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

\AtBeginDocument{{\noindent\small
\emph{Electronic Journal of Differential Equations},
Vol. 2012 (2012), No. 239, pp. 1--10.\newline
ISSN: 1072-6691. URL: http://ejde.math.txstate.edu or http://ejde.math.unt.edu
\newline ftp ejde.math.txstate.edu}
\thanks{\copyright 2012 Texas State University - San Marcos.}
\vspace{8mm}}

\begin{document}
\title[\hfilneg EJDE-2012/239\hfil Positive solutions]
{Positive solutions for nonlinear elliptic systems}

\author[A. Ben Dekhil\hfil EJDE-2012/239\hfilneg]
{Adel Ben Dekhil}  % in alphabetical order

\address{Adel Ben Dekhil \newline
D\'epartement de Math\'ematiques\\
Facult\'e des Sciences de Tunis \\
Universit\'e de Tunis El Manar,
Campus Universitaire, 2092 Tunis, Tunisia}
\email{Adel.Bendekhil@ipein.rnu.tn}

\thanks{Submitted  October 14, 2012. Published December 28, 2012.}
\subjclass[2000]{35B08, 35B09, 35J47}
\keywords{Semilinear elliptic systems; positive large solution;
 positive bounded solution}

\begin{abstract}
 In this article, we study the existence of positive solutions for the system
 \begin{gather*}
 \Delta u=H(x,u,v),\\
 \Delta v=K(x,u,v),\text{in }\mathbb{R}^n\; (n\geq 3),
 \end{gather*}
 where $H,K: \mathbb{R}^n\times[0,\infty)\times[0,\infty)\to[0,\infty)$
 are continuous functions satisfying $H(x,u,v)\leq p_1(|x|)F(u+v)$
 and $ K(x,u,v)\leq q_1(|x|)G(u+v)$.
 In terms of the growth of the variable potential functions $p_1,q_1$
 and the nonlinearities $F$ and $G$, we establish  some sufficient
 conditions for the existence of positive continuous solutions for
 this system and we discuss whether these solutions are bounded
 or blow up at infinity.
\end{abstract}

\maketitle
\numberwithin{equation}{section}
\newtheorem{theorem}{Theorem}[section]
\newtheorem{corollary}[theorem]{Corollary}
\newtheorem{example}[theorem]{Example}
\newtheorem{remark}[theorem]{Remark}
\allowdisplaybreaks

\section{Introduction}

Semilinear elliptic systems of the form
\begin{equation} \label{S1}
\begin{gathered}
\Delta u=H(x,u,v),\quad \text{in } \mathbb{R}^n\;(n\geq 3) , \\
\Delta v=K(x,u,v),\quad \text{in } \mathbb{R}^n,
\end{gathered}
\end{equation}
have been studied intensively  in the previous few years and various
results concerning the existence and nonexistence of positive entire large or
bounded solutions have been obtained. We  refer the reader to
\cite{AR, CR1, DZ, GMRZ, GR, AL1, LW2, LZZ, PS, Qi, WW, ZL, ZZ, ZSX}
and  their references for
recent results concerning the existence and qualitative analysis
of  solutions  of \eqref{S1}.

The interest in systems of nonlinear stationary equations is motivated by
applications to theory of Newtonian fluids and nonlinear optics. 
More precisely, coupled nonlinear stationary systems
arise in the description of several physical phenomena such as the 
propagation of pulses in birefringent optical
fibers and Kerr-like photorefractive media, see \cite{AA, Men}.

When $H(x,u,v)=p(|x|)v^{\alpha}$, $K(x,u,v)=q(|x|)u^{\beta}$,
$0< \alpha \leq \beta$,
Lair and Wood in \cite{LW2} considered
the existence and nonexistence of entire positive radial 
solutions to \eqref{S1} under the conditions of integrability 
or non integrability of the functions  $r \to rp(r)$
and $r\to r q(r)$  on $(0,\infty)$. 
Their results were extended by C\^{i}rstea and
 R\u{a}dulescu \cite{CR1}, Wang and Wood \cite{WW},
 Ghergu and R\u adulescu \cite{GR}, Peng
and Song \cite{PS}, Ghanmi et al
\cite{GMRZ},  Li et al \cite{LZZ},   Zhang
\cite{ZZ}. In \cite{ZSX}, the authors considered the case where
$H(x,u,v)=p(x)f(v)$, $K(x,u,v)=q(x)g(u)$ with $p,q$  nontrivial
nonnegative continuous on $[0,\infty)$ and $f, g$
continuous , nondecreasing on $[0,\infty)$ that are positive on $(0, \infty)$.
In the case where $p, q$ are radial and under the condition
\begin{eqnarray}
\int_1^{\infty}\Big(\int_0^t(f(s)+g(s))ds\Big)^{-1/2}\,dt=\infty\,,
\end{eqnarray}
they proved that \eqref{S1} has one positive solution $(u,v)$. Moreover, if
$$
\int_0^{\infty}sp(s)\,ds=\int_0^{\infty}sq(s)\,ds=\infty\,,
$$ 
then every positive radial entire solution $(u,v)$ of \eqref{S1} is
large (i.e. $\lim_{x \to \infty}
u(x)=\lim_{x \to \infty}v(x)=\infty$).
And if 
$$
\int_0^{\infty}sp(s)\,ds<\infty
\text{ and } \int_0^{\infty}sq(s)\,ds< \infty\,,
$$
then every positive radial entire solution $(u,v)$ of $\eqref{S1}$ is bounded.

