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

\AtBeginDocument{{\noindent\small
\emph{Electronic Journal of Differential Equations},
Vol. 2012 (2012), No. 67, pp. 1--13.\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{9mm}}

\begin{document}
\title[\hfilneg EJDE-2012/67\hfil Schr\"odinger systems with a convection term]
{Schr\"odinger systems with a convection term for the
$(p_1,\dots ,p_d)$-Laplacian in $\mathbb{R}^N$}

\author[D.-P. Covei\hfil EJDE-2012/67\hfilneg]
{Dragos-Patru Covei}

\address{Dragos-Patru C. Covei \newline
Department of Development, Constantin Brancusi University from Tg-Jiu,
Str. Grivitei, Nr. 1, Tg-Jiu, Romania}
\email{coveipatru@yahoo.com}

\thanks{Submitted March 12, 2012. Published May 2, 2012.}
\subjclass[2000]{35J62, 35J66, 35J92, 58J10, 58J20}
\keywords{Entire solutions; large solutions; quasilinear systems;
radial solutions}

\begin{abstract}
 The main goal is to study  nonlinear Schr\"odinger type
 problems for the $(p_1,\dots ,p_d)$-Laplacian  with
 nonlinearities satisfying Keller- Osserman conditions.
 We establish the existence of infinitely many positive entire radial
 solutions by an application of a fixed point theorem and the
 Arzela-Ascoli theorem. An important aspect in this article is that
 the solutions are obtained by successive approximations and hence
 the proof can be implemented in a computer program.
\end{abstract}

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


\section{Introduction}

Nonreactive scattering of atoms and molecules, and  related bound
state energy eigenvalue problems can be formulated by the radial
Schr\"{o}dinger system
\begin{equation}
\begin{gathered}
U''+\frac{N-1}{r}U'=A(r) U(r) \\
U(r) \to U_{\infty }\quad\text{as }r\to \infty
\end{gathered}\label{S}
\end{equation}
where $r:=| x|$ ($| \cdot |$ is the Euclidean norm),
the wave function $U(r) $ is a $d\times 1$ vector and the
potential function $A(r) $ is a $d\times d$ symmetric matrix.
We refer the reader to \cite{L,GG} for some additional details.

In recent years, much effort has been devoted to the problems which arise in
connection with the system \eqref{S} and that are related to nonlinear
differential equations. However, most of the treatments are either for
coupled systems of equations or for scalar equations
(see \cite{BEC4}-\cite{LZZ}).

The object of this work is to develop an existence theory for radial
solutions of the basic nonlinear elliptic system
\begin{equation}
\begin{gathered}
\Delta _{p_1}u_1+h_1(| x| ) |
\nabla u_1| ^{p_1-1}=a_1(| x|
) g_1(u_1,\dots ,u_d) \quad\text{for }x\in \mathbb{R}^{N},
\\
\dots  \\
\Delta _{p_d}u_d+h_d(| x| ) |
\nabla u_d| ^{p_d-1}=a_d(| x|) g_d(u_1,\dots ,u_d)
 \quad\text{for }x\in \mathbb{R}^{N},
\end{gathered} \label{11}
\end{equation}
where $d\geq 1$, $1<p_i\leq $ $N-1$, $i=1,\dots ,d$
 and $\Delta _{p_i}$ is the so called $p_i$-Laplacian operator defined by
\[
\Delta _{p_i}u_i:=\operatorname{div}(| \nabla u_i|^{p_i-2}\nabla u_i) .
\]
The operator $\Delta _{p_i}$ with $p_i\neq 2$ occurs in many
mathematical models of physical processes: it is used in non-Newtonian
fluids where the shear stress $\overrightarrow{\tau }$ and the velocity
gradient $\nabla u_i$ of the fluid are related in the manner that
$\overrightarrow{\tau }(x)  = r_i(x)| \nabla u_i| ^{p_i-2}\nabla u_i$
(for $p_i=2$ (respectively, $p_i<2$, $p_i>2$) if the fluid is Newtonian (respectively, pseudoplastic, dilatant)), in some reaction-diffusion problems,
in nonlinear elasticity ($p_i>2$), glaciology ($1<p_i<\frac{4}{3}$), of
flow through porous media ($p_i=\frac{3}{2}$), in petroleum extraction
as well as in torsional creep problems, see the book of Diaz \cite{D} and Lions \cite{LI} where are
collected detailed references on physical background and presented
mathematical treatments of free boundary problem associated with the
operator $\Delta _{p_i}$.

In the mathematical context several interesting results about blowup
theorems for solutions of nonlinear system like \eqref{11} are known and
have been obtained by several authors. For further discussion, examples and
references, in the particular case $d=2$ and $p_1=p_2=2$ we refer to
\cite{CD2}, \cite{LZZ} and \cite{PW}.
 The case $p_1\neq \dots \neq p_d\neq 2$ is not yet well understood, but is the subject of much current research. Some very recent existence results on the $p_i$-Laplacian can be
found in a recent paper of Hamydy-Massar and Tsouli \cite{H} where they study
entire large solutions to the system \eqref{11} when $p_j\geq 2$
($d=2 $, $j=1,2$) and for the functions $h_j$, $a_j$, $g_j$ that satisfy
\begin{itemize}
\item[(A1)]  $h_j$, $a_j:[ 0,\infty ) \to [0,\infty ) $ are radial continuous functions;

\item[(G1)]  $g_j:[ 0,\infty ) ^{d}\to [ 0,\infty) $ are continuous in all variables;

\item[(G2)]  $g_j$ are non-decreasing on $[ 0,\infty ) ^{d}$ in
all variables;

\item[(G3)] for all $M>0$
\[
\lim_{t\to \infty }\frac{g_1(Mg_2(t) ^{\frac{1}{p_2-1}}) }{t^{p_1-1}}=0.
\]
\end{itemize}
Under these hypotheses and the integral condition
\[
\int_1^{\infty }(r^{1-N}e^{-\int_0^{r}h_j(t)
dt}\int_0^{t}r^{N-1}e^{\int_0^{r}h_j(t) dt}a_j(
s) ds) ^{\frac{1}{p_j-1}}dt=\infty , \quad j=1,\dots ,d
\]
they proved that the system \eqref{11} has infinitely many positive entire
large solutions.

Regarding the case $d=2$ and $p_1=p_2=2$, Zhang and Liu \cite{LZZ}
studied the existence of entire large positive solutions of the system
\begin{gather*}
\Delta u_1+| \nabla u_1| =a_1(r) g_1(u_1,u_2), \\
\Delta u_2+| \nabla u_2| =a_2(r) g_2(u_1,u_2) ,
\end{gather*}
where $r:=| x|$, $x\in \mathbb{R}^{N}$.
They generalized the results of several authors by considering $a_1$,
$a_2$, $g_1$ and $g_2$ satisfying (A1), (G1), (G2) and instead of (G3)
the condition
\begin{equation}
\int_{a}^{\infty }\frac{ds}{g_1(s,s) +g_2(s,s) }
=\infty \quad\text{for }r\geq a>0.  \label{LZZ}
\end{equation}
It is interesting that for a single equation of the form
$\Delta u=g(u) $ where $g(u) $ is positive, real continuous function
defined for all real $u$ and nondecreasing the existence of entire large
solutions is equivalent to a condition on $g$ known as the Keller-Osserman
condition
\begin{equation}
\int_{u_0}^{\infty }\Big(\int_0^{t}g(s) ds\Big)^{-1/2}dt
=\infty \quad\text{for  }u_0>0,  \label{KO1}
\end{equation}
(see \cite{K,O}) and that for systems, no such a result exists
yet.

