\documentclass[reqno]{amsart}
\usepackage{hyperref}

\AtBeginDocument{{\noindent\small
\emph{Electronic Journal of Differential Equations},
Vol. 2011 (2011), No. 39, pp. 1--9.\newline
ISSN: 1072-6691. URL: http://ejde.math.txstate.edu or http://ejde.math.unt.edu
\newline ftp ejde.math.txstate.edu}
\thanks{\copyright 2011 Texas State University - San Marcos.}
\vspace{9mm}}

\begin{document}
\title[\hfilneg EJDE-2011/39\hfil Existence of entire solutions]
{Existence of entire solutions for semilinear
elliptic systems under the Keller-Osserman condition}

\author[Z. Zhang, Y. Shi, Y. Xue\hfil EJDE-2011/39\hfilneg]
{Zhijun Zhang, Yongxiu Shi, Yanxing Xue}  % in alphabetical order

\address{Zhijun Zhang \newline
School of Mathematics and Information Science, Yantai University,
Yantai, Shandong, 264005,  China}
\email{zhangzj@ytu.edu.cn}

\address{Yongxiu Shi \newline
School of Mathematics and Information Science, Yantai University,
Yantai, Shandong, 264005,  China}
\email{syxiu0926@126.com}

\address{Yanxing Xue \newline
School of Mathematics and Information Science, Yantai University,
Yantai, Shandong, 264005,  China}
\email{xiaoxue19870626@163.com}

\thanks{Submitted January 22, 2011. Published March 9, 2011.}
\thanks{Supported by grants 10671169 from NNSF of China
and  2009ZRB01795 from NNSF  of \hfill\break\indent
Shandong Province}
\subjclass[2000]{35J55, 35J60, 35J65}
\keywords{Semilinear elliptic systems; entire solutions; existence}

\begin{abstract}
 Under the Keller-Osserman  condition on $f+g$, we show the existence 
 and nonexistence  of entire  solutions  for the semilinear elliptic system
 $\Delta u =p(x)f(v), \quad \Delta v =q(x)g(u),\quad x\in \mathbb{R}^N$,
 where  $p,q:\mathbb{R}^N\to [0,\infty)$ are continuous functions.
\end{abstract}

\maketitle
\numberwithin{equation}{section}
\newtheorem{theorem}{Theorem}[section]
\newtheorem{lemma}[theorem]{Lemma}
\newtheorem{remark}[theorem]{Remark}
\newtheorem{corollary}[theorem]{Corollary}


\section{Introduction}

The purpose of this paper is to investigate the existence and
nonexistence of entire solutions to the  semilinear elliptic
system
\begin{equation}\label{e1.1}
\begin{gathered}
\Delta u=p(x)f(v),\quad x \in \mathbb{R}^N \; (N\geq3), \\
\Delta v=q(x)g(u),\quad x \in \mathbb{R}^N.
\end{gathered}
\end{equation}
By an entire large solution $(u, v)$, we mean a pair of functions
 $u, v\in C^2(\mathbb{R}^N)$ that satisfies \eqref{e1.1} and
\begin{equation}\label{e1.2}
\lim_{|x|\to \infty}u(x)=\lim_{|x|\to \infty}v(x)=+\infty.
\end{equation}
In this article, we assume that $p,q,f$ and $g$ satisfy the following
hypotheses:
\begin{itemize}
\item[(H1)]   $p,q:\mathbb{R}^N\to [0,\infty)$ and
$f,g:[0,\infty)\to  [0,\infty)$ are continuous and
nontrivial;
\item[(H2)]   $f$ and $g$ are nondecreasing on $[0,
\infty)$ and $f(t)>0$, $g(t)>0$ for all $t>0$;
\item[(H3)]  $H(\infty):=\lim _{r\to \infty}H(r)=\infty$,
\end{itemize}
where
\begin{gather}\label{e1.3}
 H(r):=\int_a^r\frac{dt}{\sqrt{2(F(t)+G(t))}},\quad   r\geq a>0,\\
\label{e1.4}
 F(t):=\int_0^t f(s)ds,\quad G(t):=\int_0^t g(s)ds.
\end{gather}
We see that
$$
H'(r)=\frac {1}{\sqrt{2(F(r)+G(r))}}>0,\quad \forall r>a
$$
and $H$ has the inverse function $H^{-1}$ on $[a, \infty)$.
Denote
\begin{equation}\label{e1.5}
 \begin{gathered}
\phi_1(r):=\max_{|x|=r} p(x),\quad  \phi_2 (r):=\min_{|x|=r} p(x),\\
\psi_1(r):=\max_{|x|=r} q(x),\quad  \psi_2 (r):=\min_{|x|=r} q(x).
\end{gathered}
\end{equation}
First we review the  single elliptic equation
\begin{equation}\label{e1.6}
 \Delta u=p(x)f(u),\quad x  \in  \mathbb{R}^{N}.
\end{equation}
For $p \equiv 1$  on $\mathbb{R}^N$ and $f$ satisfying
(H1) and (H2),  Keller-Osserman
\cite{KE,OS} first supplied the necessary and sufficient
condition
  \begin{equation}\label{e1.7}
   \int_1^\infty
\frac{dt}{\sqrt{2F(t)}}=\infty
\end{equation}
  for the existence of entire  radial large solutions to
  \eqref{e1.6}.
For the weight   $p(x)=p(|x|)$ and $f(u)=u^\alpha$ with
$\alpha\in (0, 1]$, Lair and Wood \cite{LA2} proved that
\eqref{e1.6}  has a non-negation entire radial large solution if and
only if
\begin{equation}\label{e1.8}
\int_0^\infty rp(r)dr=\infty.
\end{equation}
Recently, Lair \cite {LA3} obtained the following results.