By using the sub-supersolution method, they establish some conditions 
on $p$, $q$ in order to prove the existence of positive bounded solutions 
in the case where $p, q$ are non radial. Their results extend partially  
those of Zhang \cite{ZZ}, where the existence of
entire positive radial large solutions or bounded ones for \eqref{S1}
was studied under the condition $\int_1^{\infty}\frac{ds}{f(s)+g(s)}=
\infty$.

Their results do not cover the cases where 
\begin{gather*}
H(x,u,v)=a_1(x)v^{\alpha_1}+b_1(x)u^{\gamma_1}+c_1(x)(u+v)^{\beta_1}
+d_1(x)u^{\delta_1} v^{\lambda_1},\\
K(x,u,v)=a_2(x)v^{\alpha_2}+b_2(x)u^{\gamma_2}+c_2(x)(u+v)^{\beta_2}
+d_2(x)u^{\delta_2} v^{\lambda_2}\,,
\end{gather*} 
where
$\alpha_i$, $\beta_i$, $\gamma_i$, $\delta_i$, $\lambda_i \in(0,\infty)$.

Our aim in this paper is to extend the results in \cite{DZ} for the 
particular case of the Dirichlet laplacian
and to extend those in \cite{ZSX} to a wider class
of functions $H(x,u,v)$ and $K(x,u,v)$. 
More precisely, our results apply in particular to the previous
examples of functions $H$, $K$ and to the case where
 $H(x,u,v)=p(x)f(u,v)$ , $K(x,u,v)=q(x)g(u,v)$ with
$f$ and $g$ are nondecreasing with respect to first and the second variables.
To this aim, we assume that $H$ and $K$ satisfy the following hypotheses:
\begin{itemize}
\item[(H1)] $H$, $K:\mathbb{R}^n\times[0,\infty)\times
[0,\infty)\to[0,\infty)$ are continuous.

\item[(H2)] There exist nonnegative functions $p_i$, $q_i$,
$f_i$, $g_i$, $1\leq i\leq 2$, $F$ and $G$ satisfying
for each $x\in\mathbb{R}^n$ and $(u,v)\in[0,\infty)\times[0,\infty)$,
\begin{gather*}
p_2(|x|)g_1(v)g_2(u)\leq H(x,u,v)\leq p_1(|x|)F(u+v),\\
q_2(|x|)f_1(u)f_2(v)\leq K(x,u,v)\leq q_1(|x|)G(u+v),
\end{gather*}
with $f_i$, $g_i$, $1\leq i\leq 2$, $F$, $G:[0,\infty)\to[0,\infty)$ are
nondecreasing continuous, positive on  $(0,\infty)$ and 
$p_i$, $q_i:[0,\infty)\to[0,\infty)$ are continuous.

\item[(H3)] There exist $c>0$ such  that
$$
\int_0^t\sqrt{\Lambda(s)}ds<L_{c}(\infty):=\lim_{r \to \infty}L_{c}(r),
$$
for all $t>0$, where for $\alpha>0$,
\begin{equation}\label{E2}
\begin{gathered}
L_\alpha(t)=\int_\alpha^t\frac{ds}{\sqrt{2(\mathcal{F}(s)+\mathcal{G}(s))}},\quad
  t \geq \alpha, \\
\Lambda(r)=\max_{s\in[0,r]}(p_1(s)+q_1(s)),\quad r\geq0\,,\\
\mathcal{F}(r)=\int_0^rF(s)ds\,, \quad  
\mathcal{G}(r)=\int_0^rG(s)ds\,,\quad r\geq 0.
\end{gathered}
\end{equation}
\end{itemize}
We note that $L_\alpha$ has an inverse function $L_\alpha^{-1}$ 
from $[\alpha, \infty)$ to $[0,L_{\alpha}(\infty))$,
where $L_{\alpha}(\infty)=\lim_{r\to \infty}L_{\alpha}(r) \in (0,\infty]$.

