{-\lambda_i\alpha_{i-1} -\lambda_{i-1}}f_{i+1}(r)^{(\alpha_i+2)\alpha_{i-1}+2},\quad R\leq r\leq 3R.}$$ From \eqref{double f}, we see that for $R\leq r\leq 3R$, \begin{align*} & f_{i+1}'(r)^{2\alpha_{i-1}\alpha_{i-2}+2\alpha_{i-2}}f_{i-2}''(r)\\ &\geq CR^{-\lambda_i\alpha_{i-1}\alpha_{i-2}-\lambda_{i-1}\alpha_{i-2} -\lambda_{i-2}}f_{i+1}(r)^{\alpha_i\alpha_{i-1}\alpha_{i-2} +2\alpha_{i-1}\alpha_{i-2}+2\alpha_{i-2}}. \end{align*} By repeating this procedure, we obtain $$f_{i+1}'(r)^{K_i}f_{i-(m-1)}''(r) =f_{i+1}'(r)^{K_i}f_{i+1}''(r) \geq CR^{-L_i}f_{i+1}(r)^{M_i}, \label{aaaa}$$ where \begin{gather*} K_i=2\sum_{j=1}^{m-1}\prod_{k=j}^{m-1}\alpha_{i-k},\\ L_i=\sum_{j=1}^{m-1}\Big\{\lambda_{i-(j-1)}\prod_{k=j}^{m-1} \alpha_{i-k}\Big\}+\lambda_{i+1},\\ M_i=\prod_{k=0}^{m-1}\alpha_{i-k}+2\sum_{j=1}^{m-1}\prod_{k=j}^{m-1}\alpha_{i-k}=A+K_i. \end{gather*} Multiplying the inequality \eqref{aaaa} by $f_{i+1}'(r)\geq 0$ and integrating on $[R,r]$, we obtain $${f_{i+1}'(r)f_{i+1}(r)^{-\frac{M_i+1}{K_i+2}}\geq CR^{-\frac{L_i}{K_i+2}}, \quad R1}, we may set {(M_i+1)/(K_i+2)=\delta_i+1},\\ \delta_i=(A-1)/(K_i+2). Integrating this inequality on [2R,3R] we get$$ f_{i+1}(2R)^{-\delta_i}\geq CR^{-\frac{L_i}{K_i+2}+1}. $$From \eqref{estimate u} with r=2R and this inequality, we have \bar u_{i+2}(R)\leq CR^{\tau_i}, where$$ \tau_i=\frac{1}{\alpha_{i+1}\delta_i}\Big\{\frac{L_i}{K_i+2}-1+(\lambda_{i+1}-2) \delta_i\Big\}. From the definitions of K_i, L_i, and \delta_i, we see that \begin{align*} \tau_i &= \frac{1}{\alpha_{i+1}\delta_i(K_i+2)} \Big[\sum_{j=1}^{m-1}\big\{\lambda_{i-j+1}\prod_{k=j}^{m-1}\alpha_{i-k}\big\} -2\sum_{j=1}^{m-1}\prod_{k=j}^{m-1}\alpha_{i-k}\\ &\quad +(\lambda_{i+1}-2)\prod_{k=0}^{m-1}\alpha_{i-k}\Big]\\ &= \frac{1}{\alpha_{i+1}(A-1)}\Big[\sum_{j=1}^{m-2}\big\{(\lambda_{i-j+1}-2) \prod_{k=j}^{m-1}\alpha_{i-k}\big\}+(\lambda_{i-m+2}-2)\alpha_{i-m+1}\\ &\quad +(\lambda_{i+1}-2)\prod_{k=0}^{m-1}\alpha_{i-k}\Big]\\ &= \frac{1}{\alpha_{i+1}(A-1)}\Big[\sum_{j=0}^{m-2}\big\{(\lambda_{i-j+1}-2) \prod_{k=j}^{m-1}\alpha_{i-k}\big\}+(\lambda_{i+2}-2)\alpha_{i+1}\Big]\\ &= \frac{1}{A-1}\Big[\sum_{j=0}^{m-2}\big\{(\lambda_{i-j+1}-2) \prod_{k=j}^{m-2}\alpha_{i-k}\big\}+\lambda_{i+2}-2\Big]\\ &= \frac{1}{A-1}\Big[(\lambda_{i+1}-2)\alpha_i\alpha_{i-1}\dots\alpha_{i-(m-2)} +(\lambda_i-2)\alpha_{i-1}\alpha_{i-2}\dots\alpha_{i-m+2}+\dots\\ &\quad +(\lambda_{i-m+3}-2)\alpha_{i-m+2}+\lambda_{i+2}-2\Big]\\ &= \frac{1}{A-1}\Big[\lambda_{i+2}-2+\sum_{j=1}^{m-1}\Big\{(\lambda_{i+2+j}-2) \prod_{k=0}^{j-1}\alpha_{i+2+k}\Big\}\Big] =\frac{\Lambda_{i+2}}{A-1}. \end{align*} Therefore, we obtain \eqref{mean sol estimate} by the definition of \beta_i. Put B_\rho(x)=\{y\in\mathbb{R}^N:|y-x|\leq\rho\}. Since u_i, i=1,2,\dots,m, are subharmonic functions in \mathbb{R}^N, we have \begin{align*} u_i(x) & \leq {\frac{1}{|B_{|x|/2}(x)|}\int_{B_{|x|/2}(x)}u_i(y)dy}\\ & \leq {\frac{C}{|x|^N}\int_{B_{3|x|/2}(0)\backslash B_{|x|/2}(0)}u_i(y)dy}\\ & = {\frac{C}{|x|^N}\int_{|x|/2}^{3|x|/2}\int_{|y|=r}u_i(y)dSdr}\\ & = {\frac{C}{|x|^N}\int_{|x|/2}^{3|x|/2}r^{N-1}\bar u_i(r)dr}\\ & \leq {\frac{C}{|x|^N}\int_{|x|/2}^{3|x|/2}r^{N-1+\beta_i}dr}\\ & = {\frac{C}{|x|^N}\big[\big(\frac{3|x|}{2}\big)^{N+\beta_i} -\big(\frac{|x|}{2}\big)^{N+\beta_i}\big]}\\ & = {C|x|^{\beta_i}\quad \mbox{at }\infty,} \end{align*} where C>0 is a constant. Thus the proof is complete. \end{proof} \begin{remark}\label{twosystem} {\rm In \cite{Bidaut-Veron-Grillot}, M-F. Bidaut-Veron and P. Grillot have obtained important estimates of solutions on singularities for the case m=2. In the case m=2, by using Kelvin transformation, the estimates which they obtained become the same as those which we got in Theorem \ref{estimate}. Furthermore, it is important that these estimates hold without assumptions \alpha_1\geq 1 and \alpha_2\geq 1.} \end{remark} \subsection{Radially symmetric system} In this subsection we study the nonexistence of nonnegative nontrivial radial entire solutions of \eqref{problem}. Through this subsection we always assume that P_i, i=1,2,\dots,m, have radial symmetry. \begin{theorem}\label{nonexistence-radial-1} Let N\geq 3. Suppose that P_i, i=1,2,\dots,m, satisfy $$\label{assumption n3} {P_i(r)\geq\frac{C_i}{r^{\lambda_i}},\quad r\geq R_0>0,}$$ where C_i>0 and \lambda_i are constants. Moreover, \Lambda_i defined by \eqref{constant Lambda} satisfy $$\label{condition N3} \Lambda_i\leq 0\quad \mathit{for~some}~i\in\{1,2,\dots,m\}.$$ If (u_1,u_2,\dots,u_m) is a nonnegative radial entire solution of \eqref{problem}, then(u_1,u_2,\dots,u_m)\equiv(0,0,\dots,0)\,.$$\end{theorem} \begin{theorem}\label{nonexistence-radial-2} Let N=2. Suppose that P_i, i=1,2,\dots,m, satisfy $$\label{assumption n2} {P_i(r)\geq\frac{C_i}{r^2(\log r)^{\lambda_i}},\quad r\geq R_0>1,}$$ where C_i>0 and \lambda_i, i=1,2,\dots,m, are constants. Moreover $$\label{condition N2} \Lambda_i\leq A-1\quad \mathit{for~some}~i\in\{1,2,\dots,m\}\,.