\begin{lemma} \label{lem1.1}
 Let  $f$ and $b$ satisfy {\rm (H1)} and {\rm (H2)}
with $f(0)=0$. Suppose
\begin{itemize}
\item[(i)]  \eqref{e1.7} holds;
\item[(ii)]  there exists a positive constant $\varepsilon$
such that $\int_0^\infty r^{1+\varepsilon} \phi_1(r) dr <\infty$,
\item[(iii)]    $r^{2N-2}\phi_1(r)$ is nondecreasing near $\infty$.
\end{itemize}
Then \eqref{e1.6} has one nonnegative nontrivial entire bounded
solution. If, on the other hand, $p$ satisfies
$$
\int_0^\infty r\phi_2(r)dr=\infty
$$
and (iii) holds, then \eqref{e1.6} has no nonnegative nontrivial
entire bounded  solution.
\end{lemma}

\begin{lemma}\label{lem1.2}
Let  $f$ and $b$ satisfy {\rm (H1)} and {\rm (H2)}
with $f(0)=0$ and $p(x)=p(|x|)$. Suppose \eqref{e1.7} holds. Then
\eqref{e1.6} has one nonnegative nontrivial entire  solution.
Suppose further that (iii) and \eqref{e1.8} hold, then any
nonnegative nontrivial entire  solution of \eqref{e1.6} is large.
Conversely, if \eqref{e1.6} has a nonnegative nontrivial entire
large solution, then $p$ satisfies
$$
 \int_0^\infty r^{1+\varepsilon} \phi_1(r) dr =\infty,\quad
 \forall\varepsilon>0.
$$
\end{lemma}
For more works, see for example
\cite{BZ, CR1,GM,LA1,LA2,LA3,TZ,YANG, YZ1,YZ2} and
the references therein.

Now let us return to \eqref{e1.1}.

When  $p(x)=p(|x|)$, $q(x)=q(|x|)$, $f(v)=v^\alpha$,
$g(u)=u^\gamma$, and $0<\alpha\leq \gamma$,
Lair and Wood \cite {LA4} considered the existence and
nonexistence of entire positive
radial solutions to system \eqref{e1.1}. Moreover, when
$0<\alpha\leq 1$ and $ 0\leq \gamma\leq 1$,
Lair \cite {LA5} showed that \eqref{e1.1} has a nonnegative
entire radial large solution if and only if $p$ and $q$
satisfy both of the following conditions
\begin{gather}\label{e1.9}
\int_0^\infty tp(t)
 \Big(t^{2-N}\int_0^t  s^{N-3}Q_1(s) ds\Big)^\alpha dt=\infty,\\
\label{e1.10}
\int_0^\infty tq(t)  \Big(t^{2-N}\int_0^t
  s^{N-3}P_1(s) ds\Big)^\gamma dt=\infty,
\end{gather}
where
$$
P_1(r)=\int_0^r \tau p(\tau)d\tau,\quad
Q_1(r)=\int_0^r \tau q(\tau)d\tau.
$$