\begin{remark} \label{rmk1.1} \rm
If $L_\alpha(\infty)=\infty$, then 
$$
\int_{\alpha}^\infty\frac{ds}{\sqrt{\mathcal{F}(s)}}=
\int_{\alpha}^\infty\frac{ds}{\sqrt{\mathcal{G}(s)}}=\infty.
$$
\end{remark}

\begin{remark} \label{rmk1.2} \rm
By \cite{AL1}, we see that if 
$\int_1^\infty\frac{ds}{\sqrt{\mathcal{F}(s)}}<\infty$, then  
$\int_1^\infty\frac{ds}{F(s)}<\infty$. In other words, if
$\int_1^\infty\frac{ds}{F(s)}=\infty$, then
$\int_1^\infty\frac{ds}{\sqrt{\mathcal{F}(s)}}=\infty$. 
Conversely, if $\int_1^\infty\frac{ds}{\sqrt{\mathcal{F}(s)}}=\infty$, then
$\int_1^\infty\frac{ds}{F(s)}=\infty$ does not hold. 
For example, for $\beta>0$ and
$F(t)=2(1+t)(\ln(t+1))^{2\beta-1}(\ln(t+1)+\beta)$, we have
 $\mathcal{F}(t)=(t+1)^2(\ln(t+1))^{2\beta}$.
So we can see that $\int_1^\infty\frac{ds}{F(s)}=\infty$
if and only if $0<\beta\leq\frac{1}{2}$ and
$\int_1^\infty\frac{ds}{\sqrt{\mathcal{F}(s)}}=\infty$  if and only if
$\;0<\beta\leq1.$
\end{remark}

To discuss the existence of positive radial solutions to these 
nonlinear systems,
we study the  system of nonlinear differential equations
\begin{equation}  \label{M}
\begin{gathered}
\frac{1}{A}(A y')'=g(t,y,z), \quad \text{in }  (0,\infty), \\
\frac{1}{B}(B z')'=f(t,y,z),\quad \text{ in }  (0,\infty),\\
\lim_{t\to0^+}A(t)y'(t)=\lim_{t\to0^+}B(t)z'(t)=0,\\
y(0)=a>0,\quad z(0)=b>0,
\end{gathered}
\end{equation}
where the continuous functions $A, B:[0,\infty) \to [0,\infty)$
are  nondecreasing, differentiable and positive on $(0,\infty)$.
We assume that $f$ and $g$ satisfy the following hypotheses.
\begin{itemize}
\item[(A1)] $f$, $g:(0,\infty)\times[0,\infty)\times[0,\infty)\to[0,\infty)$
are continuous, nondecreasing with respect
to the second and the third variables.

\item[(A2)] There exist nonnegative functions $h_i$, $k_i$, $\xi_i$, $\omega_i$,
$1\leq i\leq 2$, $\phi$ and $\psi$ satisfying
for each $(t,u,v)\in(0,\infty)\times[0,\infty)\times[0,\infty)$,
\begin{gather*}
h_2(t)\omega_1(v)\omega_2(u)\leq g(t,u,v)\leq h_1(t)\phi(u+v),
\\
k_2(t)\xi_1(u)\xi_2(v)\leq f(t,u,v)\leq k_1(t)\psi(u+v),
\end{gather*}
with $\xi_i$, $\omega_i$, $1\leq i\leq 2$, $\phi$,
$\psi:[0,\infty)\to[0,\infty)$ are
nondecreasing continuous, positive on $(0,\infty)$ and
$h_i$, $k_i:(0,\infty)\to[0,\infty)$
continuous and satisfying
$$
\int_0^1\frac{1}{A(t)}\Big(\int_0^tA(s)\,h_i(s)\,ds\Big)dt<\infty,\quad
\int_0^1\frac{1}{B(t)}\Big(\int_0^tB(s)\,k_i(s)\,ds\Big)dt<\infty.
$$

\item[(A3)] For all $ r>0$, we suppose that  $$\sqrt{2} \int_0^r
\sqrt{\widetilde{\Lambda}(s)}ds<M_{a+b}(\infty) \,,$$
where,  for $\alpha>0$,
\begin{gather*}
M_\alpha(t)=\int_\alpha^t\frac{ds}{\sqrt{2(\Phi(s)+\Psi(s))}}\,,\quad
 t\geq \alpha\\
\Phi(r)=\int_0^r\phi(s)ds,\quad \Psi(r)=\int_0^r\psi(s)ds\,,\quad r\geq 0\\
\widetilde{\Lambda}(r)=\max_{s\in[0,r]}(h_1(s)+k_1(s)),\quad r\geq0.
\end{gather*}
\end{itemize}
For any nonnegative measurable functions $\varphi$ in $(0,\infty)$, we define
\begin{gather*}
S_A\varphi(t)=\int_0^t\frac{1}{A(r)}\Big(\int_0^rA(s)\varphi(s)ds\Big)dr\,,\quad
S_B\varphi(t)=\int_0^t\frac{1}{B(r)}\Big(\int_0^rB(s)\varphi(s)ds\Big)dr.
\end{gather*}
Now, we are ready to give our existence result for \eqref{M}.