$$ If (u_1,u_2,\dots,u_m) is a nonnegative radial entire solution of \eqref{problem}, then$$(u_1,u_2,\dots,u_m)\equiv(0,0,\dots,0). $$\end{theorem} \begin{theorem}\label{nonexistence-radial-3} Let N=1. Suppose that P_i, i=1,2,\dots,m, satisfy \eqref{assumption n3} with some constants C_i>0 and \lambda_i,~i=1,2,\dots,m. Moreover$$\Lambda_i\leq A-1\quad \mathit{for~some}~i\in\{1,2,\dots,m\}.$$If (u_1,u_2,\dots,u_m) is a nonnegative radial entire solution of \eqref{problem}, then$$(u_1,u_2,\dots,u_m)\equiv(0,0,\dots,0).$$\end{theorem} \begin{proof}[Proof of Theorem \ref{nonexistence-radial-1}] Let (u_1,u_2,\dots,u_m) be a nonnegative nontrivial radial entire solution of \eqref{problem}. Then (u_1,u_2,\dots,u_m) satisfies the system of ordinary differential equations $$\label{radial n} \begin{gathered} {(r^{N-1}u_i'(r))'=r^{N-1}P_i(r)u_{i+1}(r)^{\alpha_i},\quad r>0,}\\ u_i'(0)=0, \end{gathered}\quad i=1,2,\dots,m.$$ Integrating \eqref{radial n} over [0,r], we have$$ {u_i'(r)=r^{1-N}\int_0^r s^{N-1}P_i(s)u_{i+1}(s)^{\alpha_i}ds,\quad i=1,2,\dots,m.} $$Hence, we see that u_i, i=1,2,\dots,m, are nondecreasing on r\geq 0. Integrating \eqref{radial n} twice over [R,r], for R\geq 0 and i=1,2,\dots,m, we have $$\label{int Rn} {u_i(r)\geq u_i(R)+\frac{1}{N-2}\int_R^rs\big[1-(\frac{s}{r})^{N-2}\big] P_i(s)u_{i+1}(s)^{\alpha_i}ds,}$$ Since u_i, i=1,2,\dots,m, are nonnegative, nontrivial and nondecreasing functions, there exists an r_*>0 such that u_{i_0}(r_*)>0 for some i_0\in\{1,2,\dots,m\}. We may assume that r_*\geq R_0. We see from \eqref{int Rn} with R=r_* that u_i(r)>0 for r>r_*,~i=1,2,\dots,m. Using similar arguments as in the proof of Theorem \ref{estimate}, we obtain $$\label{contra} {u_i(r)\leq C_i r^{\beta_i}\quad\mbox{at }\infty,\quad i=1,2,\dots,m,}$$ where C_i>0 are constants and \beta_i are defined by \eqref{constant beta}. Note that our assumption \eqref{condition N3} shows \beta_i\leq 0 for some i\in\{1,2,\dots,m\}. If there exists an i_0\in\{1,2,\dots,m\} such that \Lambda_{i_0}<0, then we see that \beta_{i_0}<0 in \eqref{contra}. This shows that u_{i_0} tends to 0 as r\to\infty. On the other hand, from \eqref{int Rn} with R=r_* we see that$$ {u_{i_0}(r)>u_{i_0}(r_*)>0,\quad r>r_*+1.} $$This is a contradiction. It remains only to discuss the case that \Lambda_i\geq 0, i=1,2,\dots,m. From the assumption of \Lambda_i, there exists an i_0\in\{1,2,\dots,m\} such that \Lambda_{i_0}=0. Without loss of generality we may assume that i_0=m, that is,$$ \Lambda_i\geq 0,\quad i=1,2,\dots,m-1 \quad\mbox{and}\quad \Lambda_m=0\,. From the definition of \beta_i it follows that \beta_i\geq 0 and \beta_m=0. We first observe that $$\label{lambda m-1} \lambda_{m-1}\leq 2$$ and $$\label{lambda i} {\lambda_i\leq-\sum_{j=1}^{m-i-1}\Big\{(\lambda_{i+j}-2)\prod_{k=0}^{j-1}\alpha_{i+k}\Big\} +2,\quad i=1,2,\dots,m-2.}$$ In fact, from the definition of \Lambda_i, we obtain \begin{align*} \lambda_i&\geq {-\sum_{j=1}^{m-1}\Big\{(\lambda_{i+j}-2)\prod_{k=0}^{j-1}\alpha_{i+k}\Big\}+2}\\ &= -\Big(\sum_{j=1}^{m-i-1} +\sum_{j=m-i+1}^{m-1}\Big) \Big\{(\lambda_{i+j}-2)\prod_{k=0}^{j-1}\alpha_{i+k}\Big\} -(\lambda_m-2)\prod_{k=0}^{m-i-1}\alpha_{i+k}+2\\ &\equiv -S_1-S_2-S_3+2\,. \end{align*} From the assumption on \Lambda_m, we have \lambda_m-2=-\sum_{j=1}^{m-1}\Big\{(\lambda_{m+j}-2)\prod_{k=1}^{j-1}\alpha_{m+k}\Big\}. Substituting this relation to S_3 we have \begin{align*} S_3&={-\sum_{j=1}^{m-1}\Big\{(\lambda_{m+j}-2)\prod_{k=0}^{j-1}\alpha_{m+k}\Big\} \prod_{k=0}^{m-i-1}\alpha_{i+k}}\\ &= {-\sum_{j=1}^{m-1}\Big\{(\lambda_{m+j}-2)\prod_{k=0}^{j+m-i-1}\alpha_{i+k}\Big\}}\\ &= {-\Big(\sum_{j=m-i+1}^{m-1}+\sum_{j=m}^{2m-i-1}\Big) \Big\{(\lambda_{i+j}-2)\prod_{k=0}^{j-1}\alpha_{i+k}\Big\}}\\ &= {-S_2-\sum_{j=0}^{m-i-1}\Big\{(\lambda_{i+m+j}-2)\prod_{k=0}^{j+m-1}\alpha_{i+k}\Big\}}\\ &= -S_2-S_1A-(\lambda_i-2)A. \end{align*} Thus we obtain \lambda_i\geq S_1(A-1)+(\lambda_i-2)A+2, namely \begin{align*} 0&\geq (A-1)(\lambda_i-2+S_1)\\ &= (A-1)\Big[\lambda_i-2+\sum_{j=1}^{m-i-1}\Big\{(\lambda_{i+j}-2) \prod_{k=0}^{j-1}\alpha_{i+k}\Big\}\Big]. \end{align*} Since A>1, we see that \eqref{lambda i} holds. Similarly we can get \eqref{lambda m-1}. From the above computation we see that {\lambda_i<-\sum_{j=1}^{m-i-1}\Big\{(\lambda_{i+j}-2)\prod_{k=0}^{j-1}\alpha_{i+k}\Big\}+2} \quad\mbox{if } \Lambda_i>0 $$and$$ {\lambda_i=-\sum_{j=1}^{m-i-1}\Big\{(\lambda_{i+j}-2)\prod_{k=0}^{j-1}\alpha_{i+k}\Big\}+2} \quad\mbox{if }\Lambda_i=0\,. For the rest of this article C denotes various positive constants. Integrating \eqref{radial n} twice over [r_*,r], from \eqref{assumption n3}, we have \begin{aligned} u_i(r) & \geq {u_i(r_*)+\frac{1}{N-2}\int_{r_*}^rs\big[1-(\frac{s}{r})^{N-2}\big] P_i(s)u_{i+1}(s)^{\alpha_i}\,ds,} \\ & \geq {u_i(r_*)+\frac{C_i}{N-2}\big[1-(\frac{1}{2})^{N-2}\big] \int_{r_*}^{r/2}sP_i(s)u_{i+1}(s)^{\alpha_i}\,ds} \\ & \geq {C\int_{r_*}^{r/2}s^{1-\lambda_i}u_{i+1}(s)^{\alpha_i}ds,} \end{aligned}\label{contra int} where r\geq 2r_*, i=1,2,\dots,m. We first consider the case that \Lambda_{m-1}=0. From \eqref{lambda m-1} we see that \lambda_{m-1}=2. From \eqref{contra int} with i=m-1, we have $u_{m-1}(r)\geq Cu_m(r_*)^{\alpha_{m-1}}\int_{r_*}^{r/2}s^{-1}ds \geq C\log r,\quad r\geq r_1>2r_*.