  Ghanmi,  M\^{a}agli,  R\u{a}dulescu and  Zeddini \cite
{GMRZ} generalized the  results  in \cite {LA4} to  the case when
$f$ and $g$ are satisfy the condition that:
 For all $c>0$, there exists $L_c>0$ such
that for all $s_1, s_2\in[c,\infty)$,
\begin{equation}\label{e1.11}
|f(s_2)-f(s_1)|+|g(s_2)-g(s_1)|\leq L_c|s_2-s_1|.
\end{equation}
Recently, the authors in \cite{LZZ} showed the  existence of entire
positive  radial  large solutions for \eqref{e1.1}  under the
condition
\begin{equation}\label{e1.12}
\int_1^\infty\frac {ds}{f(s)+g(s)}=\infty.
\end{equation}
For related works, see
\cite{CR2,GM,GMRZ,PS,WW,YZ1,YZ2,ZH} and the references therein.

In this paper,  we extend some of the existence results for entire
positive solutions in Keller \cite {KE}, Osserman  \cite {OS} and
Lair  \cite {LA3}  to  \eqref{e1.1}.
Our main results are as the following.

\begin{theorem}\label{thm1.1}
Under the hypotheses {\rm (H1)--(H3)}.  Suppose  that
\begin{itemize}
\item[(H4)] $r^{2N-2}\big(\phi_1(r)+\psi_1(r)\big)$ is nondecreasing
 for large  $r$;
\item[(H5)] there exists a positive constant $\varepsilon$
such that
$$
\int_0^\infty r^{1+\varepsilon}
\big(\phi_1(r)+\psi_1(r)\big) dr <\infty,
$$
\end{itemize}
then \eqref{e1.1} has a positive entire bounded solution $(u,v)$.
\end{theorem}

 From Theorem \ref{thm1.1}, we have the following corollaries for
the spherically symmetric case $p(x)=p(|x|)$ and $q(x)=q(|x|)$.

\begin{corollary}\label{cor1.1}
Under  hypotheses {\rm (H1)--(H3)},  \eqref{e1.1} has one positive
solution $(u,v)$. Suppose furthermore that
\begin{itemize}
\item[(H6)]  $ P(\infty)=Q(\infty)=\infty$, where
\begin{gather*}
P(\infty):=\lim _{r\to \infty}P(r),\quad
P(r):=\int_0^{r}t^{1-N}
 \Big(\int_0^t s^{N-1}p(s) ds\Big)dt,\quad r\geq 0,\\
Q(\infty):=\lim _{r\to \infty}Q(r),\quad
Q(r):=\int_0^{r}t^{1-N} \Big(\int_0^t s^{N-1}q(s)ds\Big)dt,\quad
 r\geq 0.
\end{gather*}
\end{itemize}
Then every positive radial entire solution $(u, v)$ of  \eqref{e1.1} is
 large and satisfies
$$
u(r)\geq u(0) +f(v(0))P(r),\quad
v(r)\geq v(0) +g(u(0))Q(r),\quad \forall r\geq 0.
$$
\end{corollary}

\begin{corollary}\label{cor1.2}
Assume {\rm (H1)--(H4)}.
 If   \eqref{e1.1}  has a non-negative radial entire large solution,
then
 \begin{equation}\label{e1.13}
 \int_0^\infty r^{1+\varepsilon}
\big(p(r)+q(r)\big) dr =\infty, \quad \forall  \varepsilon >0.
\end{equation}
\end{corollary}

\begin{corollary}\label{cor1.3}
  Under hypotheses {\rm (H1)--(H3)}, \eqref{e1.1} has no radial
entire large solutions  if $p+q$ satisfies one of the following
 conditions:
\begin{itemize}
\item[(i)]  $ p(r)+q(r) \leq Cr^{2-2N}$ for large $r$;
\item[(ii)]  $r^{2N-2}\big(p(r)+q(r)\big)$ is nondecreasing
near $\infty$ and
$$
\int_0^\infty \sqrt{p(r)+q(r)}dr <\infty;
$$
\item[(iii)]   $\int_0^\infty \sqrt{\Lambda(r)}dr <\infty$,
where
\begin{equation}\label{e1.14}
\Lambda(r)=\max_{t\in [0, r]}\big(p(t)+q(t)\big),\quad r\geq 0.
\end{equation}
\end{itemize}
\end{corollary}