Motivated by the references mentioned above it is interested whether
similar results can be obtained for nonlinearities $g_i$ ($i=1,\dots ,d$) of
the type \eqref{KO1}, which includes, as a special case, a similar result of
\cite{CD2}. Also, we are interested in the existence results allowing any
$p_1,\dots ,p_d>1$. The answers of these questions are certainly not trivial
and seems to be applicable to more general nonlinearities e.g., those
studied in \cite{K,JM1} or more suggestive in the works of \cite{GS}
respectively \cite{PW}.

Let us finish our presentation to announce our main result that can be
stated as follows.

\begin{theorem} \label{thm1}
Suppose the functions $a_j$, $h_j$ satisfy {\rm (A1)}, $g_j$
satisfy {\rm (G1), (G2)} and the ``Keller-Osserman type'' condition
\begin{equation}
\quad I(\infty ) :=\lim_{r\to \infty }I(r)=\infty  \label{KO}
\end{equation}
where
\[
I(r) :=\int_{a}^{r}[G(s) ]^{-1/\min \{p_1,\dots ,p_d\}}ds
\]
for $r\geq a>0$, and
 $G(s) :=\int_0^s  \sum_{i=1}^d g_i(t,\dots ,t) dt+1$. Under
these hypotheses  there are infinitely many positive entire radial
solutions of  \eqref{11}. Suppose furthermore that
\[
\frac{p_j}{p_j-1}s^{\frac{p_j(N-1) }{p_j-1}}e^{\frac{
p_j}{p_j-1}\int_0^s h_j(t)dt}a_j(s) \quad \text{for }
j=1,\dots ,d,
\]
 is nondecreasing for large $s$. Then
\begin{itemize}
\item[(i)] The solutions are bounded if there exists a positive number
$\varepsilon $  such that
\begin{equation}
\int_0^{\infty }t^{1+\varepsilon }(e^{\frac{p_j}{p_j-1}
\int_0^{t}h_j(t)dt}a_j(t) ) ^{2/p_j}dt<\infty \quad
\text{for all }j=1,\dots ,d,  \label{5}
\end{equation}
\item[(ii)] The solutions are large if
\begin{equation}
\int_0^{\infty }(\frac{e^{-\int_0^{t}h_j(s) ds}}{
t^{N-1}}\int_0^{t}s^{N-1}e^{\int_0^s h_j(t)
dt}a_j(s) ds) ^{1/(p_j-1) }dt=\infty \label{12}
\end{equation}
for  $j=1,\dots ,d$.  
\end{itemize}
\end{theorem}

As far as we know, there is no such a result in any work from the
literature, because no solutions have been detected yet for the system of
the form \eqref{11} under the Keller-Osserman conditions \eqref{KO}.

\section{Proof of the Theorem \ref{thm1}} \label{ss1}

In this section, we show the existence of positive radial solutions
of \eqref{11}. The proof is inspired by \cite{CD2} with some new ideas.
 Now we remark that \eqref{11} has a solution $(u_1,\dots ,u_d) :=(u_1(
r) ,\dots ,u_d(r) ) $ if and only if $(u_1,\dots ,u_d) $
solves the system of second-order ordinary
differential equations
\begin{equation}
\begin{gathered}
(p_1-1) (u_1') ^{p_1-2}u_1''+\frac{N-1}{r}(u_1') ^{p_1-1}+h_1(
r) | u_1'| ^{p_1-1}=a_1(
r) g_1(u_1,\dots ,u_d) , \\
\dots  \\
(p_d-1) (u_d') ^{p_d-2}u_d''+\frac{N-1}{r}(u_d') ^{p_d-1}+h_d(
r) | u_d'| ^{p_d-1}=a_d(
r) g_d(u_1,\dots ,u_d) , \\
u_i'(0) =0\quad\text{for }i=1,\dots ,d
\end{gathered}  \label{66}
\end{equation}
where we can assume in the next that $u_i'(r) \geq 0$ for  $i=1,\dots ,d$.

However, in view of the symmetry of $(u_1,\dots ,u_d) $, we
have that radial solutions of \eqref{66} are  positive solutions
 $(u_1,\dots ,u_d) $ of the integral equations
\begin{equation}
\begin{gathered}
u_1(r) =u_1(0)+\int_0^{r}\Big(\frac{e^{-
\int_0^{t}h_1(s) ds}}{t^{N-1}}\int_0^{t}s^{N-1}e^{
\int_0^s h_1(s) dt}a_1(s) g_1(
u_1,\dots ,u_d) ds\Big) ^{\frac{1}{p_1-1}}dt, \\
\dots  \\
u_d(r) =u_d(0)+\int_0^{r}\Big(\frac{e^{-
\int_0^{t}h_d(s) ds}}{t^{N-1}}\int_0^{t}s^{N-1}e^{
\int_0^s h_d(s) dt}a_d(s) g_d(
u_1,\dots ,u_d) ds\Big) ^{\frac{1}{p_d-1}}dt.
\end{gathered}  \label{non}
\end{equation}
Our first idea in the proof of the main result is to regard \eqref{non} as
an operator equation
\[
S(u_1(r),\dots ,u_d(r) ) =(u_1(r),\dots ,u_d(r) )
\]
with
\[
S:C[ 0,\infty ) \times \dots \times C[ 0,\infty )
\to C[ 0,\infty ) \times \dots \times C[ 0,\infty
) \text{ }
\]
defined by
\begin{equation}
\begin{split}
& S(u_1(r) ,\dots ,u_d(r) ) \\
&=  \begin{pmatrix}
u_1(0)+\int_0^{r}(\frac{e^{-\int_0^{t}h_1(s) ds}}{
t^{N-1}}\int_0^{t}s^{N-1}e^{\int_0^s h_1(s)
dt}a_1(s) g_1(u_1,\dots ,u_d) ds) ^{\frac{
1}{p_1-1}}dt \\
\dots  \\
u_d(0)+\int_0^{r}(\frac{e^{-\int_0^{t}h_d(s) ds}}{
t^{N-1}}\int_0^{t}s^{N-1}e^{\int_0^s h_d(s)
dt}a_d(s) g_d(u_1,\dots ,u_d) ds) ^{\frac{
1}{p_d-1}}dt
\end{pmatrix}
\end{split}\label{op}
\end{equation}
where $u_1(0)=\dots =u_d(0)=b/d$ with $b\geq a>0$ are the central
values for the system. The integration in this operator implies that a fixed
point
\[
(u_1,\dots ,u_d) \in C[ 0,\infty ) \times \dots \times C [ 0,\infty )
\]
is in fact in the space
 $C^{1}[ 0,\infty ) \times \dots \times C^{1} [ 0,\infty ) $.
Then a solution of \eqref{66} will be obtained as
a fixed point of the operator \eqref{op}. To establish a solution to this
operator, we use successive approximation which constitutes an indispensable
tool for solving nonlinear systems \eqref{11} at this point. We define,
recursively, sequences $\{ u_i^k\} _{i=\overline{1,\dots ,d}}^{k\geq 1}$
on $[ 0,\infty ) $ by
\[
u_1^{0}=\dots =u_d^{0}=\frac{b}{d}\quad\text{for all $r\geq 0$ and
$b\geq a>0$}
\]
and
\begin{equation}
\begin{split}
& (u_1^k,\dots ,u_d^k) =S(
u_1^{k-1}(r),\dots ,u_d^{k-1}(r) ) \\
&=  \begin{pmatrix}
\frac{b}{d}+\int_0^{r}(\frac{e^{-\int_0^{t}h_1(s) ds}
}{t^{N-1}}\int_0^{t}s^{N-1}e^{\int_0^s h_1(s)
dt}a_1(s) g_1(u_1^{k-1},\dots ,u_d^{k-1})
ds) ^{\frac{1}{p_1-1}}dt \\
\dots  \\
\frac{b}{d}+\int_0^{r}(\frac{e^{-\int_0^{t}h_d(s) ds}
}{t^{N-1}}\int_0^{t}s^{N-1}e^{\int_0^s h_d(s)
dt}a_d(s) g_d(u_1^{k-1},\dots ,u_d^{k-1})
ds) ^{\frac{1}{p_d-1}}dt
\end{pmatrix} ^{T}.
\end{split} \label{oop}
\end{equation}
It is easy to see that, for all $r\geq 0$, $i=1,\dots ,d$ and
 $k\in N$ we have \[
u_i^k(r) \geq \frac{b}{d},
\]
and that $\{ u_i^k\} _{i=1,\dots ,d}^{k\geq 1}$ is an increasing
sequence of nonnegative and non-decreasing functions.