$ On the other hand, we can see that \beta_{m-1}=0 in \eqref{contra}; that is, u_{m-1} is bounded near infinity. This is a contradiction. Next we consider the case that \Lambda_{m-2}=0. Then we see from \eqref{lambda m-1} and \eqref{lambda i} with i=m-2 that \lambda_{m-1}<2\quad\mathrm{and}\quad \lambda_{m-2}=-(\lambda_{m-1}-2)\alpha_{m-2}+2. From \eqref{contra int} with i=m-1 we have $u_{m-1}(r)\geq Cu_{m}(r_*)^{\alpha_{m-1}} \int_{r_*}^{r/2}s^{1-\lambda_{m-1}}ds \geq Cr^{2-\lambda_{m-1}},\quad r\geq r_1>2r_*.$ From this estimate and \eqref{contra int} with i=m-2 we obtain \begin{align*} u_{m-2}(r)&\geq C\int_{r_1}^{r/2}s^{1-\lambda_{m-2}+(2-\lambda_{m-1}) \alpha_{m-2}}ds \\ &= C\int_{r_1}^{r/2}s^{-1}ds \\ &\geq C\log r,\quad r\geq r_2>2r_1\,. \end{align*} On the other hand, we can see that \beta_{m-2}=0 in \eqref{contra}; that is, u_{m-2} is bounded near infinity. This is a contradiction. Similarly, suppose that there exists an i_0\in\{1,2,\dots,m\} such that {\Lambda_{i_0}=0} and \Lambda_i>0,~i=i_0+1,\dots,m-1. Then we see from \eqref{lambda m-1} and \eqref{contra int} with i=m-1 that u_{m-1}(r)\geq Cr^{2-\lambda_{m-1}},\quad r\geq r_1>2r_*. From this estimate, \eqref{lambda i} with i=m-2, \eqref{contra int} with i=m-2, we have \begin{align*} u_{m-2}(r)&\geq {C\int_{r_*}^{r}s^{1-\lambda_{m-2} +\alpha_{m-2}(2-\lambda_{m-1})}ds}\\ &\geq {Cr^{2-\lambda_{m-2}+\alpha_{m-2}(2-\lambda_{m-1})},\quad r\geq r_2>2r_1.} \end{align*} By repeating the above procedure, we get a sequence {\{r_j\}_{j=2}^{m-{i_0}-1}} such that {u_i(r)\geq Cr^{\tau_i},\quad r\geq r_j>2r_{j-1},\quad i=m-2,m-3,\dots,i_0+1,} where \begin{align*} \tau_i&= 2-\lambda_i+\alpha_i\tau_{i+1}\\ &= {2-\lambda_i+\sum_{j=1}^{m-i-1} \Big\{(2-\lambda_{i+j})\prod_{k=0}^{j-1}\alpha_{i+k}\Big\}>0.} \end{align*} From \eqref{lambda i} with i=i_0 and \eqref{contra int} with i=i_0, we have \begin{align*} u_{i_0}(r)&\geq {C\int_{r_{m-i_0-1}}^{r/2}s^{1-\lambda_{i_0}+\alpha_{i_0} \tau_{i_0 +1}}ds}\\ &= {C\int_{r_{m-i_0-1}}^{r/2}s^{-1}ds}\\ &\geq C\log r,\quad r\geq r_{m-i_0}>2r_{m-i_0-1}. \end{align*} On the other hand, since \Lambda_{i_0}=0, we have \beta_{i_0}=0 in \eqref{contra}. This yields a contradiction. Thus the proof of Theorem \ref{nonexistence-radial-1} is complete. \end{proof} \begin{proof}[Proof of Theorem \ref{nonexistence-radial-2}] Suppose to the contrary that \eqref{problem} has a nonnegative nontrivial radial entire solution (u_1,u_2,\dots,u_m). Then (u_1,u_2,\dots,u_m) satisfies \eqref{radial n}. Integrating \eqref{radial n} twice over [0,r], we have $$\label{integral n2} {u_i(r)=u_i(0)+\int_0^r s\log(\frac{r}{s})P_i(s)u_{i+1}(s)^{\alpha_i}ds,\quad i=1,2,\dots,m.}$$ Let r\geq e. Then from \eqref{integral n2}, we have \begin{aligned} u_i(r) & = {u_i(0)+\int_0^1 s\log(\frac{r}{s})P_i(s)u_{i+1}(s)^{\alpha_i}ds} \\ & {+\int_1^e s\log(\frac{r}{s})P_i(s)u_{i+1}(s)^{\alpha_i}ds +\int_e^r s\log(\frac{r}{s})P_i(s)u_{i+1}(s)^{\alpha_i}ds} \\ & \geq {u_i(0)+u_{i+1}(0)^{\alpha_i}\int_0^1 sP_i(s)ds\log r +\int_e^r s\log(\frac{r}{s})P_i(s)u_{i+1}(s)^{\alpha_i}ds} \\ & \geq {\tilde C_i\log r+\int_e^r s\log(\frac{r}{s})P_i(s)u_{i+1}(s)^{\alpha_i}ds, \quad r\geq e,} \end{aligned}\label{sol esti intn2} where i=1,2,\dots,m and \tilde C_i\geq 0 are constants. Let u_i(r)=v_i(r)\log r. Then from \eqref{sol esti intn2}, we have $$\label{sol esti v} {v_i(r)\geq \tilde C_i+\int_e^r s\left(1-\frac{\log s}{\log r}\right)P_i(s) (\log s)^{\alpha_i}v_{i+1}(s)^{\alpha_i}ds.}$$ Let t=\log s, \eta=\log r, and v_i(r)=v_i(e^\eta)=\tilde v_i(\eta). Then \eqref{sol esti v} becomes {\tilde v_i(\eta)\geq\tilde C_i+\int_1^\eta t\big(1-\frac{t}{\eta}\big) \tilde P_i(t)\tilde v_i(t)^{\alpha_i}dt,\quad i=1,2,\dots,m,} $$where \tilde P_i, i=1,2,\dots,m, are given by \tilde P_i(t)=e^{2t}P_i(e^t)t^{\alpha_i-1}. From \eqref{assumption n2}, we have$$ {\tilde P_i(t)\geq e^{2t}\frac{C_i}{e^{2t}(\log e^t)^{\lambda_i}}t^{\alpha_i-1} =\frac{C_i}{t^{\lambda_i-\alpha_i+1}},\quad t\geq \log{R_0},\quad i=1,2,\dots,m.} From \eqref{condition N2} and the definition of \Lambda_i, \begin{align*} \lambda_i-\alpha_i+1 &={\Lambda_i+2-\sum_{j=1}^{m-1}\Big\{(\lambda_{i+j}-2)\prod_{k=0}^{j-1}\alpha_{i+k}\Big\} -\alpha_i+1}\\ &\leq 2-\sum_{j=1}^{m-1}\Big\{\big((\lambda_{i+j}-\alpha_{i+j}+1)-2\big) \prod_{k=0}^{j-1}\alpha_{i+k}\Big\} \\ &\quad +A-\alpha_i-\sum_{j=1}^{m-1}\Big\{(\alpha_{i+j}-1)\prod_{k=0}^{j-1}\alpha_{i+k}\Big\}\\ &= {2-\sum_{j=1}^{m-1}\Big\{\big((\lambda_{i+j}-\alpha_{i+j}+1)-2\big) \prod_{k=0}^{j-1}\alpha_{i+k}\Big\}}, \end{align*} namely, for some i\in\{1,2,\dots,m\}, (\lambda_i-\alpha_i+1)-2+\sum_{j=1}^{m-1} \Big\{((\lambda_{i+j}-\alpha_{i+j}+1)-2)\prod_{k=0}^{j-1}\alpha_{i+k}\Big\}\leq 0\,. Using similar arguments as in the proof of Theorem \ref{nonexistence-radial-1}, we obtain a contradiction. Thus the proof is complete. \end{proof} \begin{proof}[Proof of Theorem \ref{nonexistence-radial-3}] Let (u_1,u_2,\dots,u_m) be a nonnegative