\begin{theorem}\label{thm1.2}
Under  hypotheses {\rm (H1)--(H3)}, \eqref{e1.1} has no radial
entire large solutions  if $p+q$ satisifes
\begin{equation}\label{e1.15}
0 < \liminf_{ r\to \infty} \frac {p(r)+q(r)}{ r^\beta}\leq
\limsup _{ r\to \infty} \frac {p(r)+q(r)}{ r^\beta}<\infty,
\quad  \beta<-2.
\end{equation}
 \end{theorem}

 \begin{remark}\label{rmk1.1} \rm
By (H1) and   (H2), we see that (H3) implies
$$
\int_a^\infty\frac {ds}{\sqrt{F(s)}}
=\int_a^\infty\frac {ds}{\sqrt{G(s)}}=\infty.
$$
\end{remark}

 \begin{remark}\label{rmk1.2}\rm
By \cite {LA2},  we see that
$P(\infty)=\infty$
 if and only if $\int_0^\infty rp(r)dr=\infty$.
\end{remark}

 \begin{remark}\label{rmk1.3} \rm
By \cite {LA1},  we see that if
$\int_1^\infty\frac {dt}{\sqrt{F(t)}}<\infty$, then
$\int_1^\infty\frac {dt}{f(t)}<\infty$.
In other words, if
$\int_1^\infty\frac {dt}{f(t)}=\infty$, then
$\int_1^\infty\frac {dt}{\sqrt{F(t)}}=\infty$.
Conversely, if   $\int_1^\infty\frac
{dt}{\sqrt{F(t)}}=\infty $, then $\int_1^\infty\frac
{dt}{f(t)}=\infty $ does not hold. For example,
$$
f(t)=2(1+t)(\ln (t+1)\big)^{2\sigma-1}\big(\ln (t+1)+\sigma\big),\quad
F(t)=(t+1)^2\big(\ln(t+1)\big)^{2\sigma},
$$
where $\sigma>0$.
We can see that
$\int_1^\infty\frac {dt}{f(t)}=\infty $ if and only if
$\sigma \in (0, 1/2]$ and
$\int_1^\infty\frac {dt}{\sqrt{F(t)}}=\infty $ if and
only if $\sigma \in (0, 1]$.
\end{remark}