We note that $\{ u_i^k\} _{i=1,\dots ,d}^{k\geq 1}$ satisfy
\begin{equation}
\begin{aligned}
&(p_1-1) [ (u_1^k) ']^{p_1-2}(u_1^k) ''+(\frac{N-1}{r}
+h_1(r) ) [ (u_1^k) '] ^{p_1-1} \\
&=a_1(r) g_1(u_1^{k-1}(r),\dots ,u_d^{k-1}(r) ) , \\
&\quad \dots  \\
&(p_d-1) [ (u_d^k) ']^{p_d-2}(u_d^k) ''+(\frac{N-1}{r}
+h_1(r) ) [ (u_d^k) '] ^{p_d-1} \\
&=a_d(r) g_d(u_1^{k-1}(r),\dots ,u_d^{k-1}(r) ) .
\end{aligned}  \label{sis1}
\end{equation}
Using the monotonicity of $\{ u_i^k\} _{i=1,\dots ,d}^{k\geq 1}$
we have
\begin{equation} \label{8}
\begin{aligned}
a_1(r) g_1(u_1^{k-1}(r),\dots ,u_d^{k-1}(r) )
&\leq a_1(r) g_1(u_1^k,\dots ,u_d^k)   \\
&\leq a_1(r) \sum_{i=1}^d g_i\Big(
\sum_{i=1}^d u_i^k,\dots ,\sum_{i=1}^d u_i^k\Big) ,   \\
&\quad \dots   \\
a_d(r) g_d(u_1^{k-1}(r),\dots ,u_d^{k-1}(r) )
 &\leq a_d(r) g_d(u_1^k,\dots ,u_d^k)   \\
&\leq a_d(r) \sum_{i=1}^d g_i\Big(
\sum_{i=1}^d u_i^k,\dots , \sum_{i=1}^d u_i^k\Big) ;
\end{aligned}
\end{equation}
 moreover,
\begin{equation}
\begin{aligned}
&(p_1-1) [ (u_1^k(r) )'] ^{p_1-1}(u_1^k) ''+(\frac{N-1}{r}+h_1(r) ) [ (u_1^k(
r) ) '] ^{p_1} \\
&\leq a_1(r) \sum_{i=1}^d g_i\Big(
\sum_{i=1}^d u_i^k,\dots ,\sum_{i=1}^d u_i^k\Big) (u_1^k(r) ) ',
\\
&\quad \dots  \\
&(p_d-1) [ (u_d^k(r) )'] ^{p_d-1}(u_d^k) ''+(
\frac{N-1}{r}+h_d(r) ) [ (u_d^k(r) ) '] ^{p_d} \\
&\leq a_d(r) \sum_{i=1}^d g_i\Big(
\sum_{i=1}^d u_i^k,\dots ,\sum_{i=1}^d u_i^k\Big) (u_d^k(r) ) ',
\end{aligned}  \label{88}
\end{equation}
which implies
\begin{equation}
\begin{aligned}
&(p_1-1) [ (u_1^k(r) )'] ^{p_1-1}(u_1^k) ''+(
\frac{N-1}{r}+h_1(r) ) [ (u_1^k(r) ) '] ^{p_1} \\
&\leq a_1(r) \sum_{i=1}^d g_i\Big(
\sum_{i=1}^d u_i^k,\dots ,\sum_{i=1}^d u_i^k\Big)
\Big(\sum_{i=1}^d u_i^k(r)\Big) ', \\
&\quad \dots  \\
&(p_d-1) [ (u_d^k(r) )'] ^{p_d-1}(u_d^k) ''
+(\frac{N-1}{r}+h_d(r) ) [ (u_d^k(r) ) '] ^{p_d} \\
&\leq a_d(r) \sum_{i=1}^d g_i\Big(
\sum_{i=1}^d u_i^k,\dots ,\sum_{i=1}^d u_i^k\Big) \Big(\sum_{i=1}^d u_i^k(r) \Big) '.
\end{aligned}  \label{sum}
\end{equation}
Now if we let
\begin{equation}
a_i^{R}=\max \{a_i(r) :0\leq r\leq R\},\quad i=1,\dots ,d ,  \label{not}
\end{equation}
we can prove that $u_i^k(R) $ and $(u_i^k(R) ) '$, both of them are
nonnegative and bounded above independent of $k$. Using \eqref{not} and
the fact that $(u_i^k) '\geq 0$ for $i=1,\dots ,d$, we observe that
\eqref{sum} yields
\begin{gather*}
(p_1-1) [ (u_1^k) ']^{p_1-1}(u_1^k) ''
\leq a_1^{R} \sum_{i=1}^d g_i\Big(\sum_{i=1}^d u_i^k,\dots ,\sum_{i=1}^d u_i^k
\Big) \Big(\sum_{i=1}^d u_i^k(r) \Big)' \\
\dots  \\
(p_d-1) [ (u_d^k) ']^{p_d-1}(u_d^k) ''
\leq a_d^{R}\sum_{i=1}^d g_i\Big(\sum_{i=1}^d
u_i^k,\dots ,\sum_{i=1}^d u_i^k\Big) \Big(\sum_{i=1}^d u_i^k(r) \Big)'
\end{gather*}
or, equivalently
\begin{equation}
\begin{gathered}
\frac{p_1-1}{p_1}\{ [ (u_1^k) ']^{p_1}\} '\leq a_1^{R}\sum_{i=1}^d
g_i\Big(\sum_{i=1}^d u_i^k,\dots ,\sum_{i=1}^d 
u_i^k\Big) \Big( \sum_{i=1}^d u_i^k(r) \Big) ', \\
\dots  \\
\frac{p_d-1}{p_d}\{ [ (u_d^k) ']^{p_d}\} '\leq a_d^{R}\sum_{i=1}^d
g_i\Big(\sum_{i=1}^d u_i^k,\dots ,\sum_{i=1}^d u_i^k\Big) \Big( \sum_{i=1}^d 
u_i^k(r) \Big) '.
\end{gathered}  \label{l}
\end{equation}
An integration of \eqref{l} in $(0,r) $ gives
\begin{gather}
\begin{aligned}
\big[ (u_1^k(r) ) '\big] ^{p_1}
&\leq \frac{p_1}{p_1-1}a_1^{R}\int_{b}^{\sum_{i=1}^d u_i^k(r) }
\sum_{i=1}^dg_i(s,\dots ,s) ds   \\
&\leq \frac{p_1}{p_1-1}a_1^{R}\int_0^{\sum_{i=1}^d u_i^k(r) }
\sum_{i=1}^d g_i(s,\dots ,s) ds,  
\end{aligned} \label{e1} \\
\dots   \notag \\
\begin{aligned}
\big[ (u_d^k(r) ) '\big] ^{p_d}
&\leq \frac{p_d}{p_d-1}a_d^{R}\int_{b}^{\sum_{i=1}^d u_i^k(r) }
\sum_{i=1}^d g_i(s,\dots ,s) ds  \\
&\leq \frac{p_d}{p_d-1}a_d^{R}\int_0^{\sum_{i=1}^d u_i^k(r) }\sum_{i=1}^d
g_i(s,\dots ,s) ds.
\end{aligned}  \label{e2}
\end{gather}
At this stage, it is clear that
\begin{gather}
(u_1^k(r) ) ' \leq \sqrt[p_1]{\frac{
p_1}{p_1-1}a_1^{R}}\Big(\int_0^{\sum_{i=1}^d u_i^k(r) }
\sum_{i=1}^d g_i(s,\dots ,s) ds+1\Big) ^{1/\min \{p_1,\dots ,p_d\}},  \label{1i}
\\
\dots   \notag \\
(u_d^k(r) ) ' \leq \sqrt[p_d]{\frac{
p_d}{p_d-1}a_d^{R}}\Big(\int_0^{\sum_{i=1}^d u_i^k(r) }\sum_{i=1}^d g_i(
s,\dots ,s) ds+1\Big) ^{1/\min \{p_1,\dots ,p_d\}},  \label{2i}
\end{gather}
Summing \eqref{1i}-\eqref{2i} and simplifying, we obtain
\begin{equation}
\begin{aligned}
&\Big(\sum_{i=1}^d u_i^k(r)\Big) '
\Big(\int_0^{\sum_{i=1}^d u_i^k(r) }
\sum_{i=1}^d g_i(s,\dots ,s) ds+1\Big) ^{-1/\min \{ p_1,\dots ,p_d\} }  \\
&\leq \sum_{j=1}^d \sqrt[p_j]{\frac{p_j}{p_j-1}
a_j^{R}}\quad\text{for }0\leq r\leq R.  \label{9}
\end{aligned}
\end{equation}
Integrating \eqref{9} between $0$ and $R$, we have
\begin{align*}
&\int_{b}^{\sum_{i=1}^d u_i^k(R) }
\Big[ \int_0^{t}\sum_{i=1}^d g_i(
s,\dots ,s) ds+1\Big] ^{-1/\min \{p_1,\dots ,p_d\}}dt \\
&= I\Big(\sum_{i=1}^d u_i^k(R)\Big) -I(b)
\leq R\sum_{j=1}^d \sqrt[p_j]{\frac{p_j}{p_j-1}a_j^{R}}.
\end{align*}
Since $I$ is a bijection with $I^{-1}$ increasing we obtain
\begin{equation}
\sum_{i=1}^d u_i^k(R) \leq
I^{-1}\Big(R\sum_{j=1}^d \sqrt[p_j]{\frac{p_j}{
p_j-1}a_j^{R}}+I(b) \Big) \quad\text{for all }r\geq 0,
\label{bound1}
\end{equation}
as in \cite{YH}. We are now in the position to observe that from the
Keller-Osserman condition \eqref{KO} we can conclude that
 $\sum_{i=1}^d u_i^k(R) $ is uniformly bounded
above independent of $k$ and using this fact in \eqref{9} shows that the
same is true of $\big(\sum_{i=1}^d u_i^k(R) \big) '$. Then,
since $u_i^k(r) \leq u_i^k(R) $ ($r\leq R$ and $u_i^k(r) $ is
non-decreasing sequence!) for $i=1,\dots ,d$ we obtain the conclusion that the
sequences $u_i^k(r) $ are uniformly bounded above
independent of $k$. Also, we clearly have $u_i^k(r) >0$ for
all $r\geq 0$ and so our sequence is equi-continuous on $[ 0,R] $
for arbitrary $R>0$.  A recapitulation of the above information says that
$u_i^k(r) $ ($i=1,\dots ,d$) is a monotonic, uniformly bounded,
equi-continuous sequence of functions on $[ 0,R] $ and then there
exists a function
\[
(u_1,\dots ,u_d) \in C([ 0,R] ) \times
\dots \times C([ 0,R] )
\]
such that $u_i^k(r) \to u_i(r) $ ($
i=1,\dots ,d$) uniformly. Therefore, by an argument of a Fixed Point Theorem,
it follows that $(u_1,\dots ,u_d) $ is a fixed point of \eqref{oop} in
$C([ 0,R] ) \times \dots \times C([0,R] ) $.