\begin{theorem} \label{AX}
Under the hypotheses {\rm (A1)--(A3)}, system \eqref{M} has a positive
solution $(y,z)\in\left(C([0,\infty))\cap C^1((0,\infty))\right)^2$ 
satisfying for each $t\in[0,\infty)$ 
\begin{gather*}
a+ \omega_1(b)\omega_2(a)S_A(h_2)(t)\leq y(t)
\leq M_{a+b}^{-1}\Big(\sqrt{2}\int_0^t \sqrt{\widetilde{\Lambda}(s)}ds\Big), \\
b+\xi_1(a)\xi_2(b)S_B(k_2)(t)\leq z(t)
 \leq M_{a+b}^{-1}\Big(\sqrt{2}\int_0^t\sqrt{\widetilde{\Lambda}(s)}ds\Big)\,,
\end{gather*}
where $M_{a+b}^{-1}$ is the inverse function of $M_{a+b}$ which is
 defined from $[0,M_{a+b}(\infty))$ to $[\alpha,\infty)$.
\end{theorem}

\begin{remark} \label{rmk1.4} \rm 
In the case $A(t)=B(t)$, the  solution $(y,z)$ of  system \eqref{M} satisfies
\begin{gather*}
a+\omega_1(b)\omega_2(a)S_A(h_2)(t)\leq y(t)
\leq M_{a+b}^{-1}\Big(\int_0^t \sqrt{\widetilde{\Lambda}(s)}ds\Big),\\
 b+\xi_1(a)\xi_2(b)S_A(k_2)(t)\leq z(t)
\leq M_{a+b}^{-1}\Big(\int_0^t \sqrt{\widetilde{\Lambda}(s)}ds\Big)
 \,,\quad \text{for } t\geq 0.
\end{gather*}
\end{remark}

Now we give the existence result for system \eqref{S1} under the following
hypotheses.
\begin{itemize}
\item[(H4)] The function $r\to r^{2(n-1)}\,(p_1(r)+q_1(r))$ is 
nondecreasing for large $r$.

\item[(H5)] There exists a positive constant $\varepsilon$ such that
$$
\int_0^\infty r^{1+\varepsilon}(p_1(r)+q_1(r))dr<\infty,
$$
\end{itemize}

\begin{theorem} \label{AA}
Under assumptions {\rm (H1)--(H5)}, system \eqref{S1} has a positive
 entire bounded continuous solution.
\end{theorem}

\begin{example} \label{examp1.1} \rm
Let $\alpha_i$, $\beta_i$, $\gamma_i$, $\delta_i$, $\lambda_i \in (0,\infty)$  
satisfying  $\max(\alpha_i,\beta_i, \gamma_i,
\delta_i+\lambda_i)<1$  for $1\leq i\leq 2$ and $2<\sigma<2(n-1)$. 
Put  
\begin{gather*}
H(x,u,v)=a_1(x)v^{\alpha_1}+b_1(x)u^{\gamma_1}+c_1(x)(u+v)^{\beta_1}
+d_1(x)u^{\delta_1} v^{\lambda_1}, \\
K(x,u,v)=a_2(x)v^{\alpha_2}+b_2(x)u^{\gamma_2}+c_2(x)(u+v)^{\beta_2}
+d_2(x)u^{\delta_2} v^{\lambda_2}\,,
\end{gather*}
 where and $a_i$, $b_i$, $c_i$, $d_i$ are
nontrivial nonnegative continuous function in $\mathbb{R}^n$ satisfying
$$
\sum_{i=1}^2\Big(\max_{|y|\leq s}a_i(y)+\max_{|y|\leq s}b_i(y)
+\max_{|y|\leq s}c_i(y)+\max_{|y|\leq s}d_i(y)\Big)
\leq \frac{1}{1+s^{\sigma}}.
$$
Then problem \eqref{S1} has a positive continuous bounded solution $(u,v)$. 
Indeed, the hypotheses (H1), (H2), (H4) and (H5) are clearly satisfied. 
Since  $\max(\alpha_i,\beta_i, \gamma_i,
\delta_i+\lambda_i)<1$ for $1\leq i\leq 2$ and
\begin{gather*}
H(x,u,v)\leq \frac{1}{1+|x|^{\sigma}} 
[(u+v)^{\alpha_1}+(u+v)^{\beta_1}+(u+v)^{\gamma_1}+(u+v)^{\delta_1
 +\lambda_1}],\\
K(x,u,v)\leq \frac{1}{1+|x|^{\sigma}} 
[(u+v)^{\alpha_2}+(u+v)^{\beta_2}+(u+v)^{\gamma_2}+(u+v)
 ^{\delta_2+\lambda_2}]
\end{gather*}
we deduce that $L_1(\infty)=\infty$ and so hypothesis (H3) is satisfied.
\end{example}