\section{Proof of main theorems}

\begin{proof}[Proof of Theorem \ref{thm1.1}]
  Suppose (H4) holds. We will show that
\eqref{e1.1}  has a solution by finding a supersolution,
 $(\bar{u},\bar{v})$ and a subsolution,
$(\underline{u}, \underline{v})$, for
which $\underline{u} \leq \bar{u}$ and $\underline{v} \leq \bar{v}$.
To do this, we first prove the existence of $(\underline{u},
\underline{v})$ to  \eqref{e1.1} by
considering  the system of the integral equations
\begin{equation}\label{e2.1}
\begin{gathered}
 \underline{u}(r)=\beta+\int_0^{r}t^{1-N}
 \Big(\int_0^t s^{N-1}\phi_1(s)f(\underline{v}(s)) ds\Big)dt,\quad
 r\geq 0, \\
 \underline{v}(r)=\beta+\int_0^{r}t^{1-N}
 \Big(\int_0^t s^{N-1}\psi_1(s)g(\underline{u}(s)) ds\Big)dt,\quad
 r\geq 0,
\end{gathered}
\end{equation}
where $\beta\geq a>0$, $a$ is in \eqref{e1.3}.
Let $\{\underline{v}_{m}\}_{m\geq 0}$
and  $\{\underline{u}_{m}\}_{m\geq 1}$ be the sequences of positive
continuous functions defined on $[0,\infty)$ by
\begin{equation}\label{e2.2}
\begin{gathered}
 \underline{v}_0(r)=\beta, \\
 \underline{u}_{m}(r)=\beta+\int_0^{r} t^{1-N}
 \Big(\int_0^t s^{N-1}\phi_1(s)f(\underline{v}_{m-1}(s)) ds\Big)dt,\quad
 r\geq 0,\\
 \underline{v}_{m}(t)=\beta+\int_0^{r} t^{1-N}
 \Big(\int_0^t s^{N-1}\psi_1(s)g(\underline{u}_m(s)) ds\Big)dt,\quad
 r\geq 0.
\end{gathered}
\end{equation}
 Obviously,  for all $ r\geq 0$ and $m\in {\mathbb{N}}$,
$\underline{u}_{m}(r)\geq \beta$,   $\underline{v}_{m}(r)\geq \beta$
and $\underline{v}_0\leq \underline{v}_1$. $\mathbf{(H_2)}$ yields
$u_1(r)\leq u_2(r)$ for all $r\geq 0$, then
$\underline{v}_1(r)\leq \underline{v}_2(r)$ for all $r\geq 0$.
By the same argument, we
obtain that the sequences $\{\underline{u}_m(r)\}$ and
$\{\underline{v}_m(r)\}$ are increasing with respect to $m$ for
$r\in[0, \infty)$. Moreover, for each $r>0$,
\begin{gather*}
\underline{u}_m'(r)=r^{1-N}
 \Big(\int_0^{r} s^{N-1}\phi_1(s)f(\underline{v}_{m-1}(s))
 ds\Big)\geq 0,\\
\underline{v}_m'(r)=r^{1-N}
 \Big(\int_0^{r} s^{N-1}\psi_1(s)g(\underline{u}_{m}(s)) ds\Big)\geq 0
\end{gather*}
and
\begin{align*}
 &\Big(r^{N-1}\big(\underline{u}_m(r)+\underline{v}_m(r)\big)'\Big)'\\
&=r^{N-1}\big( \phi_1(r)f(\underline{v}_{m-1}(r))+\psi_1(r)
g(\underline{u}_{m}(r)) \big)\\
&\leq r^{N-1}\big(\phi_1(r)+\psi_1(r)\big)\Big(
 f(\underline{v}_m(r)+\underline{u}_m(r))+g(\underline{v}_m(r)
+\underline{u}_m(r))\Big).
\end{align*}
Let
$$