Next, we extend this result to show that $S$ has a fixed point in
 $C^{1}([ 0,\infty ) ) \times \dots \times C^{1}([ 0,\infty ) ) $. Let 
$\{ u_i^k(r)\} _{i=1,\dots ,d}^{k\geq 1}$ be a sequence of fixed points defined by 
\begin{equation}
\begin{gathered}
(u_1^k(r) ,\dots ,u_d^k(r) )
=S(u_1^k(r) ,\dots ,u_d^k(r) )\quad \text{on }[ 0,k] ,  \\
(u_1^k(r) ,\dots ,u_d^k(r) ) \in C([ 0,k] ) \times \dots \times C([ 0,k]) ,
\end{gathered}  \label{c11}
\end{equation}
for $k=1,2,3,\dots $. As earlier, we may show that both 
$u_1^k(r) ,\dots $ and $u_d^k(r) $ are bounded and
equi-continuous on $[ 0,1] $. Thus by applying the Arzela-Ascoli
Theorem to each sequence separately, we can derive that 
$\{ (u_1^k(r) ,\dots ,u_d^k(r) ) \}^{k\geq 1}$ contains a convergent 
subsequence,
$(u_1^{k_1^{1}}(r) ,\dots ,u_d^{k_d^{1}}(r) )$,
that converges uniformly on $[ 0,1]\times \dots \times [ 0,1]$.
Let
\[
(u_1^{k_1^{1}}(r) ,\dots ,u_d^{k_d^{1}}(
r) ) \to (u_1^{1},\dots ,u_d^{1}) \quad \text{uniformly on }
[ 0,1] \times \dots \times [ 0,1] 
\]
as $k_1^{1},\dots ,k_d^{1}\to \infty $. Likewise, the subsequences 
$u_1^{k_1^{1}}(r) ,\dots, u_d^{k_d^{1}}(r)$ are  bounded 
and equi-continous on $[ 0,2] $ so there
exists a subsequence
$(u_1^{k_1^{2}}(r) ,\dots ,u_d^{k_d^{2}}(r) )$  of 
$(u_1^{k_1^{1}}(r),\dots ,u_d^{k_d^{1}}(r) )$
such that
$(u_1^{k_1^{2}}(r) ,\dots ,u_d^{k_d^{2}}(
r) ) \to (u_1^{2},\dots ,u_d^{2})$
uniformly on \\
$[ 0,2] \times \dots \times [ 0,2]$
as $k_1^{2},\dots ,k_d^{2}\to \infty$.
Note that
\[
\{ (u_1^{k_1^{2}}(r) ,\dots ,u_d^{k_d^{2}}(
r) ) \} \subseteq \{ (u_1^{k_1^{1}}(
r) ,\dots ,u_d^{k_d^{1}}(r) ) \} \subseteq
\{ (u_1^k(r) ,\dots ,u_d^k(r)
) \} _{k\geq 1}^{\infty }
\]
so
\[
(u_1^{2},\dots ,u_d^{2}) =(u_1^{1},\dots ,u_d^{1})\quad
\text{on }[ 0,1] \times \dots \times [ 0,1] .
\]
Continuing this  reasoning, we obtain a sequence, denoted
$(u_1^k(r) ,\dots ,u_d^k(r) ) $, such that
\begin{gather*}
(u_1^k(r) ,\dots ,u_d^k(r) )  \in
C([ 0,k] ) \times \dots \times C([ 0,k] ),\quad  k=1,2,\dots \\
(u_1^k(r) ,\dots ,u_d^k(r) )
=(u_1^{1}(r) ,\dots ,u_d^{1}(r) ) \quad\text{for }r\in [ 0,1] \\
(u_1^k(r) ,\dots ,u_d^k(r) )
=(u_1^{2}(r) ,\dots ,u_d^{2}(r) ) \quad\text{for }r\in [ 0,2] \\
\dots  \\
(u_1^k(r) ,\dots ,u_d^k(r) )
=(u_1^{k-1}(r) ,\dots ,u_d^{k-1}(r) )\quad\text{for }r\in [ 0,k-1] ,
\end{gather*}
and these functions are radially symmetric. Therefore 
$(u_1^k(r) ,\dots ,u_d^k(r) )$ converges
pointwise to some $(u_1(r) ,\dots ,u_d(r)) $ which satisfies
\[
(u_1(r) ,\dots ,u_d(r) ) =(u_1^k(r) ,\dots ,u_d^k(r) ) \quad\text{ if }
0\leq r\leq k.
\]
Hence, $(u_1(r) ,\dots ,u_d(r) ) $ is radially symmetric. Further,
 since $(u_1^k(r) ,\dots ,u_d^k(r) ) $ is in the form \eqref{c11},
 we have that $(u_1^k(r) ,\dots ,u_d^k(r) ) $
is also equi-continuous.
 Pointwise convergence and equi-continuity
imply uniform convergence and thus the convergence is uniform on bounded
sets. Thus
\[
(u_1(r) ,\dots ,u_d(r) ) \in C^{1}([ 0,\infty ) ) \times \dots
 \times C^{1}([ 0,\infty ) )
\]
is a fixed point of \eqref{oop} and a solution to \eqref{11} with central
value $(\frac{b}{d},\dots ,\frac{b}{d}) $. Since $b\geq a>0$ was
chosen arbitrarily, it follows that \eqref{11} has infinitely many positive
entire solutions and so the first part of our theorem is proved.