nontrivial radial entire solution of \eqref{problem}. Then by integrating \eqref{problem} over [0,r], we have \begin{align*} u_i(r) &= {u_i(0)+\int_0^1(r-s)P_i(s) u_{i+1}(s)^{\alpha_i}ds +\int_1^r(r-s)P_i(s)u_{i+1}(s)^{\alpha_i}ds}\\ &\geq {u_i(0)+u_{i+1}(0)^{\alpha_i}\int_0^1r\left(1-\frac{s}{r}\right)P_i(s)ds +\int_1^r(r-s)P_i(s)u_{i+1}(s)^{\alpha_i}ds}\\ &\geq \tilde C_ir+\int_1^r(r-s)P_i(s)u_{i+1}(s)^{\alpha_i}ds, \end{align*} where i=1,2,\dots,m, r\geq 2, and \tilde C_i\geq 0 are constants. Setting u_i(r)=rv_i(r) for r\geq 2 and i=1,2,\dots,m, we obtain v_i(r)\geq \tilde C_i+\int_1^rs\left(1-\frac{s}{r}\right) \tilde P_i(s)v_{i+1}(s)^{\alpha_i}ds, $$where \tilde P_i(s)=P_i(s)s^{\alpha_i-1}. From \eqref{assumption n3}, we have$$ {\tilde P_i(s)\geq \frac{C_i}{s^{\lambda_i-\alpha_i+1}},\quad s\geq R_0, \quad i=1,2,\dots,m.} $$Using the same computation as in the proof of Theorem \ref{nonexistence-radial-2}, we can see that for some i\in\{1,2,\dots,m\},$$ (\lambda_i-\alpha_i+1)-2+\sum_{j=1}^{m-1}\Big\{\big((\lambda_{i+j}-\alpha_{i+j}+1)-2\big) \prod_{k=0}^{j-1}\alpha_{i+k}\Big\}\leq 0\,. $$From the proof of Theorem \ref{nonexistence-radial-1}, we get a contradiction. Thus the proof is complete. \end{proof} \subsection{System \eqref{problem} without radial symmetry} In this subsection we consider the nonexistence of nonnegative nontrivial entire solutions of \eqref{problem} without radial symmetry. Through this subsection we always assume that \alpha_i\geq 1, i=1,2,\dots,m, and A>1. \begin{theorem}\label{nonexistence-1} Let N\geq 3. Suppose that P_i, i=1,2,\dots,m, satisfy $$\label{condition n3 P} {\liminf_{|x|\to\infty}|x|^{\lambda_i}P_i(x)>0,}$$ where \lambda_i, i=1,2,\dots,m, are constants. Also \Lambda_i\leq 0 for some i\in\{1,2,\dots,m\}. If (u_1,u_2,\dots,u_m) is nonnegative entire solution of \eqref{problem}, then$$(u_1,u_2,\dots,u_m)\equiv(0,0,\dots,0).$$\end{theorem} \begin{theorem}\label{nonexistence-2} Let N=2. Suppose that P_i, i=1,2,\dots,m, satisfy $$\label{condition n2 P} {\liminf_{|x|\to\infty}|x|^2(\log|x|)^{\lambda_i}P_i(x)>0,}$$ where \lambda_i are constants. Moreover \Lambda_i\leq A-1 for some i\in\{1,2,\dots,m\}. If (u_1,u_2,\dots,u_m) is nonnegative entire solution of \eqref{problem}, then$$(u_1,u_2,\dots,u_m)\equiv(0,0,\dots,0).$$\end{theorem} \begin{theorem}\label{nonexistence-3} Let N=1. Suppose that P_i, satisfy \eqref{condition n3 P} with some constants \lambda_i, i=1,2,\dots,m. Moreover \Lambda_i\leq A-1 for some i\in\{1,2,\dots,m\}. If (u_1,u_2,\dots,u_m) is nonnegative entire solution of \eqref{problem}, then$$(u_1,u_2,\dots,u_m)\equiv(0,0,\dots,0).$$\end{theorem} Suppose that \eqref{condition n3 P} holds. Then there exist some constants C_i>0 and R_0>0 such that$$ P_i(x)\geq \frac{C_i}{|x|^{\lambda_i}}, \quad |x|\geq R_0,\quad i=1,2,\dots,m\,. $$So we can see that \hat P_i defined by \eqref{mean p} satisfy$$ \hat P_i(r)\geq \frac{C_i}{r^{\lambda_i}},\quad r\geq R_0\,. $$Similarly, suppose that \eqref{condition n2 P} holds. Then \hat P_i satisfy$$ \hat P_i(r)\geq\frac{C_i}{r^2(\log r)^{\lambda_i}},\quad r\geq R_0>1, $$where i=1,2,\dots,m, and C_i>0 are some constants. The proof of Theorem \ref{nonexistence-1} follows from Lemma \ref{lemma-1}, Theorem \ref{estimate} and the proof of Theorem \ref{nonexistence-radial-1}. Similarly, the proofs of Theorems \ref{nonexistence-2} and \ref{nonexistence-3} follow from Lemma \ref{lemma-1} and the proofs of Theorems \ref{nonexistence-radial-2} and \ref{nonexistence-radial-3}, respectively. \begin{remark} {\rm When m=2, our nonexistence results (Theorems \ref{nonexistence-1}--\ref{nonexistence-3}) reduce to those obtained in \cite{Teramoto}. However, the proofs presented here are simpler than in \cite{Teramoto}.} \end{remark} \section{Existence results} In this section we consider existence of positive radial entire solutions of the semilinear elliptic system $$\label{radial} \begin{gathered} \Delta u_1=P_1(|x|)u_2^{\alpha_1},\\ \Delta u_2=P_2(|x|)u_3^{\alpha_2},\\ \vdots\\ \Delta u_m=P_m(|x|)u_{m+1}^{\alpha_m},\quad u_{m+1}=u_1\,. \end{gathered}$$ Through this section, we assume that P_i(r), r=|x|, i=1,2,\dots,m, are nonnegative continuous functions and \alpha_i>0 are constants satisfying A>1. \begin{theorem}\label{existence-1} Let N\geq 3. Suppose that P_i satisfy $$\label{condition P n3} {P_i(r)\leq\frac{C_i}{r^{\lambda_i},}\quad r\geq R_0>0,}$$ where ~i=1,2,\dots,m, and C_i>0, \lambda_i are constants. Moreover \Lambda_i>0,~i=1,2,\dots,m. Then \eqref{radial} has infinitely many positive radial entire solutions. \end{theorem} \begin{theorem}\label{existence-2} Let N=2. Suppose that P_i satisfy $$\label{condition P n2} {P_i(r)\leq\frac{C_i}{r^2(\log r)^{\lambda_i}},\quad r\geq R_0>1,}$$ where i=1,2,\dots,m, and C_i>0 and \lambda_i are constants. Moreover $$\label{lambda n2} \Lambda_i>A-1,\quad i=1,2,\dots,m\,.