\Lambda(r)=\max_{t\in [0, r]}\big(\phi_1(t)+\psi_1(t)\big),\quad
r\geq 0.
$$
 Multiplying this by
$2r^{N-1}\big(\underline{u}_m(r)+\underline{v}_m(r)\big)'$ and
integrate on $[0, r]$, we obtain
\begin{align*}
&\Big(r^{N-1}\big(\underline{u}_m(r)+\underline{v}_m(r)\big)'\Big)^2\\
& \leq  2 \int_0^r t^{2(N-1)}\big(\phi_1(t)+\psi_1(t)\big)\\
& \Big( f(\underline{v}_m(t)+\underline{u}_m(t))+
 g(\underline{v}_m(t)+\underline{u}_m(t))\Big)
\big(\underline{u}_m(t)+\underline{v}_m(t)\big)'dt\\
&\leq 2 r^{2(N-1)}\Lambda(r)\int_{2\beta}^{\underline{u}_m(r)
 +\underline{v}_m(r)} \big( f(\sigma)+g(\sigma)\big)d \sigma\\
&\leq  2 r^{2(N-1)}\Lambda(r) \big(F(\underline{u}_m(r)
 +\underline{v}_m(r))
 +G(\underline{u}_m(r)+\underline{v}_m(r))  \big),
\end{align*}
and
\begin{equation}\label{e2.3}
 \big(\underline{u}_m(r)+\underline{v}_m(r)\big)'  \leq
\sqrt{2\Lambda(r)}\Big(\big(F(\underline{v}_m(r)+\underline{u}_m(r))+
G(\underline{v}_m(r)+\underline{u}_m(r))\big)\Big)^{1/2}.
\end{equation}
 Thus
\begin{align*}
& \int_0^{r}\frac {\underline{u}_m'(t)+\underline{v}_m'(t)}
{\sqrt{2}\big({F(\underline{u}_m(t)+\underline{v}_m(t))
+G(\underline{u}_m(t)+\underline{v}_m(t))\big)}^{1/2}}dt \\
&=  \int_{2 \beta}^{\underline{u}_m(r)+\underline{v}_m(r)}
 \frac {d\tau} { \sqrt{2(F(\tau)+G(\tau))} }  \\
& =   H(\underline{u}_m(r)+\underline{v}_m(r))- H(2 \beta) \leq
\int_0^r\sqrt{M(t)}dt.
\end{align*}
Since $H^{-1}$ is  increasing on $[0, \infty)$,  we have
\begin{equation}\label{e2.4}
\underline{u}_m(r)+\underline{v}_m(r)\leq H^{-1}\Big
(H(2\beta)+\int_0^r \sqrt{M(t)}dt\Big), \quad \forall r\geq 0.
\end{equation}
It follows by
(H3) and \eqref{e2.2} that  the sequences $\{\underline{u}_m\}$
and $\{\underline{v}_m\}$ are bounded  and equi-continuous on
$[0,c_0]$ for arbitrary $c_0>0$. By Arzela-Ascoli theorem,
$\{\underline{u}_m\}$ and $\{\underline{v}_m\}$ have subsequences
converging uniformly to $\underline{u}$ and $\underline{v}$ on
$[0, c_0]$. By the arbitrariness of $c_0>0$, we see that
$(\underline{u}, \underline{v})$
is  a positive entire solution of
\begin{equation}\label{e2.5}
\begin{gathered}
\Delta \underline{u}=\phi_1(r)f(\underline{v})\geq p(x)f(\underline{v}),
\quad x \in \mathbb{R}^N, \\
\Delta \underline{v}=\psi_1(r)g(\underline{u})\geq
q(x)g(\underline{u}),\quad x \in \mathbb{R}^N;
\end{gathered}
\end{equation}
i.e., $(\underline{u}, \underline{v})$
 is  a positive entire subsolution of \eqref{e1.1}.