\noindent\textbf{Proof of (i)}
 Assume that \eqref{5} holds. Finally, we show
that any entire positive radial solution\textit{\ }$(
u_1,\dots ,u_d) $\ of system \eqref{11} is bounded. We choose $R>0$
so that
\[
\frac{p_j}{p_j-1}r^{\frac{p_j(N-1) }{p_j-1}}e^{\frac{
p_j}{p_j-1}\int_0^{r}h_j(t) dt}a_j(r)
\]
are non-decreasing for $r\geq R$ and $j=1,\dots ,d$.
Multiply  each line of the system
\begin{gather*}
\begin{aligned}
&(p_1-1) [ (u_1(r) ) '] ^{p_1-1}(u_1) ''+(\frac{N-1}{r}
+h_1(r) ) [ (u_1(r) ) '] ^{p_1} \\
&\leq a_1(r) \sum_{i=1}^d g_i\Big(
\sum_{i=1}^d u_i,\dots ,\sum_{i=1}^d u_i\Big) 
\Big(\sum_{i=1}^d u_i(r) \Big) ',
\end{aligned} \\
 \dots  \\
\begin{aligned}
&(p_d-1) [ (u_d(r) ) '] ^{p_d-1}(u_d) ''+(\frac{N-1}{r}
+h_d(r) ) [ (u_d(r) )'] ^{p_d} \\
&\leq a_d(r) \sum_{i=1}^d g_i\Big(\sum_{i=1}^d u_i,\dots ,
\sum_{i=1}^d u_i\Big) \Big(\sum_{i=1}^d u_i(r) \Big) '.
\end{aligned}
\end{gather*}
 by
\[
\frac{p_i}{p_i-1}r^{\frac{p_i(N-1) }{p_i-1}}e^{\frac{p_i}{p_i-1}\int_0^{r}h_i(t)dt}
\quad i=1,\dots ,d,
\]
where $i$ represent the equation of the system that will be multiplied by.
Then summing we have
\begin{align*}
&\Big[ r^{\frac{p_1(N-1) }{p_1-1}}e^{\frac{p_1}{p_1-1}
\int_0^{r}h_1(t)dt}(u_1') ^{p_1}\Big]
' \\
&\leq r^{\frac{p_1(N-1) }{p_1-1}}
\frac{p_1}{p_1-1}e^{\frac{p_1}{p_1-1}\int_0^{r}h_1(t)dt}a_1(r)
\sum_{i=1}^d g_i\Big(\sum_{i=1}^d u_i,\dots ,\sum_{i=1}^d u_i\Big) 
\Big(\sum_{i=1}^d u_i\Big) ' \\
&\quad \dots  \\
&\Big[ r^{\frac{p_d(N-1) }{p_d-1}}e^{\frac{p_d}{p_d-1}
\int_0^{r}h_d(t)dt}(u_d') ^{p_d}\Big]' \\
&\leq r^{\frac{p_d(N-1) }{p_d-1}} 
\frac{p_d}{ p_d-1}e^{\frac{p_d}{p_d-1}\int_0^{r}h_d(t)dt}a_d(r)
\sum_{i=1}^d g_i\Big(\sum_{i=1}^d u_i,\dots ,\sum_{i=1}^d u_i\Big) 
\Big(\sum_{i=1}^d u_i\Big) '.
\end{align*}
Integrating this gives
\begin{gather} \label{s2}
\begin{aligned}
&\int_{R}^{r}\Big[ s^{\frac{p_1(N-1) }{p_1-1}}(e^{
\frac{1}{p_1-1}\int_0^s h_1(t)dt}u_1') ^{p_1}\Big] 'ds   \\
&\leq \int_{R}^{r}s^{\frac{p_1(N-1) }{p_1-1}}\frac{p_1}{
p_1-1}e^{\frac{p_1}{p_1-1}\int_0^s h_1(t)dt}a_1(s)
\sum_{i=1}^d g_i\Big(\sum_{i=1}^d u_i,\dots ,\sum_{i=1}^d u_i\Big)
\Big(\sum_{i=1}^d u_i\Big) 'ds, 
\end{aligned}  \\
\dots \notag   \\
\label{ss2}
\begin{aligned}
&\int_{R}^{r}\Big[ s^{\frac{p_d(N-1) }{p_d-1}}(e^{
\frac{1}{p_d-1}\int_0^s h_d(t)dt}u_d') ^{p_d}
\Big] 'ds   \\
&\leq \int_{R}^{r}s^{\frac{p_d(N-1) }{p_d-1}}\frac{p_d}{
p_d-1}e^{\frac{p_d}{p_d-1}\int_0^s h_d(t)dt}a_d(s)
\sum_{i=1}^d g_i\Big(\sum_{i=1}^d u_i,\dots ,\sum_{i=1}^d u_i\Big)
\Big(\sum_{i=1}^d u_i\Big) 'ds.
\end{aligned}
\end{gather}
With the use of \eqref{s2}-\eqref{ss2} we obtain
\begin{align*}
&r^{\frac{p_1(N-1) }{p_1-1}}\Big(e^{\frac{1}{p_1-1}
\int_0^{r}h_1(t)dt}u_1'(r)\Big) ^{p_1}-R^{
\frac{p_1(N-1) }{p_1-1}}\Big(e^{\frac{1}{p_1-1}
\int_0^{R}h_1(t)dt}(u_1'(R) ) \Big)^{p_1} \\
&\leq \int_{R}^{r}s^{\frac{p_1(N-1) }{p_1-1}}\frac{p_1}{
p_1-1}e^{\frac{p_1}{p_1-1}\int_0^s h_1(t)dt}a_1(s)
\sum_{i=1}^d g_i\Big(\sum_{i=1}^d u_i,\dots ,\sum_{i=1}^d u_i\Big) \Big(
\sum_{i=1}^d u_i\Big) 'ds, \\
&\quad \dots  \\
&r^{\frac{p_d(N-1) }{p_d-1}}\Big(e^{\frac{1}{p_d-1}
\int_0^{r}h_d(t)dt}u_d'(r) \Big) ^{p_d}-R^{
\frac{p_d(N-1) }{p_d-1}}\Big(e^{\frac{1}{p_d-1}
\int_0^{R}h_d(t)dt}u_d'(R) \Big) ^{p_d} \\
&\leq \int_{R}^{r}s^{\frac{p_d(N-1) }{p_d-1}}\frac{p_d}{
p_d-1}e^{\frac{p_d}{p_d-1}\int_0^s h_d(t)dt}a_d(s)
\sum_{i=1}^d g_i\Big(\sum_{i=1} u_i,\dots ,\sum_{i=1}^d u_i\Big) 
\Big(\sum_{i=1}^d u_i\Big) 'ds,
\end{align*}
for $r\geq R$.