$$ Then \eqref{radial} has infinitely many positive radial entire solutions. \end{theorem} \begin{theorem}\label{existence-3} Let N=1. Suppose that P_i satisfy \eqref{condition P n3} with some constants C_i>0 and \lambda_i, i=1,2,\dots,m. Moreover \Lambda_i>A-1,~i=1,2,\dots,m. Then \eqref{radial} has infinitely many positive entire solutions. \end{theorem} We give an example that shows the sharpness of our results. \subsection*{Example} Let us consider the elliptic system $$\label{example} \begin{gathered} \Delta u_1={\frac{1}{(1+|x|)^{\lambda_1}}u_2^{\alpha_1},}\\ \Delta u_2={\frac{1}{(1+|x|)^{\lambda_2}}u_3^{\alpha_2},}\\ \vdots\\ \Delta u_m={\frac{1}{(1+|x|)^{\lambda_m}}u_1^{\alpha_m},} \end{gathered}$$ where x\in\mathbb{R}^N, N\geq 3, and \alpha_i>0,i=1,2,\dots,m, are constants satisfying \alpha_1\alpha_2\cdots \alpha_m>1. We can completely characterize the existence of positive radial entire solutions of this system in terms of \alpha_i and \lambda_i, i=1,2,\dots,m. In fact, we can see that the inequalities$$ {\frac{C_i}{|x|^{\lambda_i}}\leq\frac{1}{(1+|x|)^{\lambda_i}} \leq\frac{\tilde C_i}{|x|^{\lambda_i}},\quad |x|\geq 1,\quad i=1,2,\dots,m} $$hold for some constants C_i>0 and \tilde C_i>0, i=1,2,\dots,m. Then, from Theorem \ref{nonexistence-radial-1} and Theorem \ref{existence-1}, a necessary and sufficient condition for \eqref{example} to have positive radial entire solution is$$\Lambda_i>0,\quad i=1,2,\dots,m.$$\begin{proof}[Proof of Theorem \ref{existence-1}] Without loss of generality, we assume that R_0=1 in \eqref{condition P n3}. We first observe that (u_1,u_2,\dots,u_m) is a positive radial entire solution of \eqref{radial} if and only if the function (v_1(r),v_2(r),\dots,v_m(r))=(u_1(x),u_2(x),\dots,u_m(x)), r=|x|, satisfies the system of second order ordinary differential equations $$\label{radial2} \begin{gathered} r^{1-N}(r^{N-1}v_i')'=P_i(r)v_{i+1}^{\alpha_i},\quad r>0,\\ v_i'(0)=0, \end{gathered}$$ where i=1,2,\dots,m, and '=d/dr. Integrating \eqref{radial2} twice, we obtain the following system of integral equations equivalent to \eqref{radial2}: $$\label{integral n3} v_i(r)=a_i+\frac{1}{N-2}\int_0^rs\big[1-(\frac{s}{r})^{N-2}\big] P_i(s)v_{i+1}(s)^{\alpha_i}ds\,,$$ where r\geq 0, i=1,2,\dots,m, and a_i=v_i(0). Therefore, it suffices to solve \eqref{integral n3}. Choose constants a_i>0, i=1,2,\dots,m, so that $$\label{constant} \begin{gathered} {\frac{(2a_{i+1})^{\alpha_i}}{N-2}\int_0^1 sP_i(s)ds\leq\frac{a_i}{2},}\\ {\frac{C_i(2a_{i+1})^{\alpha_i}}{(N-2)(2-\lambda_i+\alpha_i\beta_{i+1})} \leq\frac{a_i}{2},} \end{gathered}$$ where \beta_i, i=1,2,\dots,m, are defined by \eqref{constant beta}. It is possible to choose such a_i's by the assumption A>1. We note that 2-\lambda_i+\alpha_i\beta_{i+1}=\beta_i by the definitions of \Lambda_i and \beta_i. Define the functions F_i,~i=1,2,\dots,m, by$$ F_i(r)=\begin{cases} 2a_i & \mbox{for } 0\leq r\leq 1,\\ 2a_ir^{\beta_i} & \mbox{for } r\geq 1. \end{cases} $$We regard the space (C[0,\infty))^m as a Fr\'echet space equipped with the topology of uniform convergence of functions on each compact subinterval of [0,\infty). Let X\subset (C[0,\infty))^m denotes the subset defined by$$ X=\{(v_1,v_2,\dots,v_m)\in (C[0,\infty))^m: a_i\leq v_i(r)\leq F_i(r),~r\geq 0,~~1\leq i\leq m\}. Clearly, X is a non-empty closed convex subset of (C[0,\infty))^m. Define the mapping \mathcal{F}:X\to (C[0,\infty))^m by \mathcal{F}(v_1,v_2,\dots,v_m)=(\tilde v_1,\tilde v_2,\dots,\tilde v_m), where $\tilde v_i(r)=a_i+\frac{1}{N-2}\int_0^rs\big[1-(\frac{s}{r})^{N-2}\big] P_i(s)v_{i+1}(s)^{\alpha_i}ds,\quad r\geq 0\,.$ To apply the Schauder-Tychonoff fixed point theorem, we show that \mathcal{F} is a continuous mapping from X into itself such that \mathcal{F}(X) is relatively compact. \noindent\textbf{(I)} \mathcal{F} maps X into itself. Let (v_1,v_2,\dots,v_m)\in X. Clearly, \tilde v_i\geq a_i, i=1,2,\dots,m. For 0\leq r\leq 1, we have \begin{align*} \tilde v_i(r) &\leq {a_i+\frac{1}{N-2}\int_0^rsP_i(s)v_{i+1}(s)^{\alpha_i}ds}\\ &\leq {a_i+\frac{(2a_{i+1})^{\alpha_i}}{N-2}\int_0^1sP_i(s)ds}\\ &\leq {a_i+\frac{a_i}{2}<2a_i,\quad i=1,2,\dots,m.} \end{align*} For r\geq 1, from \eqref{condition P n3}, we have \begin{align*} \tilde v_i(r) &\leq {a_i+\frac{1}{N-2}\int_0^1sP_i(s)v_{i+1}(s)^{\alpha_i}ds +\frac{1}{N-2}\int_1^rsP_i(s)v_{i+1}(s)^{\alpha_i}ds}\\ &\leq {\frac{3a_i}{2}+\frac{(2a_{i+1})^{\alpha_i}C_i}{N-2}\int_1^rs^{1-\lambda_i +\alpha_i\beta_{i+1}}ds}\\ &\leq {\frac{3a_i}{2}+\frac{(2a_{i+1})^{\alpha_i}C_i}{(N-2)(2-\lambda_i +\alpha_i\beta_{i+1})}r^{2-\lambda_i+\alpha_i\beta_{i+1}}}\\ &\leq {\frac{3a_i}{2}+\frac{a_i}{2}r^{\beta_i}\leq 2a_ir^{\beta_i},\quad i=1,2,\dots,m.