 Next we prove that $(\underline{u}, \underline{v})$ is bounded.
Since $(\underline{u}, \underline{v})$ satisfies
\begin{gather}\label{e2.6}
 \big(r^{N-1}\underline{u}'(r)\big)' = r^{N-1}\phi_1(r)f
(\underline{v}),\\
\label{e2.7}
\big(r^{N-1}\underline{v}'(r)\big)' = r^{N-1}\psi_1(r)g
(\underline{u}).
\end{gather}
Choose $R > 0$ so that
$r^{2N-2}\big(\phi_1(r)+\psi_1(r)\big)$
 is  nondecreasing  on $[R, \infty)$ and
$$
\underline{u}(r)>0, \quad \underline{v}(r)>0,\quad \forall r\geq R.
$$

 Now, since $\underline{u}'(r)\geq 0$ and $\underline{v}'(r)\geq 0$
for $r\geq 0$, and (H2) holds,
multiplying \eqref{e2.6} and \eqref{e2.7} by
$r^{N-1}\underline{u}'(r)$ and
$r^{N-1}\underline{v}'(r)$, respectively, and integrating from $0$
to $r$, we have
\begin{align*}
\big(r^{N-1}\underline{u}'(r)\big)^2
&\leq \big(R^{N-1}\underline{u}'(R)\big)^2
+2\Big(\int_{R}^{r}t^{2(N-1)}p(t)
 f(\underline{v}(t))\underline{u}'(t)dt\Big)\\
&\leq  C+2r^{2(N-1)}\big(\phi_1(r)+\psi_1(r)\big)
 \Big(\int_{R}^{r}  \frac {d}{dt}F(\underline{v}(t)+\underline{u}(t))dt\Big)\\
&\leq C+2r^{2(N-1)}\big(\phi_1(r)+\psi_1(r)\big)F(\underline{v}(r)
 +\underline{u}(r)),
\end{align*}
 and
\[
 \big(r^{N-1}\underline{v}'(r)\big)^2\leq
C+2r^{2(N-1)}\big(\phi_1(r)+\psi_1(r)\big)
G(\underline{v}(r)+\underline{u}(r)),
\]
for $r>R$, where
$C=\big(R^{N-1}\big(\underline{u}'(R)+\underline{v}'(R))\big)^2$,
which yields
 \begin{align*}
&\underline{u}'(r)+\underline{v}'(r)\\
&\leq \sqrt{2C}r^{-(N-1)}+\sqrt{2(\phi_1(r)+\psi_1(r))}{\big(G(\underline{u}(r)
+\underline{v}(r))+F(\underline{v}(r)+\underline{u}(r))\big)}^{1/2},
\end{align*}
and
\begin{align*}
& \frac {d}{dr}\int_{\underline{u}(R)
+\underline{v}(R)}^{\underline{u}(r)+\underline{v}(r)}\frac
{d\tau}{\sqrt{2\big(F(\tau)+G(\tau)\big)}}\\
&\leq \sqrt{C}r^{1-N}\big(G(\underline{u}(r)+\underline{v}(r))
 +F(\underline{v}(r)+\underline{u}(r))\big)^{-1/2}
+\sqrt{\phi_1(r)+\psi_1(r)}.
\end{align*}
 Integrating the above inequality and using the facts that
 $$
G(\underline{u}(r)+\underline{v}(r))+F(\underline{v}(r)
 +\underline{u}(r))\geq G(\underline{u}(R)+\underline{v}(R))
+F(\underline{v}(R)+\underline{u}(R))=C_1,
$$
for all $r\geq R$,  and
$$
\sqrt{\phi_1(r)+\psi_1(r)}\leq
\sqrt{2r^{1+\varepsilon}\big(\phi_1(r)+\psi_1(r)\big)r^{-1-\varepsilon}}
\leq r^{1+\varepsilon}\big(\phi_1(r)+\psi_1(r)\big) +
r^{-(1+\varepsilon)}
$$
for $\varepsilon>0$,
we have
\begin{align*}
H(\underline{u}(r)+\underline{v}(r))
&\leq H(\underline{u}(R)+\underline{v}(R))+\int_R^r
 s^{1+\varepsilon}\big(\phi_1(s)+\psi_1(s)\big)ds
+(\varepsilon R^\varepsilon)^{-1}\\
&\quad +\sqrt{CC_1^{-1}}(NR^N)^{-1}.
\end{align*}
Letting $r\to\infty$,
we find that $(\underline{u}, \underline{v})$ is bounded  since
$\phi_1+\psi_1$  satisfies (H5)
 and $f+ g$ satisfies (H3). Thus, Since
 $(\underline{u}, \underline{v})$ is nondecreasing, we have
$$
\lim_{r\to \infty}\underline{u}(r)=M_1>0,\quad \
 \ \lim_{r\to \infty}\underline{v}(r)=M_2>0.
$$
In the same way, we can see that the  system
\begin{equation}\label{e2.8}
\begin{gathered}
 \bar{u}(0)=\bar{v}(0)=\max\{M_1, M_2\},\quad
\bar{u}'(r)=\bar{v}'(r)=0,\\
 \Delta \bar{u}(x)=\bar{u}''(r)+\frac {N-1}{r}\bar{u}'(r)=
 \phi_2(r)f(\bar{v}(r)) ,\quad r>0,\\
 \Delta \bar{v}(x)=\bar{v}''(r)+\frac {N-1}{r}\bar{v}'(r)=
 \psi_2(r)g(\bar{u}(r)) ,\quad r>0
\end{gathered}
\end{equation}
has a bounded solution $(\bar{u}, \bar{v})$ which is a supersolution
for \eqref{e1.1}. It is also clear that
$$
\bar{u}(r)\geq M_1\geq \underline{u}(r) ,\quad
\bar{v}(r)\geq M_2\geq \underline{v}(r),\quad \forall r\geq 0.
$$
Hence the standard super-sub
solution principle (see \cite {SA,HE}) implies that \eqref{e1.1}
has a bounded solution $(u, v)$ such that
$\underline{u}(x) \leq u(x)\leq
\bar{u}(x)$ and $\underline{v}(x) \leq v(x) \leq\bar{v}(x)$
 on $\mathbb{R}^N$.
 This completes the proof.
\end{proof}

\begin{proof}[Proof of Theorem \ref{thm1.2}]
 We follow  the arguments  in  (\cite[Theorem 4.3]{GS}  and
\cite[Theorem 3.4]{YZ2}) for  studying the nonexistence of
entire radial large solutions to \eqref{e1.6}. Let
\begin{equation}\label{e2.9}
a(r) = r^\theta\int_r^\infty t\big(p(t)+q(t)\big)dt,\quad r\geq 0.
\end{equation}
By \eqref{e1.15}, there
exist $R_0>0, C_2 > C_1> 0$ such that
$$
C_1 r^\beta \leq p(r)+q(r)\leq  C_2 r^\beta,\quad
 r\geq R_0,
$$
so
\begin{align*}
a'(r)
&= \theta r^{\theta-1}\int_r^\infty t\big(p(t)+q(t)\big)dt
-r^{\theta+1}\big(p(r)+q(r)\big)\\
&= -r^{\beta+\theta+1} \Big(C_1-\frac {C_2\theta}{-\beta-2}\Big)<0
\end{align*}
provided  $\theta \in \big(0, C_1C_2^{-1}(-\beta-2) \big)$;
i.e., $a$ is decreasing in $[R_0,\infty)$.  Define
\begin{equation}\label{e2.10}
b(r) = \int_r^\infty t\big(p(t)+q(t)\big)dt,\quad r\geq 0.
\end{equation}