Noting that, by the monotonicity of
\[
\frac{p_j}{p_j-1}s^{\frac{p_j(N-1) }{p_j-1}}e^{\frac{
p_j}{p_j-1}\int_0^s h_j(t)dt}a_j(s) 
\]
for $j=1,\dots ,d$  and $r\geq s\geq R$, we obtain
\begin{align*}
&r^{^{\frac{p_1(N-1) }{p_1-1}}}\Big(e^{\frac{1}{p_1-1}
\int_0^{r}h_1(t)dt}u_1'\Big) ^{p_1} \\
&\leq C+\frac{p_1}{p_1-1}r^{^{\frac{p_1(N-1) }{p_1-1}
}}e^{\frac{p_1}{p_1-1}\int_0^{r}h_1(t)dt}a_1(r)
G\Big(\sum_{i=1}^d u_i\Big) , \\
&\quad \dots , \\
& r^{^{\frac{p_d(N-1) }{p_d-1}}}\Big(e^{\frac{1}{p_d-1}
\int_0^{r}h_d(t)dt}u_d'\Big) ^{p_d} \\
&\leq C+\frac{p_d}{p_d-1}r^{^{\frac{p_d(N-1) }{p_d-1}
}}e^{\frac{p_d}{p_d-1}\int_0^{r}h_d(t)dt}a_d(r)
G\Big(\sum_{i=1}^d u_i\Big) ,
\end{align*}
which yields
\begin{gather}
\begin{aligned}
& r^{^{\frac{N-1}{p_1-1}}}e^{\frac{1}{p_1-1}
\int_0^{r}h_1(t)dt}u_1'  \\
&\leq \Big[ C+\frac{p_1}{p_1-1}r^{^{\frac{p_1(N-1) }{
p_1-1}}}e^{\frac{p_1}{p_1-1}\int_0^{r}h_d(t)dt}a_1(
r) G\Big(\sum_{i=1}^d u_i(r)\Big) \Big] ^{1/p_1} 
\end{aligned}   \label{100} \\
\quad \dots    \notag \\
\begin{aligned}
&r^{^{\frac{N-1}{p_d-1}}}e^{\frac{1}{p_d-1}
\int_0^{r}h_1(t)dt}u_d'   \\
&\leq \big[ C+\frac{p_d}{p_d-1}r^{^{\frac{p_d(N-1) }{
p_d-1}}}e^{\frac{p_d}{p_d-1}\int_0^{r}h_d(t)dt}a_d(
r) G\Big(\sum_{i=1}^d u_i(r)
\Big) \Big] ^{1/p_d}
\end{aligned} \label{101}
\end{gather}
where
\begin{align*}
C=\max \Big\{& R^{\frac{p_1(N-1) }{p_1-1}}
\big[ e^{\frac{1}{p_1-1}\int_0^{R}h_1(t)dt}(u_1(R) ) '
\big] ^{p_1},\dots ,\\
&R^{\frac{p_d(N-1) }{p_d-1}}
\big[ e^{ \frac{1}{p_d-1}\int_0^{R}h_d(t)dt}(u_d(R) )
'\big] ^{p_d}\Big\} .
\end{align*}
We need to recall an important inequality which is the key ingredient of
our next proof. Since $(1/p_i) <1$ we know that
\[
(b_1+b_2) ^{1/p_i}\leq b_1^{1/p_i}+b_2^{1/p_i}
\]
for any non-negative constants $b_i$ and $i=1,2$. Therefore, by applying
these inequalities in \eqref{100} and \eqref{101} we obtain
\begin{align*}
u_1' 
&\leq e^{\frac{1}{p_1-1}\int_0^{r}h_1(t)dt}u_1' \\
&\leq \sqrt[p_1]{C}r^{\frac{1-N}{p_1-1}}+r^{\frac{1-N}{p_1-1}}
\Big[\frac{p_1}{p_1-1}r^{^{\frac{p_1(N-1) }{p_1-1}}}e^{
\frac{p_1}{p_1-1}\int_0^{r}h_1(t)dt}a_1(r) \Big]^{1/p_1}
\big[G(\sum_{i=1}^d u_i) \big] ^{1/p_1}
\\
&\quad \dots  \\
u_d' &\leq e^{\frac{1}{p_d-1}\int_0^{r}h_d(t)dt}u_d' \\
&\leq \sqrt[p_d]{C}r^{\frac{1-N}{p_d-1}}+r^{\frac{1-N}{p_d-1}}
\Big[ \frac{p_d}{p_d-1}r^{^{\frac{p_d(N-1) }{p_d-1}}}e^{
\frac{p_d}{p_d-1}\int_0^{r}h_d(t)dt}a_d(r) \Big]^{1/p_d}
\big[G(\sum_{i=1}^d u_i) \big] ^{1/p_d}.
\end{align*}
Summing the above inequalities and integrating, we obtain
\begin{equation} \label{sis3}
\begin{aligned}
&\frac{d}{dr}\int_{\sum_{i=1}^d u_i(R)}^{\sum_{i=1}^d u_i(r) }
[ G(t) ] ^{-1/\min \{p_1,\dots ,p_d\}}dt   \\
&\leq \sum_{j=1}^d  \sqrt[p_j]{C}r^{\frac{1-N}{
p_j-1}}\Big[ G(\sum_{i=1}^d u_i(r) ) \Big] ^{-1/\min \{p_1,\dots ,p_d\}}\\
&\quad +\sum_{i=1}^d \Big(\frac{p_i}{p_i-1}e^{\frac{p_i}{p_i-1}
\int_0^{r}h_i(t)dt}a_i(r) \Big) ^{1/p_i}.
\end{aligned}
\end{equation}
Inequality \eqref{sis3} combined with
\begin{align*}
\Big(e^{\frac{p_i}{p_i-1}\int_0^{r}h_i(t)dt}a_i(s)
\Big) ^{1/p_i}
 &= \Big(s^{p_i(1+\varepsilon ) /2}e^{
\frac{p_i}{p_i-1}\int_0^{r}h_i(t)dt}a_i(s)
s^{-p_i(1+\varepsilon ) /2}\Big) ^{1/p_i} \\
&\leq (\frac{1}{2}) ^{1/p_i}\Big[ s^{1+\varepsilon }(
e^{\frac{p_i}{p_i-1}\int_0^{r}h_i(t)dt}a_i(r) )
^{2/p_i}+s^{-1-\varepsilon }\Big] ,
\end{align*}
for each $\varepsilon >0$, yields
\begin{equation}
\begin{aligned}
&\int_{\sum_{i=1}^d u_i(R) }^{\sum_{i=1}^d u_i(r) }
[ G(t) ] ^{-1/\min \{p_1,\dots ,p_d\}}dt   \\
&\leq \int_{R}^{r} \sum_{j=1}^d  \sqrt[p_j]{C}t^{ \frac{1-N}{p_j-1}}
\Big[ G\Big(\sum_{i=1}^d
u_i(t) \Big) \Big] ^{-1/\min \{p_1,\dots ,p_d\}}dt
\\
&\quad +\sum_{i=1}^d (\frac{1}{2}) ^{1/p_i}
\sqrt[p_i]{\frac{p_i}{p_i-1}}
\Big[ \int_{R}^{r}t^{1+\varepsilon
}\Big(e^{\frac{p_i}{p_i-1}\int_0^{t}h_i(s)ds}a_i(t)
\Big) ^{2/p_i}dt+\int_{R}^{r}t^{-1-\varepsilon }dt\Big]
 \\
&\leq \sum_{j=1}^d \sqrt[p_j]{C}\Big[ G\Big(\sum_{i=1}^d u_i(R) \Big) \Big]
^{-1/\min \{p_1,\dots ,p_d\}}\frac{p_j-1}{p_j-N}R^{\frac{p_j-N}{
p_j-1}}   \\
&\quad +\sum_{i=1}^d (\frac{1}{2}) ^{1/p_i}
\sqrt[p_i]{\frac{p_i}{p_i-1}}\Big[ \int_{R}^{r}t^{1+\varepsilon
}(e^{\frac{p_i}{p_i-1}\int_0^{t}h_i(s)ds}a_i(t)
) ^{2/p_i}dt+\frac{1}{\varepsilon R^{\varepsilon }}\Big]
. \end{aligned} \label{111}
\end{equation}
Since the right side of this inequality is bounded (note that
 $u_i(t) \geq b/d$), so is the left side and hence, in light of
 Keller-Osserman condition, the sequence $\sum_{i=1}^d
u_i(r) $ is bounded and finally $u_i(r) $ ($i=1,\dots ,d$) is a bounded function.
 Thus, for every $x\in \mathbb{R}^{N}$
the function $(u_1(| x| ),\dots ,u_d(| x| ) ) $ is a positive
bounded solution of \eqref{11}.