\begin{example} \label{examp1.2} \rm
Let $2<\sigma<2(n-1)$, and let $\beta_1, \beta_2>0$ be such that
 $\beta_0=\max(\beta_1,\beta_2)>1$
and
\begin{equation} \label{cond-ex2}
1+\frac{2}{\sigma-2}<\frac{1}{2(\beta_0-1)(Log(3))^{\beta_0}}.
\end{equation}
Let $p, q$ be two nontrivial nonnegative continuous functions
in $\mathbb{R}^n$ such that
$$
\max_{|y|\leq s}p(y)+\max_{|y|\leq s}q(y)\leq \frac{1}{1+s^{\sigma}}.
$$
Then the  problem
\begin{gather*}
\Delta u=2p(x)(u+v+2)(Log(u+v+2))^{2\beta_1-1}(Log(u+v+2)+\beta_1)\\
\Delta v=2q(x)(u+v+2)(Log(u+v+2))^{2\beta_2-1}(Log(u+v+2)+\beta_2)\,,
\end{gather*}
 in  $\mathbb{R}^n$ with $n\geq 3$,
has a positive bounded continuous solution $(u,v)$.
Indeed, the hypotheses (H1),
(H2), (H4) and (H5) are clearly satisfied. Now,  (H3) follows
from \eqref{cond-ex2} and the fact that
for each $t\geq1$,
$$
\int_0^t\frac{ds}{\sqrt{1+s^{\sigma}}}
\leq \int_0^1\frac{ds}{\sqrt{1+s^{\sigma}}}+
\int_1^t\frac{ds}{\sqrt{1+s^{\sigma}}}
\leq 1+\int_1^t\frac{ds}{s^{\frac{\sigma}{2}}}\leq 1+\frac{2}{\sigma-2}
$$
and
\begin{align*}
\int_1^{\infty}\frac{ds}{\sqrt{2(s+2)^2
((\log(s+2))^{2\beta_1}+(\log(s+2))^{2\beta_2})}}
&\geq \int_1^{\infty}\frac{ds}{2(s+2)(Log(s+2))^{\beta_0}}\\
&=\frac{1}{2(\beta_0-1)(Log(3))^{\beta_0-1}}.
\end{align*}
 From Theorem \ref{AA}, we have the following corollaries in the
case where
$H(x,u,v)=H(|x|,u,v)$ and $K(x,u,v)=K(|x|,u,v)$.
\end{example}

\begin{corollary} \label{coro1.6} 
Under hypotheses {\rm (H1)--(H3)}, problem \eqref{S1} has one positive
solution. Under the additional hypothesis
\begin{itemize}
\item[(H6)] $\int_0^{\infty}sp_2(s)\,ds=\int_0^{\infty}sq_2(s)\,ds=\infty$,
\end{itemize}
every positive radial entire solution $(u,v)$ of \eqref{S1} 
is large and satisfies
$$ 
u(0)+g_1(v(0))g_2(u(0))P_2(r)\leq u(r),\quad 
v(0)+f_1(u(0))f_2(v(0))Q_2(r)\leq v(r) \,,
$$
for all $r\geq 0$, where 
$$
P_2(r)=\int_0^rt^{1-n}\Big(\int_0^ts^{n-1}p_2(s)\,ds\Big)\,dt,\quad 
Q_2(r)=\int_0^rt^{1-n}\Big(\int_0^ts^{n-1}q_2(s)\,ds\Big)\,dt.
$$
\end{corollary}

For the next corollary, we use the assumption
\begin{itemize}
\item[(H7)] $L_\alpha(\infty)=\infty$.
\end{itemize}

\begin{corollary} \label{coro1.7}
Assume that {\rm (H1), (H2), (H4), (H7)}  are satisfied.
If \eqref{S1} has a nonnegative radial entire large solution, then
$$
\int_0^\infty r^{1+\varepsilon}(p_1(r)+q_1(r))dr
=\infty,\quad \forall\;\varepsilon>0.
$$
\end{corollary}

An other result for the radial case is given under the assumption
\begin{itemize}
\item[(H8)] $\sqrt{p_1+q_1}\in L^1(0,\infty)$.
\end{itemize}

\begin{theorem} \label{BB}
Assume that {\rm (H1), (H2), (H4), (H8)} are satisfied.
If $(u,v)$ is a positive entire large radial solution of \eqref{S1},
then $F$ and
$G$ satisfy the Keller-Osserman condition
$$
L_1(\infty)=\int_1^{\infty}\frac{ds}{\sqrt{2
(\mathcal{F}(s)+\mathcal{G}(s))}}<\infty.
$$
\end{theorem}