} \end{align*} Therefore, \mathcal{F}(X)\subset X. \noindent\textbf{(II)} \mathcal{F} is continuous. Let \{(v_{1,l},v_{2,l},\dots,v_{m,l})\}_{l=1}^{\infty} be a sequence in X which converges to (v_1,v_2,\dots,v_m)\in X uniformly on each compact subinterval of [0,\infty). Then \begin{align*} |\tilde v_{i,l}(r)-\tilde v_i(r)| &\leq \frac{1}{N-2}\int_0^rs\big[1-(\frac{s}{r})^{N-2}\big] P_i(s)|v_{i+1,l}(s)^{\alpha_i}-v_{i+1}(s)^{\alpha_i}|ds\\ &\leq \frac{1}{N-2}\int_0^rsP_i(s)|v_{i+1,l}(s)^{\alpha_i}-v_{i+1}(s)^{\alpha_i}|ds, \quad i=1,2,\dots,m\,. \end{align*} Since the functions h_{i,l}(s)=sP_i(s)|v_{i+1,l}(s)^{\alpha_i}-v_{i+1}(s)^{\alpha_i}|, l\in\mathbb{N},~1\leq i\leq m, satisfy h_{i,l}(s)\leq 2sP_i(s)F_{i+1}(s)^{\alpha_i}, s\geq 0, and \{h_{i,l}(s)\}_{l=1}^{\infty}, i=1,2,\dots,m, converge to 0 at every point s, the Lebesgue dominated convergence theorem implies that \{\tilde v_{i,l}\}_{l=1}^{\infty}, i=1,2,\dots,m, converge to \tilde v_i uniformly on each compact subinterval of [0,\infty). These imply the continuity of \mathcal{F}. \noindent\textbf{(III)} \mathcal{F}(X) is relatively compact. It suffices to show the local equicontinuity of \mathcal{F}(X), since \mathcal{F}(X) is locally uniformly bounded by the fact that \mathcal{F}(X)\subset X. Let (v_1,v_2,\dots,v_m)\in X and R>0. Then we have \tilde v_i'(r) = \int_0^r(\frac{s}{r})^{N-1}P_i(s)v_{i+1}(s)^{\alpha_i}ds \leq \int_0^RP_i(s)F_{i+1}(s)^{\alpha_i}ds\,. $$These imply the local boundedness of the set \{(\tilde v_1',\tilde v_2',\dots,\tilde v_m');~(v_1,v_2,\dots,v_m)\in X\}. Hence the relative compactness of \mathcal{F}(X) is shown by the Ascoli-Arzel\`{a} theorem. Therefore, applying the Schauder-Tychonoff fixed point theorem, there exists an element (v_1,v_2,\dots,v_m)\in X such that (v_1,v_2,\dots,v_m)=\mathcal{F}(v_1,v_2,\dots,v_m), that is, (v_1,v_2,\dots,v_m) satisfies the system of integral equations \eqref{integral n3}. The function (u_1(x),u_2(x),\dots,u_m(x))=(v_1(|x|),\dots,v_m(|x|)) then gives a solution of \eqref{radial2}. Since infinitely many (a_1,a_2,\dots,a_m) satisfy \eqref{constant}, we can construct an infinitude of positive radial entire solutions of \eqref{radial}. This completes the proof. \end{proof} \begin{proof}[Proof of Theorem \ref{existence-2}] Without loss of generality, we may assume that R_0=e in \eqref{condition P n2}. As before, it suffices to solve the following system of integral equations:$$ {v_i(r)=a_i+\int_0^rs\log(\frac{r}{s})P_i(s)v_{i+1}(s)^{\alpha_i}ds,\quad r\geq 0, \quad i=1,2,\dots,m,} $$where a_i=v_i(0). Choose constants a_i>0 so that \begin{gather*} (2a_{i+1})^{\alpha_i}e\int_0^e P_i(s)ds\leq\frac{a_i}{2},\\ \frac{(2a_{i+1})^{\alpha_i}C_i}{1-\lambda_i+\alpha_i\beta_{i+1}} \leq\frac{a_i}{2}, \end{gather*} where \beta_i, i=1,2,\dots,m, are defined by \eqref{constant beta}. It is possible to choose such a_i's by the assumption A>1. We notice that \beta_i>1 by the assumption \eqref{lambda n2}. Define the functions F_i, i=1,2,\dots,m, by$$F_i(r)=\begin{cases} 2a_i & \mbox{for } 0\leq r\leq e,\\ 2a_i(\log r)^{\beta_i} & \mbox{for } r\geq e. \end{cases} $$Consider the set$$ Y=\{(v_1,v_2,\dots,v_m)\in(C[0,\infty))^m: a\leq v_i(r)\leq F_i(r),~r\geq 0,~~1\leq i\leq m\}, $$which is a closed convex subset of (C[0,\infty))^m. Define the mapping \mathcal{F}: Y\to(C[0,\infty))^m by \mathcal{F}(v_1,v_2,\dots,v_m)=(\tilde v_1,\tilde v_2,\dots,\tilde v_m), where$$ \tilde v_i(r)=a_i+\int_0^r s\log(\frac{r}{s})P_i(s)v_{i+1}(s)^{\alpha_i}ds, \quad r\geq 0,\quad i=1,2,\dots,m\,. We will verify that \mathcal{F} is a continuous mapping from Y into itself such that \mathcal{F}(Y) is relatively compact. We first show that \mathcal{F} maps Y into itself. Let (v_1,v_2,\dots,v_m)\in Y. It is clear that \tilde v_i\geq a_i, i=1,2,\dots,m. Let 0\leq r\leq e. Then, using the inequality 0\leq s\log(r/s)\leq r/e for 0\leq s\leq r, we have \begin{align*} \tilde v_i(r)&\leq {a_i+\frac{r}{e}\int_0^r P_i(s)v_i(s)^{\alpha_i}ds}\\ &\leq a_i+(2a_{i+1})^{\alpha_i}\int_0^e P_i(s)ds\\ &\leq a_i+\frac{a_i}{2}< 2a_i,~i=1,2,\dots,m. \end{align*} Let r\geq e. Then we write \tilde v_i(r)= {a_i+\Big(\int_0^1+\int_1^e+\int_e^r\Big)s \log(\frac{r}{s})P_i(s)v_{i+1}(s)^{\alpha_i}ds} \equiv a_i+I_1+I_2+I_3. The inequality 0\leq s\log(r/s)\leq\log r for 0\leq s\leq 1 implies that $$\label{estimate I_1} I_1 \leq \int_0^1 P_i(s)v_{i+1}(s)^{\alpha_i}ds\log r \leq (2a_{i+1})^{\alpha_i}e\int_0^1 P_i(s)ds\log r.$$ The integrals I_2 and I_3 are estimated as follows: \label{estimate I_2} \begin{aligned} I_2 & \leq {\int_1^e sP_i(s)v_{i+1}(s)^{\alpha_i}ds\log r} \\ & \leq {(2a_{i+1})^{\alpha_i}\int_1^e sP_i(s)ds\log r} \\ & \leq {(2a_{i+1})^{\alpha_i}e\int_1^e P_i(s)ds\log r}~; \end{aligned} \label{estimate I_3} \begin{aligned} I_3 & \leq {\int_e^r sP_i(s)v_{i+1}(s)^{\alpha_i}ds\log r} \\ & \leq {(2a_{i+1})^{\alpha_i}C_i\int_e^rs^{-1}(\log s)^{-\lambda_i +\alpha_i\beta_{i+1}}ds\log r} \\ & = {(2a_{i+1})^{\alpha_i}C_i\int_1^{\log r}t^{-\lambda_i +\alpha_i\beta_{i+1}}dt\log r} \\ & \leq {\frac{(2a_{i+1})^{\alpha_i}C_i}{1-\lambda_i +\alpha_i\beta_{i+1}}(\log r)^{2-\lambda_i+\alpha_i\beta_{i+1}}} \\ & \leq {\frac{a_i}{2}(\log r)^{\beta_i}.} \end{aligned} From \eqref{estimate I_1} and \eqref{estimate I_2}, we have $$I_1+I_2 \leq {(2a_{i+1})^{\alpha_i}e\int_0^e P_i(s)ds\log r} \leq {\frac{a_i}{2}(\log r)^{\beta_i}.