\noindent \textbf{Proof of (ii)}
Suppose that $a_i$ ($i=1,\dots ,d$) satisfies \ref{12}). Now, 
let $(u_1,\dots ,u_d) $ be any positive entire
radial solution of \eqref{11} determined in the first step of the proof.
Clearly
\[
(u_1(r) ,\dots ,u_d(r) ) \geq (\frac{b}{d},\dots ,\frac{b}{d})
\]
and since $g_j$ are non-decreasing on $[ 0,\infty ) ^{d}$ in
all variables it follows
\begin{equation}
g_j(u_1(r) ,\dots ,u_d(r) ) \geq
g_j(\frac{b}{d},\dots ,\frac{b}{d}) .  \label{in}
\end{equation}
On the other hand, substituting \eqref{in} in the system \eqref{66} we obtain
\begin{gather*}
(p_1-1) (u_1') ^{p_1-2}u_1''+\frac{N-1}{r}(u_1') ^{p_1-1}+h_1(
r) | u_1'| ^{p_1-1}\geq a_1(
r) g_1(\frac{b}{d},\dots ,\frac{b}{d}) , \\
\dots  \\
(p_d-1) (u_d') ^{p_d-2}u_d''+\frac{N-1}{r}(u_d') ^{p_d-1}+h_d(
r) | u_d'| ^{p_d-1}\geq a_d(
r) g_d(\frac{b}{d},\dots ,\frac{b}{d}) ,
\end{gather*}
or, equivalently
\begin{gather*}
\big[ r^{N-1}e^{\int_0^{r}h_1(t) dt}(u_1') ^{p_1-1}\big] '
\geq r^{N-1}e^{\int_0^{r}h_1(t)dt}a_1(r) g_1(\frac{b}{d},\dots ,
\frac{b}{d}) , \\
\dots  \\
\big[ r^{N-1}e^{\int_0^{r}h_d(t) dt}(u_d') ^{p_d-1}\big] '
\geq r^{N-1}e^{ \int_0^{r}h_d(t)dt}a_d(r) g_d(\frac{b}{d},\dots ,
\frac{b}{d}) .
\end{gather*}
However, this system of inequalities may be written as
\begin{gather*}
u_1(r) \geq \frac{b}{d}+g_1^{\frac{1}{p_1-1}}(\frac{b
}{d},\dots ,\frac{b}{d}) \int_0^{r}
\Big(\frac{e^{-\int_0^{t}h_1
(s) ds}}{t^{N-1}}\int_0^{t}s^{N-1}e^{\int_0^s h_1(
s) ds}a_1(s) ds\Big) ^{\frac{1}{p_1-1}}dt, \\
\dots  \\
u_d(r) \geq \frac{b}{d}+g_d^{\frac{1}{p_d-1}}(\frac{b
}{d},\dots ,\frac{b}{d}) \int_0^{r}\Big(\frac{e^{-\int_0^{t}h_d
(s) ds}}{t^{N-1}}\int_0^{t}s^{N-1}e^{\int_0^s h_d(
s) ds}a_d(s) ds\Big) ^{\frac{1}{p_d-1}}dt.
\end{gather*}
It is evident that $r\to \infty $ implies 
$(u_1(r) ,\dots ,u_d(r) ) \to (\infty,\dots ,\infty ) $. The proof is complete.