Now suppose that \eqref{e1.1}  has a radial entire large solution
$(u, v)$ with $u(r)>0$ and $v(r)>0$ for all $r\geq R$, then for
$r\geq R_0$
\begin{align*}
u(r)+v(r)
&=  u(0)+v(0) +\frac {1}{N-2}\int_0^{r} \Big(1-\big(\frac {\tau}{
r}\big)^{N-2}\Big)\tau
\big(p(\tau)f(v(\tau))\\
&\quad +q(\tau)g(u(\tau))\big)d\tau,\\
&\leq  u(0)+v(0) + \frac {1}{N-2}
\int_0^{r} \Big(1-\big(\frac {\tau}{ r}\big)^{N-2}\Big)\tau
\big(p(\tau)+q(\tau)\big)\\
&\quad\times \big(f(v(\tau)+u(\tau))+g(u(\tau)+v(\tau))\big)d\tau\\
&=  C + \frac {C}{N-2}
\int_{R_0}^{r} \Big(1-\big(\frac {\tau}{ r}\big)^{N-2}\Big)\tau
\big(p(\tau)+q(\tau)\big)\\
&\quad\times \big(f(v(\tau)+u(\tau))+g(u(\tau)+v(\tau))\big)d\tau.
\end{align*}
 Let $\tau=b^{-1}(s)$, $w = (u+v) \circ b^{-1}$.
By the monotonicity of
$b$ and $a = r^\theta b$ in $[R_0, \infty)$,
$t\big(b^{-1}(t)\big)^\theta$ is increasing in $(0, t_0],$ where
$t_0=b(R_0)$,  and
\begin{equation}\label{e2.11}
1-r^\alpha\leq  C_\alpha (1-r),\quad \forall r\in  [0, 1]
\text{ and and fixed $\alpha>0$},
\end{equation}
we obtain, for $t \in (0, t_0]$,
\begin{align*}
w(t)
&= C + \frac {1}{N-2}\int_t^{ t_0} \Big(1-\big(\frac
{b^{-1}(s)}{
b^{-1}(t)}\big)^{N-2}\Big) \big(f(w(s))+g(w(s))\big)ds \\
&\leq C + \frac {1}{N-2}\int_t^{ t_0} \Big(1-\big(\frac {
t}{s}\big)^{(N-2)/\theta}\Big) \big(f(w(s))+g(w(s))\big)ds\\
&\leq  C + \frac {1}{N-2}\int_t^{ t_0} \Big(1-\frac { t}{s}\Big)
\big(f(w(s))+g(w(s))\big)ds= z(t).
\end{align*}
It is easy to see that $z'(t)\leq 0$ for $t\in (0, t_0]$ and
$$
z''(t) = \frac {C\big(f(w(t))+g(w(t))\big)}{ t} \leq
 \frac { C\big(f(z(t))+g(z(t))\big)}{ t},
$$
which yields
\begin{align*}
z'^2(t_0) - z'^2(t)
&= 2\int_t^{ t_0}  z''(s)z'(s)d s\\
&\geq  2C\int_t^{ t_0}  \frac {\big(f(z(s))+g(z(s))\big)z'(s)}{ s}
ds \\
&\geq  \frac
{2C} {t} \int_t^{ t_0} \big(f(z(s))+g(z(s))\big)z'(s)ds \\
&= \frac {2C}{ t} \big(F(z(t_0))+G(z(t_0)) - F(z(t))-G(z(t))\big) .
\end{align*}
 Since $\lim_{t\to 0}w(t)=\infty$, so is $F(z(t))+G(z(t))$.
We obtain, for $0 < t < t_1$
small enough,
$$
z'^2(t) \leq \frac {C\big(F(z(t))+G(z(t))\big)}{t},
$$
and
$$
-\frac {C}{\sqrt{t} }\leq \frac {z'(t)}{ \sqrt{F(z(t))+G(z(t))}}
\leq 0.
$$
Integrating from $t$ to $t_1$ and letting $t\to  0$, we obtain
$$
\int_{z(t_1)}^\infty \frac { d\sigma}{\sqrt{F(\sigma)+G(\sigma)}}
\leq  C \int_0^{t_1}\frac { dt}{\sqrt{t} }
=2C\sqrt{t_1}<\infty.
$$
This is a contradiction.
The proof is completed.
\end{proof}

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\end{document}