\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 $$\begin{gathered} U''+\frac{N-1}{r}U'=A(r) U(r) \\ U(r) \to U_{\infty }\quad\text{as }r\to \infty \end{gathered}\label{S}$$ 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 $$\begin{gathered} \Delta _{p_1}u_1+h_1(| x| ) | \nabla u_1| ^{p_1-1}=a_1(| x| ) g_1(u_1,\dots ,u_d) \quad\text{for }x\in \mathbb{R}^{N}, \\ \dots \\ \Delta _{p_d}u_d+h_d(| x| ) | \nabla u_d| ^{p_d-1}=a_d(| x|) g_d(u_1,\dots ,u_d) \quad\text{for }x\in \mathbb{R}^{N}, \end{gathered} \label{11}$$ where $d\geq 1$, $12$) if the fluid is Newtonian (respectively, pseudoplastic, dilatant)), in some reaction-diffusion problems, in nonlinear elasticity ($p_i>2$), glaciology ($10$ $\lim_{t\to \infty }\frac{g_1(Mg_2(t) ^{\frac{1}{p_2-1}}) }{t^{p_1-1}}=0.$ \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 $$\int_{a}^{\infty }\frac{ds}{g_1(s,s) +g_2(s,s) } =\infty \quad\text{for }r\geq a>0. \label{LZZ}$$ 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 $$\int_{u_0}^{\infty }\Big(\int_0^{t}g(s) ds\Big)^{-1/2}dt =\infty \quad\text{for }u_0>0, \label{KO1}$$ (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 $$\quad I(\infty ) :=\lim_{r\to \infty }I(r)=\infty \label{KO}$$ 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 $$\int_0^{\infty }t^{1+\varepsilon }(e^{\frac{p_j}{p_j-1} \int_0^{t}h_j(t)dt}a_j(t) ) ^{2/p_j}dt<\infty \quad \text{for all }j=1,\dots ,d, \label{5}$$ \item[(ii)] The solutions are large if $$\int_0^{\infty }(\frac{e^{-\int_0^{t}h_j(s) ds}}{ t^{N-1}}\int_0^{t}s^{N-1}e^{\int_0^s h_j(t) dt}a_j(s) ds) ^{1/(p_j-1) }dt=\infty \label{12}$$ 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 $$\begin{gathered} (p_1-1) (u_1') ^{p_1-2}u_1''+\frac{N-1}{r}(u_1') ^{p_1-1}+h_1( r) | u_1'| ^{p_1-1}=a_1( r) g_1(u_1,\dots ,u_d) , \\ \dots \\ (p_d-1) (u_d') ^{p_d-2}u_d''+\frac{N-1}{r}(u_d') ^{p_d-1}+h_d( r) | u_d'| ^{p_d-1}=a_d( r) g_d(u_1,\dots ,u_d) , \\ u_i'(0) =0\quad\text{for }i=1,\dots ,d \end{gathered} \label{66}$$ 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 $$\begin{gathered} u_1(r) =u_1(0)+\int_0^{r}\Big(\frac{e^{- \int_0^{t}h_1(s) ds}}{t^{N-1}}\int_0^{t}s^{N-1}e^{ \int_0^s h_1(s) dt}a_1(s) g_1( u_1,\dots ,u_d) ds\Big) ^{\frac{1}{p_1-1}}dt, \\ \dots \\ u_d(r) =u_d(0)+\int_0^{r}\Big(\frac{e^{- \int_0^{t}h_d(s) ds}}{t^{N-1}}\int_0^{t}s^{N-1}e^{ \int_0^s h_d(s) dt}a_d(s) g_d( u_1,\dots ,u_d) ds\Big) ^{\frac{1}{p_d-1}}dt. \end{gathered} \label{non}$$ 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 $$\begin{split} & S(u_1(r) ,\dots ,u_d(r) ) \\ &= \begin{pmatrix} u_1(0)+\int_0^{r}(\frac{e^{-\int_0^{t}h_1(s) ds}}{ t^{N-1}}\int_0^{t}s^{N-1}e^{\int_0^s h_1(s) dt}a_1(s) g_1(u_1,\dots ,u_d) ds) ^{\frac{ 1}{p_1-1}}dt \\ \dots \\ u_d(0)+\int_0^{r}(\frac{e^{-\int_0^{t}h_d(s) ds}}{ t^{N-1}}\int_0^{t}s^{N-1}e^{\int_0^s