From the above proof and the work \cite{CD2} we can easy obtain the
following remark.

\begin{remark} \rm
 Under the same assumptions as in Theorem
\ref{thm1} on $a_j$, $h_j$ and $g_j$ except for (i)-(ii).
 If \eqref{non} has a nonnegative entire large solution, then 
$a_j$ ($j=1,\dots ,d$) satisfy
\begin{equation}
\sum_{j=1}^d (\frac{1}{2}) ^{1/p_j}
\sqrt[p_j]{\frac{p_j}{p_j-1}}\int_0^{\infty }t^{1+\varepsilon
}\Big(e^{\frac{p_j}{p_j-1}\int_0^{t}h_j(s)ds}a_j(t)
\Big) ^{2/p_j}dt=\infty ,  \label{13}
\end{equation}
for every $\varepsilon >0$.
\end{remark}

\subsection*{Acknowledgements}
We would like to thank the anonymous referee for his or her very important 
comments that improved this article.

\begin{thebibliography}{99}

\bibitem{BEC4} C. Bandle, M. Marcus;
\emph{\ Large solutions of semilinear
elliptic equations: existence, uniqueness and asymptotic behavior}, Journal
d'Analyse Math\'{e}matique 58, Pages 9-24, 1992.

\bibitem{C} P. Clement, R. Manasevich, E. Mitidieri;
\emph{Positive solutions for a quasilinear system via blow up}, 
Communication Partial and Differential Equations, 18, 2071--2106, 1993.

\bibitem{CD} D.-P. Covei;
\emph{Large and Entire Large Solution for a
Quasilinear Problem}, Nonlinear Analysis: Theory, Methods \& Applications,
Volume 70, Issue 4, Pages 1738-1745, 15 February 2009.

\bibitem{CD2} D.-P. Covei;
\emph{Radial and nonradial solutions for a
semilinear elliptic system of Schr\"{o}dinger type},  Funkcialaj
Ekvacioj-Serio Internacia, Volume 54, (2011), Pages 439--449.

\bibitem{D} J. I. Diaz;
\emph{Nonlinear Partial Differential Equations and
Free Boundaries}, Pitman Research Notes in Mathematics, 106, 1985.

\bibitem{GS} J. V. A. Goncalves, Z. Jiazheng;
\emph{Remarks on Existence of Large Solutions for p-Laplacian Equations
 with Strongly Nonlinear Terms Satisfying the Keller-Osserman Condition},
 Advanced Nonlinear Studies 10 (2010), 757-769.

\bibitem{GG} H. Grosse, A. Martin;
\emph{Particle Physics and the Schrodinger Equation}, 
Cambridge Monographs on Particle Physic's, Nuclear Physics and Cosmology, 1997.

\bibitem{YH} Y. Haitao;
\emph{Existence and nonexistence of blow-up boundary
solutions for sublinear elliptic equations}, Journal of Mathematical
Analysis and Applications, 314 (2006) 85--96.

\bibitem{H} A. Hamydy, M. Massar, N. Tsouli;
\emph{Blow-up solutions to a $(p, q)$-Laplacian system with gradient term}, 
Applied Mathematics Letters,
Volume 25, Issue 4, April 2012, Pages 745-751.

\bibitem{K} J. B. Keller;
\emph{On solution of }$\Delta u=f(u)$,
Communications on Pure and Applied Mathematics, 10 (1957), 503-510.

\bibitem{K1} J. B. Keller;
\emph{Electrohydrodynamics I. The Equilibrium of
a Charged Gas in a Container}, Journal of Rational Mechanics and Analysis,
Volume 5, Nomber 4, 1956.

\bibitem{L} Z. A. Luthey;
\emph{Piecewise Analytical Solutions Method for
the Radial Schrodinger Equation}, Ph. D. Thesis in Applied Mathematics,
Harvard University, Cambridge, MA, 1974.

\bibitem{LS} A. V. Lair, A. W. Shaker;
\emph{Existence of entire large positive solutions of semilinear elliptic systems}, 
Journal of Differential Equations, 164, 380--394, 2000.

\bibitem{LS2} A. V. Lair, A. Mohammed;
\emph{Entire large solutions of semilinear elliptic equations of mixed type}, 
Communications on Pure and Applied Analysis, Volume 8, Number 5, September 2009.

\bibitem{LI} J. L. Lions;
\emph{Quelques Methodes de Resolution des
Problimes aux Limites Non-liniaires}, (Dunod, Paris, 1969).

\bibitem{JM1C4} J. Matero;
\emph{Quasilinear elliptic equations with
boundary blow-up}, Journal D'Analyse Math\'{e}\-matique, Volume 69, Pages
229-247, 1996.

\bibitem{JM1} J. Matero;
\emph{Nonlinear elliptic problems with boundary
blow-up}, Uppsala Dissertations in Mathematics, 1997.

\bibitem{O} R. Osserman;
 \emph{On the inequality }$\Delta u\geq f(u)$,
Pacific Journal of Mathematics, 7, Pages 1641-1647, 1957.

\bibitem{P} Y. Peng, Y. Song;
 \emph{Existence of entire large positive
solutions of a semilinear elliptic system}, Applied Mathematics and
Computation, 155, 687-698, 2004.

\bibitem{PW} J. D. Peterson, A. W. Wood;
 \emph{Large solutions to non-monotone semilinear elliptic systems}, 
Journal of Mathematical Analysis
and Applications, Volume 384, Issue 2, 15 December 2011, Pages 284-292.

\bibitem{W} X. Wang, A. W. Wood;
 \emph{Existence and nonexistence of
entire positive solutions of semilinear elliptic systems}, Journal of
Mathematical Analysis and Applications, 267, 361--368, 2002.

\bibitem{LZZ} X. Zhang, L. Liu;
\emph{The existence and nonexistence of entire positive solutions of 
semilinear elliptic systems with gradient term}, 
Journal of Mathematical Analysis and Applications, Volume 371, Issue 1, 1
November 2010, Pages 300-308.

\end{thebibliography}


\end{document}