\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 \begin{equation}\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} \end{equation} 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 \begin{equation}\label{pre ass} {P_i(x)\geq \frac{C_i}{|x|^{\lambda_i}},\quad |x|\geq r_0>0,\quad i=1,2,} \end{equation} where $C_i>0$ and $\lambda_i,~i=1,2$, are constants. If $(\lambda_1,\lambda_2)$ satisfies \begin{equation}\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, \end{equation} 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{equation} \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} \end{equation} and \begin{equation}\label{constant beta} {\beta_i=\frac{\Lambda_i}{A-1},\quad i=1,2,\dots,m.} \end{equation} 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 \begin{equation}\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} \end{equation} 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 \begin{equation}\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} \end{equation} 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 \begin{equation}\label{condi P} {\liminf_{|x|\to\infty}|x|^{\lambda_i}P_i(x)>0,} \end{equation} 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 \begin{equation}\label{condi hat P} {\hat P_i(r)\geq\frac{C_i}{r^{\lambda_i}},\quad r\geq R_0.} \end{equation} \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 \begin{equation}\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}. \end{equation} 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 \begin{equation}\label{mean sol estimate} {\bar u_i(r)=O(r^{\beta_i})\quad\mbox{as } r\to\infty,\quad i=1,2,\dots,m.} \end{equation} 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 \begin{equation}\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.} \end{equation} 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 \begin{equation} 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} \end{equation} From \eqref{function f} and the monotonicity of $\bar u_i$, we see that \begin{equation}\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.} \end{equation} 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 \begin{equation} 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} \end{equation} 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 \begin{equation}\label{assumption n3} {P_i(r)\geq\frac{C_i}{r^{\lambda_i}},\quad r\geq R_0>0,} \end{equation} where $C_i>0$ and $\lambda_i$ are constants. Moreover, $\Lambda_i$ defined by \eqref{constant Lambda} satisfy \begin{equation}\label{condition N3} \Lambda_i\leq 0\quad \mathit{for~some}~i\in\{1,2,\dots,m\}. \end{equation} 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)\,.$$ \end{theorem} \begin{theorem}\label{nonexistence-radial-2} Let $N=2$. Suppose that $P_i$, $i=1,2,\dots,m$, satisfy \begin{equation}\label{assumption n2} {P_i(r)\geq\frac{C_i}{r^2(\log r)^{\lambda_i}},\quad r\geq R_0>1,} \end{equation} where $C_i>0$ and $\lambda_i$, $i=1,2,\dots,m$, are constants. Moreover \begin{equation}\label{condition N2} \Lambda_i\leq A-1\quad \mathit{for~some}~i\in\{1,2,\dots,m\}\,. \end{equation} If $(u_1,u_2,\dots,u_m)$ is a nonnegative radial entire solution of \eqref{problem}, then $$(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).$$ \end{theorem} \begin{proof}[Proof of Theorem \ref{nonexistence-radial-1}] Let $(u_1,u_2,\dots,u_m)$ be a nonnegative nontrivial radial entire solution of \eqref{problem}. Then $(u_1,u_2,\dots,u_m)$ satisfies the system of ordinary differential equations \begin{equation}\label{radial n} \begin{gathered} {(r^{N-1}u_i'(r))'=r^{N-1}P_i(r)u_{i+1}(r)^{\alpha_i},\quad r>0,}\\ u_i'(0)=0, \end{gathered}\quad i=1,2,\dots,m. \end{equation} Integrating \eqref{radial n} over $[0,r]$, we have $$ {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.} $$ Hence, we see that $u_i$, $i=1,2,\dots,m$, are nondecreasing on $r\geq 0$. Integrating \eqref{radial n} twice over $[R,r]$, for $R\geq 0$ and $i=1,2,\dots,m$, we have \begin{equation}\label{int Rn} {u_i(r)\geq u_i(R)+\frac{1}{N-2}\int_R^rs\big[1-(\frac{s}{r})^{N-2}\big] P_i(s)u_{i+1}(s)^{\alpha_i}ds,} \end{equation} Since $u_i$, $i=1,2,\dots,m$, are nonnegative, nontrivial and nondecreasing functions, there exists an $r_*>0$ such that $u_{i_0}(r_*)>0$ for some $i_0\in\{1,2,\dots,m\}$. We may assume that $r_*\geq R_0$. We see from \eqref{int Rn} with $R=r_*$ that $u_i(r)>0$ for $r>r_*,~i=1,2,\dots,m$. Using similar arguments as in the proof of Theorem \ref{estimate}, we obtain \begin{equation}\label{contra} {u_i(r)\leq C_i r^{\beta_i}\quad\mbox{at }\infty,\quad i=1,2,\dots,m,} \end{equation} where $C_i>0$ are constants and $\beta_i$ are defined by \eqref{constant beta}. Note that our assumption \eqref{condition N3} shows $\beta_i\leq 0$ for some $i\in\{1,2,\dots,m\}$. If there exists an $i_0\in\{1,2,\dots,m\}$ such that $\Lambda_{i_0}<0$, then we see that $\beta_{i_0}<0$ in \eqref{contra}. This shows that $u_{i_0}$ tends to $0$ as $r\to\infty$. On the other hand, from \eqref{int Rn} with $R=r_*$ we see that $$ {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\,. $$ From the definition of $\beta_i$ it follows that $\beta_i\geq 0$ and $\beta_m=0$. We first observe that \begin{equation}\label{lambda m-1} \lambda_{m-1}\leq 2 \end{equation} and \begin{equation}\label{lambda i} {\lambda_i\leq-\sum_{j=1}^{m-i-1}\Big\{(\lambda_{i+j}-2)\prod_{k=0}^{j-1}\alpha_{i+k}\Big\} +2,\quad i=1,2,\dots,m-2.