\section{Proof of main results}

\begin{proof}[Proof of Theorem \ref{AX}]
Let $(y_m)_{m\geq0}$ and $(z_m)_{m\geq0}$ be the sequences of positive 
continuous functions defined on $[0,\infty)$ by
\begin{gather*}
y_0(t)=a,\;\;z_0(t)=b,\\
y_{m+1}(t)=a+\int_0^t\frac{1}{A(r)}
\Big(\int_0^rA(s)g(s,y_m(s),z_m(s))ds\Big)dr,\\
z_{m+1}(t)=b+\int_0^t\frac{1}{B(r)}\Big(\int_0^rB(s)f(s,y_m(s),z_m(s))ds\Big)dr.
\end{gather*}
Clearly $y_m,z_m\in C([0,\infty))\cap C^1((0,\infty))$ and are positive, 
so we deduce by
(A1) that $(y_m)_{m\geq0}$ and $(z_m)_{m\geq0}$ are nondecreasing sequences
and for each $m\in{\mathbb{N}}$, the functions $t\to y_m(t)$ and $t\to z_m(t)$
are nondecreasing. Hence, for each $t\in(0,\infty),$
\begin{gather*}
 y'_m(t) = \frac{1}{A(t)}\int_0^tA(s)g(s,y_{m-1}(s),z_{m-1}(s))ds, \\
 z'_m(t) = \frac{1}{B(t)}\int_0^tB(s)f(s,y_{m-1}(s),z_{m-1}(s))ds
\end{gather*}
Which implies 
\[
\left(A(t)y'_m(t)\right)'= A(t)g(t,y_{m-1}(t),z_{m-1}(t)),\quad
\leq A(t)g(t,y_{m}(t),z_{m}(t)) .
\]
Multiplying this expression by $2A(t)y'_m(t)$ and integrate on $[0,t]$, 
we obtain
\begin{align*}
\left(A(t)y'_m(t)\right)^2 
&\leq 2\int_0^t
A^2(s)h_1(s)\left(\phi(y_{m}(s)+z_{m}(s))\right)\,y'_m(s)\,ds \\
&\leq 2\widetilde{\Lambda}(t)A^2(t)
 \int_0^t\left(\phi(y_{m}(s)+z_{m}(s))\right)\,(y'_m(s)+z'_m(s))\,ds\\ 
&=  2\widetilde{\Lambda}(t)A^2(t)\int_{a+b}^{y_{m}(t)+z_{m}(t)}\phi(s)\,ds.
\end{align*}
Which implies 
\[
y'_m(t) \leq \sqrt{\widetilde{\Lambda}(t)}\sqrt{2\Phi({y_{m}(t)+z_{m}(t)})}.
\]
Similarly, we have
\[
z'_m(t) \leq \sqrt{\widetilde{\Lambda}(t)}\sqrt{2\Psi({y_{m}(t)+z_{m}(t)})}.
\]
Then 
\[
y'_m(t)+z'_m(t)\leq \sqrt{2}\sqrt{\widetilde{\Lambda}(t)}
\sqrt{2(\Phi(y_m(t)+z_m(t))+\Psi(y_m(t)+z_m(t)))}.
\]
Therefore,
\begin{align*}
M_{a+b}(y_m(t)+z_m(t))
&= \int_{a+b}^{y_m(t)+z_m(t)}\frac{ds}{\sqrt{2\left(\Phi(s)+\Psi(s)\right)}}\,ds\\
&= \int_0^t\frac{y'_m(s)+z'_m(s)}{\sqrt{2\left(\Phi(y_m(s)+z_m(s))+\Psi(y_m(s)
 +z_m(s))\right)}}\,ds  \\
&\leq \sqrt{2}\int_0^r\sqrt{\widetilde{\Lambda}(s)}ds.
\end{align*}
Since $M_{a+b}^{-1}$ is increasing on $[0,M_{a+b}(\infty))$, we obtain
\[
y_m(t)+z_m(t)\leq M_{a+b}^{-1}
\Big(\sqrt{2}\int_0^t\sqrt{\widetilde{\Lambda}(s)}ds\Big),\quad \forall\,r\geq0.
\]
Therefore, the sequences $(y_m)_{m\geq0}$ and $(z_m)_{m\geq0}$ converge  to
two functions $y$ and $z$ that, for each $t\in[0,\infty)$, satisfy
\begin{gather*}
y(t)=a+\int_0^t\frac{1}{A(r)}\Big(\int_0^rA(s)g(s,y(s),z(s))ds\Big)dr,\\
z(t)=b+\int_0^t\frac{1}{B(r)}\left(\int_0^rB(s)f(s,y(s),z(s))ds\right)dr.
\end{gather*}
Hence, $y$, $z\in C([0,\infty))\cap C^1((0,\infty))$ and $(y,z)$ 
is a solution of \eqref{M} satisfying
\begin{gather*}
a+g_1(b)g_2(a)S_A(h_2)(t)\leq y(t)\leq M_{a+b}^{-1}
\Big(\sqrt{2}\int_0^t\sqrt{\widetilde{\Lambda}(s)}ds\Big), \\
b+f_1(a)f_2(b)S_B(k_2)(t)\leq z(t)\leq M_{a+b}^{-1}
\Big(\sqrt{2}\int_0^t\sqrt{\widetilde{\Lambda}(s)}ds\Big).
\end{gather*}
In the case where $A(t)=B(t)$, we multiply the inequality
\[
\left(A(t)(y'_m(t)+z'_m(t))\right)'
\leq A(t)[g(t,y_{m}(t),z_{m}(t))+f(t,y_m(t),z_m(t))]
\]
by $2A(t)(y'_m(t)+z'_m(t))$, we integrate on $[0,t]$ and we proceed
 as below to obtain
\[
\left(A(t)(y'_m(t)+z'_m(t))\right)^2  
\leq (A(t))^2 \,2\widetilde{\Lambda}(t)
\int_0^{y_{m}(t)+z_{m}(t)}\left(\phi(s)+\psi(s)\right)\,ds.
\]
Which implies 
\begin{align*}
y'_m(t)+z'_m(t) \leq  
\sqrt{\widetilde{\Lambda}(t)}\sqrt{2\Phi({y_{m}(t)+z_{m}(t)})}.
\end{align*}
So in this case $(y,z)$ satisfies
\begin{gather*}
a+\omega_1(b)\omega_2(a)S_A(h_2)(t)\leq y(t)\leq M_{a+b}^{-1}\Big(\int_0^t
\sqrt{\widetilde{\Lambda}(s)}ds\Big), \\
 b+\xi_1(a)\xi_2(b)S_A(k_2)(t)\leq z(t)\leq M_{a+b}^{-1}\left(\int_0^t
\sqrt{\widetilde{\Lambda}(s)}ds\right) \,,\; \text{ for } t\geq 0.
\end{gather*}
This completes the proof.
\end{proof}