}\label{estimate I12}$$ Thus by \eqref{estimate I_3} and \eqref{estimate I12} we obtain \tilde v_i(r)\leq 2a_i(\log r)^{\beta_i},~i=1,2,\dots,m. Therefore, \mathcal{F}(v_1,v_2,\dots,v_m)\in Y. The continuity of \mathcal{F} and the relative compactness of \mathcal{F}(Y) can be verified in a routine manner. Thus there exists an element (v_1,v_2,\dots,v_m)\in Y such that (v_1,v_2,\dots,v_m)=\mathcal{F}(v_1,v_2,\dots,v_m) by the Schauder-Tychonoff fixed point theorem. It is clear that this (v_1,v_2,\dots,v_m) gives rise to a positive radial entire solution (u_1(x),u_2(x),\dots,u_m(x))=(v_1(|x|),v_2(|x|),\dots,v_m(|x|)) of \eqref{radial}. \end{proof} The proof of Theorem \ref{existence-3} is the same as that of Theorem \ref{existence-1}. So we leave the proof to the reader. \section{Liouville type theorem} Consider the semilinear elliptic system $$\label{rewrite problem} \begin{gathered} \Delta u_1=P_1(x)u_2^{\alpha_1},\\ \Delta u_2=P_2(x)u_3^{\alpha_2},\\ \vdots\\ \Delta u_m=P_m(x)u_{m+1}^{\alpha_m},~~u_{m+1}=u_1, \end{gathered}$$ where x\in\mathbb{R}^N, N\geq 3 and m\geq 2 are integers and \alpha_i>0, i=1,2,\dots,m, are constants satisfying \alpha_1\alpha_2\cdots \alpha_m>1. Suppose that {P_i(x)\geq \frac{C_i}{|x|^{\lambda_i}},\quad |x|\geq x_0>0,\quad i=1,2,\dots,m,} $$hold for some constants C_i>0 and \lambda_i\in\mathbb{R}, satisfying \Lambda_i\leq 0 for some i\in\{1,2,\dots,m\}. If, in addition, \alpha_i\geq 1, i=1,2,\dots,m, then as studied in Sections 2.2 and 2.3 one can conclude from Theorems \ref{nonexistence-radial-1} and \ref{nonexistence-1} that system \eqref{rewrite problem} has no nonnegative nontrivial entire solutions. However, if at least one of \alpha_i, i\in\{1,2,\dots,m\}, is less than 1, then one cannot derive any information about the nonnegative nontrivial entire solutions without radial symmetry. When \alpha_1\alpha_2\cdots \alpha_m>1 and the same hypothesis of Theorem \ref{nonexistence-1} hold, does not \eqref{rewrite problem} possess a nonnegative nontrivial entire solutions? To give a partial answer this question we prove a Liouville type theorem for nonnegative entire solutions of \eqref{rewrite problem}. Our result is as follows: \begin{theorem}\label{liouville} Let N\geq 3. Suppose that $$\label{assumption 2} {\liminf_{|x|\to\infty}|x|^{\lambda_i}P_i(x)>0,\quad i=1,2,\dots,m,}$$ hold for some constants \lambda_i,i=1,2,\dots,m, and there exists an i_0\in\{1,2,\dots,m\} such that \Lambda_{i_0}\leq 0. If (u_1,u_2,\dots,u_m) is a nonnegative entire solution of \eqref{rewrite problem} satisfying $$\label{order} u_{i_0}(x)=O(\exp |x|^{\rho})~as~|x|\to\infty \quad\mbox{for some }\rho>0,$$ then (u_1,u_2,\dots,u_m)\equiv(0,0,\dots,0). \end{theorem} The next lemma is needed in proving Theorem \ref{liouville}. \begin{lemma}\label{lemma-2} Let (u_1,u_2,\dots,u_m) be a nonnegative entire solution of \eqref{rewrite problem}, and b\in(0,1) be a constant. Then its spherical mean (\bar u_1,\bar u_2,\dots,\bar u_m) satisfies the ordinary differential inequalities $$\label{ine} \begin{gathered} {\bar u_i'(r)\geq \tilde C_i rP_{i*}(r)\bar u_{i+1}(br)^{\alpha_i},~r>0,}\\ \bar u_i'(0)=0, \end{gathered}\quad i=1,2,\dots,m,$$ where \tilde C_i=\tilde C_i(N,\alpha_i,b)>0, i=1,2,\dots,m, are constants and$$ P_{i*}(r)=\min_{|x|\leq r}P_i(x),\quad r\geq 0,\quad i=1,2,\dots,m. $$\end{lemma} To prove this lemma, we present the following lemma; see \cite[p.244]{Gilbarg-Trudinger} or \cite[p.225]{Taylor}. \begin{lemma}\label{lemma-3} Let D be a domain in \mathbb{R}^N. Suppose that \sigma>0 is a constant, and x_0\in D and r>0 satisfy B_{2r}(x_0)\equiv\{x\in\mathbb{R}^N;|x-x_0|\leq 2r\}\subset D. Then, we can find a constant C=C(N,\sigma)>0 satisfying$$ \Big(\max_{B_r(x_0)}u \Big)^\sigma\leq \frac{C}{r^N}\int_{B_{2r}(x_0)}u^\sigma dx, $$for any function u\in C^2(D) satisfying u\geq 0, \Delta u\geq 0 in D. \end{lemma} \begin{proof}[Proof of Lemma \ref{lemma-2}] Let (u_1,u_2,\dots,u_m) be a nonnegative entire solution of \eqref{rewrite problem}. By taking the mean value of \eqref{rewrite problem}, we have $$\label{mean value} {(r^{N-1}\bar u_i'(r))'=\frac{1}{\omega_N}\int_{|x|=r} P_i(x)u_{i+1}(x)^{\alpha_i}dS,\quad r\geq 0,\quad i=1,2,\dots,m.}$$ Since an integration of \eqref{mean value} shows that \bar u_i(r),~i=1,2,\dots,m, are nondecreasing on [0,\infty), we may assume that b>1/2 in \eqref{ine}. Put b=1-a,~a\in (0,1/2). Integrating \eqref{mean value} over [0,r], we have $$\label{prime ui} \bar u_i'(r) = \frac{1}{\omega_N r^{N-1}}\int_{|x|\leq r}P_i(x)u_{i+1}(x)^{\alpha_i}dx \geq \frac{P_{i*}(r)}{\omega_N r^{N-1}}\int_{|x|\leq r}u_{i+1}(x)^{\alpha_i}dx\,.$$ Let r>0 be fixed. We take y_{i+1}\in\mathbb{R}^N,~i=1,2,\dots,m, such that$$ u_{i+1}(y_{i+1})=\max_{|x|=(1-a)r}u_{i+1}(x) \quad \Big(=\max_{|x|\leq(1-a)r}u_{i+1}(x)\Big), $$and take z_{i+1}\in\mathbb{R}^N, i=1,2,\dots,m, such that z_{i+1}=My_{i+1}, 00 are constants. >From this estimate and \eqref{prime ui} we obtain \eqref{ine}. Thus the proof is complete. \end{proof} \begin{proof}[Proof of Theorem \ref{liouville}] Assume that \eqref{assumption 2} holds. Then there exist positive constants C_i>0, i=1,2,\dots,m, and R_0>0 such that$$ P_i(x)\geq \frac{C_i}{|x|^{\lambda_i}}\quad\mbox{for } |x|\geq R_0. So that $$\label{condition liouville} {P_{i*}(r)\geq\frac{C_i}{r^{\lambda_i}}\quad\mbox{for } r\geq R_0.