} \end{equation} In fact, from the definition of $\Lambda_i$, we obtain \begin{align*} \lambda_i&\geq {-\sum_{j=1}^{m-1}\Big\{(\lambda_{i+j}-2)\prod_{k=0}^{j-1}\alpha_{i+k}\Big\}+2}\\ &= -\Big(\sum_{j=1}^{m-i-1} +\sum_{j=m-i+1}^{m-1}\Big) \Big\{(\lambda_{i+j}-2)\prod_{k=0}^{j-1}\alpha_{i+k}\Big\} -(\lambda_m-2)\prod_{k=0}^{m-i-1}\alpha_{i+k}+2\\ &\equiv -S_1-S_2-S_3+2\,. \end{align*} From the assumption on $\Lambda_m$, we have $$ \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{equation} \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} \end{equation} 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 \begin{equation}\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.} \end{equation} Let $r\geq e$. Then from \eqref{integral n2}, we have \begin{equation} \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} \end{equation} 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 \begin{equation}\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.} \end{equation} 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 \begin{equation}\label{condition n3 P} {\liminf_{|x|\to\infty}|x|^{\lambda_i}P_i(x)>0,} \end{equation} 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 \begin{equation}\label{condition n2 P} {\liminf_{|x|\to\infty}|x|^2(\log|x|)^{\lambda_i}P_i(x)>0,} \end{equation} 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 \begin{equation}\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} \end{equation} 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 \begin{equation}\label{condition P n3} {P_i(r)\leq\frac{C_i}{r^{\lambda_i},}\quad r\geq R_0>0,} \end{equation} 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 \begin{equation}\label{condition P n2} {P_i(r)\leq\frac{C_i}{r^2(\log r)^{\lambda_i}},\quad r\geq R_0>1,} \end{equation} where $i=1,2,\dots,m$, and $C_i>0$ and $\lambda_i$ are constants. Moreover \begin{equation}\label{lambda n2} \Lambda_i>A-1,\quad i=1,2,\dots,m\,. \end{equation} 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 \begin{equation}\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} \end{equation} 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 \begin{equation}\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} \end{equation} 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}: \begin{equation}\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\,, \end{equation} 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 \begin{equation}\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} \end{equation} 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 \begin{equation} \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. \end{equation} The integrals $I_2$ and $I_3$ are estimated as follows: \begin{equation} \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} \end{equation} \begin{equation} \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} \end{equation} From \eqref{estimate I_1} and \eqref{estimate I_2}, we have \begin{equation} 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} \end{equation} 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 \begin{equation}\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} \end{equation} 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 \begin{equation}\label{assumption 2} {\liminf_{|x|\to\infty}|x|^{\lambda_i}P_i(x)>0,\quad i=1,2,\dots,m,} \end{equation} 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 \begin{equation}\label{order} u_{i_0}(x)=O(\exp |x|^{\rho})~as~|x|\to\infty \quad\mbox{for some }\rho>0, \end{equation} 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 \begin{equation}\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, \end{equation} 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 \begin{equation}\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.} \end{equation} 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 \begin{equation} \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\,. \end{equation} 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 \begin{equation}\label{condition liouville} {P_{i*}(r)\geq\frac{C_i}{r^{\lambda_i}}\quad\mbox{for } r\geq R_0.} \end{equation} 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 \begin{equation}\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.} \end{equation} 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, \begin{equation}\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,} \end{equation} 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 \begin{equation}\label{result u1} {\bar u_1'(r)\geq Cr^\tau P_{1*}(r)\bar u_1(b^{2(m-1)+1}r)^A}, \end{equation} 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 \begin{equation}\label{exponent} {L^\frac{1}{A-1}\bar u_1(\tilde r)\geq e,} \end{equation} and \begin{equation}\label{esti low u1} \bar u_1(lr)\geq L\bar u_1(r)^A,\quad r\geq \tilde r\,, \end{equation} 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 \begin{equation}\label{exponent 2} {\bar u_1(l^k\tilde r)\geq L^{-\frac{1}{A-1}}\exp A^k.} \end{equation} 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. $$ It follows therefore from \eqref{exponent 2} that \begin{equation} \begin{aligned} \bar u_1(r) & \geq \bar u_1(l^k\tilde r)\geq L^{-\frac{1}{A-1}}\exp A^k \\ & \geq L^{-\frac{1}{A-1}}\exp\left\{A^{-\frac{\log\tilde{r}}{\log l}-1}\cdot A^\frac{\log r}{\log l}\right\} \\ & = {L^{-\frac{1}{A-1}}\exp\left\{A^{-\frac{\log\tilde r}{\log l}-1} r^\frac{\log A}{\log l}\right\}.} \end{aligned}\label{aaa} \end{equation} On the other hand, because $u_1(x)=O(\exp|x|^\rho)$ as $|x|\to\infty$, we obviously have $$\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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