h_d(s) dt}a_d(s) g_d(u_1,\dots ,u_d) ds) ^{\frac{ 1}{p_d-1}}dt \end{pmatrix} \end{split}\label{op}$$ 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 $$\begin{split} & (u_1^k,\dots ,u_d^k) =S( u_1^{k-1}(r),\dots ,u_d^{k-1}(r) ) \\ &= \begin{pmatrix} \frac{b}{d}+\int_0^{r}(\frac{e^{-\int_0^{t}h_1(s) ds} }{t^{N-1}}\int_0^{t}s^{N-1}e^{\int_0^s h_1(s) dt}a_1(s) g_1(u_1^{k-1},\dots ,u_d^{k-1}) ds) ^{\frac{1}{p_1-1}}dt \\ \dots \\ \frac{b}{d}+\int_0^{r}(\frac{e^{-\int_0^{t}h_d(s) ds} }{t^{N-1}}\int_0^{t}s^{N-1}e^{\int_0^s h_d(s) dt}a_d(s) g_d(u_1^{k-1},\dots ,u_d^{k-1}) ds) ^{\frac{1}{p_d-1}}dt \end{pmatrix} ^{T}. \end{split} \label{oop}$$ 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{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} Using the monotonicity of $\{ u_i^k\} _{i=1,\dots ,d}^{k\geq 1}$ we have \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} moreover, \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} which implies \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} Now if we let $$a_i^{R}=\max \{a_i(r) :0\leq r\leq R\},\quad i=1,\dots ,d , \label{not}$$ 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 $$\begin{gathered} \frac{p_1-1}{p_1}\{ [ (u_1^k) ']^{p_1}\} '\leq a_1^{R}\sum_{i=1}^d g_i\Big(\sum_{i=1}^d u_i^k,\dots ,\sum_{i=1}^d u_i^k\Big) \Big( \sum_{i=1}^d u_i^k(r) \Big) ', \\ \dots \\ \frac{p_d-1}{p_d}\{ [ (u_d^k) ']^{p_d}\} '\leq a_d^{R}\sum_{i=1}^d g_i\Big(\sum_{i=1}^d u_i^k,\dots ,\sum_{i=1}^d u_i^k\Big) \Big( \sum_{i=1}^d u_i^k(r) \Big) '. \end{gathered} \label{l}$$ 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{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} 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 $$\sum_{i=1}^d u_i^k(R) \leq I^{-1}\Big(R\sum_{j=1}^d \sqrt[p_j]{\frac{p_j}{ p_j-1}a_j^{R}}+I(b) \Big) \quad\text{for all }r\geq 0, \label{bound1}$$ 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 $$\begin{gathered} (u_1^k(r) ,\dots ,u_d^k(r) ) =S(u_1^k(r) ,\dots ,u_d^k(r) )\quad \text{on }[ 0,k] , \\ (u_1^k(r) ,\dots ,u_d^k(r) ) \in C([ 0,k] ) \times \dots \times C([ 0,k]) , \end{gathered} \label{c11}$$ 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 \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} 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{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} 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 $$g_j(u_1(r) ,\dots ,u_d(r) ) \geq g_j(\frac{b}{d},\dots ,\frac{b}{d}) . \label{in}$$ 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 $$\sum_{j=1}^d (\frac{1}{2}) ^{1/p_j} \sqrt[p_j]{\frac{p_j}{p_j-1}}\int_0^{\infty }t^{1+\varepsilon }\Big(e^{\frac{p_j}{p_j-1}\int_0^{t}h_j(s)ds}a_j(t) \Big) ^{2/p_j}dt=\infty , \label{13}$$ 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. 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