\begin{proof}[Proof of Theorem \ref{AA}]
 We will show that \eqref{S1} has a solution by finding a
subsolution $(\underline{u},\underline{v})$ and a supersolution 
$(\overline{u},\overline{v})$,
such that  $\underline{u}\leq\overline{u}$ and $\underline{v}\leq\overline{v}$. 
To do this, we first prove the existence of a radial subsolution 
 $(\underline{u},\underline{v})$ to \eqref{S1}
by considering the  problem
\begin{gather*}
\triangle \underline{u}=p_1(|x|)F(\underline{u}+\underline{v}),
 \quad \text{in }  \mathbb{R}^n\\
\triangle \underline{v}=q_1(|x|)G(\underline{u}+\underline{v}),\quad
\text{in }  \mathbb{R}^n,
\end{gather*}
with $n\geq 3$. 
That is, $(\underline{u},\underline{v})$ satisfies
\begin{equation} \label{M2}
\begin{gathered}
\frac{1}{r^{n-1}} (r^{n-1}\underline{u}')'
 =p_1(r)F(\underline{u}+\underline{v}), \, r\in (0,\infty)\\
\frac{1}{r^{n-1}} (r^{n-1}\underline{v}')'
 =q_1(r)G(\underline{u}+\underline{v})\, r \in (0,\infty).
\end{gathered}
\end{equation}
By Theorem \ref{AX}, we conclude that  system \eqref{M2} has a positive
entire solution $(\underline{u},\underline{v})$.
Now, since
\begin{gather*}
\triangle\underline{u} = p_1(|x|)F(\underline{u}+\underline{v})
 \geq H(x,\underline{u},\underline{v}),
\quad\text{in }  \mathbb{R}^n, \\
\triangle\underline{v} = q_1(|x|)G(\underline{u}+\underline{v})
 \geq K(x,\underline{u},\underline{v}),\quad\text{in }  \mathbb{R}^n,
\end{gather*}
we deduce that $(\underline{u},\underline{v})$ is a subsolution of system \eqref{S1}.\\
Next we prove that $(\underline{u},\underline{v})$ is bounded. Since $(\underline{u},\underline{v})$ satisfies
\begin{align*}
(r^{n-1}\underline{u}'(r))'&=& r^{n-1}p_1(r)F(\underline{u}(r)+\underline{v}(r)),\\
(r^{n-1}\underline{v}'(r))'&=& r^{n-1}q_1(r)G(\underline{u}(r)+\underline{v}(r)),
\end{align*}
it follows that
\begin{equation} \label{F3}
(r^{n-1}(\underline{u}'(r)+\underline{v}'(r)))'
= r^{n-1}[p_1(r)F(\underline{u}(r)+\underline{v}(r))
+q_1(r)G(\underline{u}(r)+\underline{v}(r))].
\end{equation}
Choose $R>0$ so that $r^{2(n-1)}\,(p_1(r)+q_1(r))$ is nondecreasing
on $[R,\infty)$. Then, after
multiplying $\eqref{F3}$  by $r^{n-1}(\underline{u}'(r)+\underline{v}'(r))$
and integrating from $R$ to $r$, we obtain
\begin{align*}
& \left(r^{n-1}(\underline{u}'(r)+\underline{v}'(r))\right)^2\\
&=C+2\Big(\int_R^rt^{2(n-1)}
[p_1(t)F(\underline{u}(t)+\underline{v}(t))+q_1(r)
\mathcal{G}(\underline{u}(r)+\underline{v}(r))]dt\Big) \\
&\leq C+2r^{2(n-1)}\big(p_1(r)+q_1(r)\big)\\
&\quad\times \Big(\int_R^r[\mathcal{F}'(\underline{u}(t)
 +\underline{v}(t))+\mathcal{G}'(\underline{u}(t)+\underline{v}(t))]
 (\underline{u}'(t)+\underline{v}'(t))\,dt\Big) \\
&\leq C+2r^{2(n-1)}(p_1(r)+q_1(r))
\big[\mathcal{F}(\underline{u}(r)+\underline{v}(r))
 +\mathcal{G}(\underline{u}(r)+\underline{v}(r))\big],
\end{align*}
where $C=\left (R^{n-1}(\underline{u}'(R)+\underline{v}'(R))\right)^2$.
This  yields
\begin{align*}
&\frac{\underline{u}'(r)+\underline{v}'(r)}{\sqrt{2}(\mathcal{F}(\underline{u}(r)
+\underline{v}(r))+\mathcal{G}(\underline{u}(r)
+\underline{v}(r)))^\frac{1}{2}}\\
&\leq\frac{\sqrt{C}r^{1-n}}{\sqrt{(\mathcal{F}(\underline{u}(r)+\underline{v}(r))
 +\mathcal{G}(\underline{u}(r)+
\underline{v}(r)))}}+\sqrt{2}\,\sqrt{p_1(r)+q_1(r)}.
\end{align*}
Integrating the above inequality and using the fact that
$$
\mathcal{F}(\underline{u}(r)+\underline{v}(r))
 +\mathcal{G}(\underline{u}(r)+\underline{v}(r))\geq\mathcal{F}(\underline{u}(R)+
\underline{v}(R))+\mathcal{G}(\underline{u}(R)+\underline{v}(R))=C_1,
$$
for all $r\geq R$, and
\[
 \sqrt{p_1(r)+q_1(r)}
\leq 2\sqrt{r^{1+\varepsilon}(p_1(r)+q_1(r))r^{-1-\varepsilon}}
\leq r^{1+\varepsilon}  (p_1(r)+q_1(r))+r^{-1-\varepsilon}\,,
\]
we obtain
\begin{align*}
&L_\alpha(\underline{u}(r)+\underline{v}(r))\\
&\leq \sqrt{2}\,\int_R^rs^{1+\varepsilon}(p_1(s)+q_1(s))ds
+\sqrt{2}\, (\varepsilon R^\varepsilon)^{-1}
 +\sqrt{\frac{C}{C_1}}(2-n) (R^{2-n})^{-1}\,,
\end{align*}
where $\alpha=\underline{u}(R)+\underline{v}(R)$.
Letting $r\to\infty$, we deduce from hypothesis  (H5) that
$(\underline{u},\underline{v})$ is bounded.
Thus, since $(\underline{u},\underline{v})$ is nondecreasing, we have
$$
\lim_{r\to\infty}\underline{u}(r)=M_1>0,\quad
\lim_{r\to\infty}\underline{v}(r)=M_2>0.
$$
Now, it is clear that $(\overline{u},\overline{v})=(M_1,M_2)$
is a supersolution for \eqref{S1} and we have for $r\geq 0$,
$$
\overline{u}(r)\geq M_1\geq\underline{u}(r),\quad
\overline{v}(r)\geq M_2\geq\underline{v}(r).
$$
Hence the standard sub-supersolution method (see \cite{HER,SAT})
implies that \eqref{S1}
has a bounded solution $(u,v)$ such that $\underline{u}\leq u\leq\overline{u}$
and $\underline{v}\leq v\leq\overline{v}$. This completes the proof.
\end{proof}