}$$ Without loss of generality we may assume that i_0=1. Suppose to the contrary that \eqref{rewrite problem} has a nonnegative nontrivial entire solution (u_1,u_2,\dots,u_m) satisfying \eqref{order} with i_0=1. Then, by Lemma \ref{lemma-2}, its spherical mean (\bar u_1,\bar u_2,\dots,\bar u_m) satisfies \eqref{ine}. We choose the constant b<1 in \eqref{ine} such that 10 is some constant and l=b^{-2m}. Integrating \eqref{ine} on [0,r], we have $$\label{int 1} {\bar u_i(r)\geq \bar u_i(0)+\tilde C_i\int_0^r sP_{i*}(s) \bar u_{i+1}(bs)^{\alpha_i}ds,\quad r\geq 0,\quad i=1,2,\dots,m.}$$ Since (u_1,u_2,\dots,u_m) is nonnegative and nontrivial, for some point x_*\in \mathbb{R}^N we have u_i(x_*)>0 for some i\in\{1,2,\dots,m\}; that is \bar u_i(r_*)>0,~r_*=|x_*|. We may assume that r_*\geq R_0. Therefore, we see from \eqref{int 1} that \bar u_i(r)>0 for r>r_*. Let r\geq r_*/b be large enough. Integrating \eqref{ine} over [br,r], from \eqref{condition liouville} and the monotonicity of u_i we have \begin{align*} \bar u_i(r)-\bar u_i(br) &\geq {\tilde C_i\int_{br}^rsP_{i*}(s)\bar u_{i+1}(bs)^{\alpha_i}ds}\\ &\geq {\tilde C_i\bar u_{i+1}(b^2r)^{\alpha_i}\int_{br}^r s^{1-\lambda_i}ds}\\ &= {\tilde C_i\frac{1-b^{2-\lambda_i}}{2-\lambda_i}\bar u_{i+1}(b^2r)^{\alpha_i} r^{2-\lambda_i},} \end{align*} namely, $$\label{esti ui} {\bar u_i(r)\geq Cr^{2-\lambda_i}\bar u_{i+1}(b^2r)^{\alpha_i},\quad i=1,2,\dots,m,}$$ where C is some positive constant. Notice that \eqref{esti ui} is still valid even though \lambda_i=2 (with C=\tilde C_i\log{b^{-1}}). From \eqref{esti ui}, by iteration, it follows that{\bar u_1(r)\geq Cr^{-\Lambda_1}\bar u_1(b^{2m}r)^A,\quad r>\frac{r_*}{b^{2m}},} $$where C>0 is some constant. From the assumption \Lambda_1\leq 0, we obtain \eqref{estimate u_1}. The inequality \eqref{ine} with i=1 and \eqref{esti ui} imply $$\label{result u1} {\bar u_1'(r)\geq Cr^\tau P_{1*}(r)\bar u_1(b^{2(m-1)+1}r)^A},$$ where$$ \tau= {1+\sum_{j=1}^{m-1}\left\{(2-\lambda_{1+j})\prod_{k=0}^{j-1}\alpha_{1+k}\right\}} = \lambda_1-1-\Lambda_1. Integrating \eqref{result u1} over [r_1,r],~b^{2(m-1)+1}r_1>r_*, we have \begin{align*} \bar u_1(r) &\geq {\bar u_1(r_1)+C\int_{r_1}^r s^\tau P_{1*}(s)\bar u_1(b^{2(m-1)+1}s)^Ads}\\ &\geq {\bar u_1(r_1)+C\bar u_1(b^{2(m-1)+1}r_1)^A\int_{r_1}^rs^{\tau-\lambda_1}ds.} \end{align*} From the assumption \Lambda_1\leq 0, we can see that \tau-\lambda_1\geq -1, which implies that \eqref{limit u1} holds. Let \tilde r be large so that $$\label{exponent} {L^\frac{1}{A-1}\bar u_1(\tilde r)\geq e,}$$ and $$\label{esti low u1} \bar u_1(lr)\geq L\bar u_1(r)^A,\quad r\geq \tilde r\,,$$ where L>0 is the constant appearing in \eqref{estimate u_1}. It is possible to choose such an \tilde r by \eqref{limit u1} and \eqref{estimate u_1}. For k\in{\mathbb N}, from \eqref{esti low u1} we obtain \begin{align*} \bar u_1(l^k\tilde r) &\geq L\bar u_1(l^{k-1}\tilde r)^A\\ &\geq L^{1+A}\bar u_1(l^{k-2}\tilde r)^{A^2}\\ &\geq \dots\\ &\geq L^{1+A+\dots+A^{k-1}}\bar u_1(\tilde r)^{A^k}\\ &= L^{-\frac{1}{A-1}}\left[L^\frac{1}{A-1}\bar u_1(\tilde r)\right]^{A^k}. \end{align*} Hence we see from \eqref{exponent} that $$\label{exponent 2} {\bar u_1(l^k\tilde r)\geq L^{-\frac{1}{A-1}}\exp A^k.}$$ Let r\geq l\tilde r. Then we can find that there exists a unique positive integer k=k(r) such that l^k\tilde r\leq r\frac{\log r-\log\tilde r}{\log l}-1. It follows therefore from \eqref{exponent 2} that \begin{aligned} \bar u_1(r) & \geq \bar u_1(l^k\tilde r)\geq L^{-\frac{1}{A-1}}\exp A^k \\ & \geq L^{-\frac{1}{A-1}}\exp\left\{A^{-\frac{\log\tilde{r}}{\log l}-1}\cdot A^\frac{\log r}{\log l}\right\} \\ & = {L^{-\frac{1}{A-1}}\exp\left\{A^{-\frac{\log\tilde r}{\log l}-1} r^\frac{\log A}{\log l}\right\}.} \end{aligned}\label{aaa} On the other hand, because $u_1(x)=O(\exp|x|^\rho)$ as $|x|\to\infty$, we obviously have $$\bar u_1(r)=O(\exp r^\rho)~\textrm{as}~r\to\infty.$$ Since $\log A/\log l=\log A/\log b^{-2m}>\rho$ from our choice of $b$, \eqref{aaa} gives a contradiction. The proof is complete. \end{proof} \begin{remark} {\rm (i) When $m=2$, Theorem \ref{liouville} reduces to \cite[Theorem 1]{Teramoto-Usami}. However, the proof given here is simpler than in \cite{Teramoto-Usami}.\\ (ii) As described in Remark \ref{twosystem}, in the case $m=2$, the nonnegative entire solution $(u_1,u_2)$ of \eqref{rewrite problem} satisfies $${u_1(x)\leq C|x|^{\beta_1}~~\mathrm{and}~~u_2(x)\leq C|x|^{\beta_2} \quad\mathrm{at}~~\infty}$$ without the assumptions $\alpha_1\geq 1$ and $\alpha_2\geq 1$ under the condition \eqref{assumption 2}. From this fact and \eqref{limit u1}, we can see that if $(\lambda_1,\lambda_2)$ satisfies $\Lambda_1\leq 0$, then the system \eqref{rewrite problem} does not have nonnegative nontrivial entire solutions. Therefore, we find that Theorem \ref{nonexistence-1} holds without the assumptions $\alpha_1\geq 1$ and $\alpha_2\geq 1$. 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