\begin{proof}[Proof of Theorem \ref{BB}]
Let $(u,v)$ be a positive entire large radial solution of \eqref{S1}.
Then $(u,v)$ satisfies
\begin{gather*}
(r^{n-1}u'(r))'\leq r^{n-1}p_1(r)F(u(r)+v(r)),\\
(r^{n-1}v'(r))'\leq r^{n-1}q_1(r)G(u(r)+v(r)).
\end{gather*}
Adding these inequalities, we obtain
\begin{align*}
(r^{n-1}(u'(r)+v'(r)))'
&\leq  r^{n-1}p_1(r)F(u(r)+v(r))+ r^{n-1}q_1(r)G(u(r)+v(r))\\
&\leq r^{n-1}(p_1(r)+q_1(r))\left(F(u(r)+v(r))+ G(u(r)+v(r))\right).
\end{align*}
Multiplying the above inequality by $2r^{n-1}(u'(r)+v'(r))$
and integrating from $0$ to $r$, and using (H4), we obtain
\begin{align*}
\big(r^{n-1}(u'(r)+v'(r))\big)^2 
& \leq 2\Big(\int_0^rs^{2(n-1)}(p_1(s)+q_1(s))
\Big(F(u(s)+v(s))\\
&\quad +G(u(s)+v(s))\Big)(u'(s)+v'(s))\,ds\Big) \\
& \leq 2r^{2(n-1)}(p_1(r)+q_1(r)) \int_0^r 
\Big(F(u(s)+v(s))\\
&\quad +G(u(s)+v(s))\Big)(u'(s)+v'(s))\,ds \\
& \leq 2r^{2(n-1)}(p_1(r)+q_1(r))
\left(\mathcal{F}(u(r)+v(r))+\mathcal{G}(u(r)+v(r))\right)
\end{align*}
which implies
$$
\frac{u'(r)+v'(r)}{\sqrt{2(\mathcal{F}(u(r)+v(r))+\mathcal{G}(u(r)+v(r)))}}
\leq\sqrt{p_1(r)+q_1(r)}.
$$
By integrating  over $[0,r]$, we  obtain 
$$
L_{u(0)+v(0)}(u(r)+(v(r))\leq\int_0^r\sqrt{p_1(s)+q_1(s)}ds.
$$
Letting $r\to\infty$, we obtain that $F$ and $G$ satisfies 
the Keller-Osserman condition.
\end{proof}

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\end{document}