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\begin{document}
\title{Large and bounded solutions for a  class of nonlinear Schr\"odinger stationary
systems\thanks{Corresponding author: Vicen\c tiu R\u adulescu, Department of Mathematics,
University of Craiova,  200585 Craiova, Romania. E-mail: {\tt vicentiu.radulescu@imar.ro}  \newline E-mail addresses:
{\tt ghanmisl@yahoo.fr}\quad {\tt habib.maagli@fst.rnu.tn}\quad {\tt noureddine.zeddini@ipein.rnu.tn}}}

\author{Abdejabbar GHANMI$\,^a$, Habib M\^AAGLI$\,^a$, Vicen\c tiu R\u ADULESCU$\,^{b,c,d}$,\\ and Noureddine ZEDDINI$\,^a$\\
$ $\\
\small $^a\,$D\'epartement de Math\'ematiques, Facult\'e des Sciences, Campus Universitaire,\\ \small 2092 Tunis, Tunisia\\
\small $^b\,$Simion Stoilow Mathematics Institute, Romanian
Academy, 010702 Bucharest, Romania\\
\small $^c\,$Department of Mathematics,
University of Craiova,  200585 Craiova, Romania\\
 \small $^d\,$Institute of Mathematics,
Physics and Mechanics,  Jadranska  19,  P. O. Box 2964,\\ \small 1001 Ljubljana, Slovenia}

\maketitle
\renewcommand{\theequation}{\thesection.\arabic{equation}}
\catcode`@=11 \@addtoreset{equation}{section} \catcode`@=12
\newtheorem{theo}{Theorem}[section]
\newtheorem{prop}[theo]{Proposition}
\newtheorem{lem}[theo]{Lemma}
\newtheorem{coro}[theo]{Corollary}
\newtheorem{rem}[theo]{Remark}
\newtheorem{rems}[theo]{Remarks}
\newtheorem{Defi}[theo]{Definition}
\newtheorem{exa}{Example}[section]


{\bf Abstract.} In the present paper we are concerned with entire radially symmetric solutions of
nonlinear Schr\"odinger elliptic systems in anisotropic media. In terms of the growth of the variable
potential functions, we establish conditions such that the solutions are either
bounded or blow up at infinity.


{\bf 2000 Mathematics Subject Classification}: 35J55, 35J60, 35J65.

{\bf Key words}: differential operator,
elliptic system, large solution,  bounded solution, Green
potential.

\section{Introduction and auxiliary results}
Existence and nonexistence of solutions for the nonlinear Schr\"odinger-type
system
$$ (\mathcal{S}) \left\{
\begin{array}{ll}
\Delta u=F(x,u,v), \qquad x \in {\mathbb{R}}^n\\
\Delta v=G(x,u,v),\qquad x \in {\mathbb{R}}^n \,\\
\end{array}%
\right.$$ have been intensively studied in the last few years.
The interest in systems of nonlinear Schr\"odinger equations is motivated by applications
to nonlinear optics. More precisely, coupled nonlinear Schr\"odinger 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{ref2,ref22}.
We also refer to
\cite{c1,c3,g1,gheradbook,m1,m2,l1,p1,radbook,velin,w1} and to the references therein for some recent results on the qualitative analysis
of the solutions to systems of this type. Most of
these results concern either the existence and the nonexistence of
{\it bounded} positive solutions, or the existence and the nonexistence of
{\it large} solutions. In the literature, a {\it large} solution means a couple
$(u,v)$ of positive
smooth functions satisfying ($\mathcal{S}$) and such that both $u(x)$ and $v(x)$ tend to infinity as $|x|\ri\infty$.
We would like to quote some references in which systems of boundary blow-up solutions
related to $(\mathcal{S})$ were analyzed. Lotka-Volterra type systems were considered in \cite{ref33,ref37}
(competitive type), \cite{ref9} (predator-prey type) and \cite{ref27} (cooperative type), while in \cite{ref24} the objective
was a competitive system not of Lotka-Volterra type.

Throughout this paper we assume that  $F(x,u,v)=p(x)g(v)$,
$G(x,u,v)=q(x)f(u)$, where $p$ and $q$ are smooth potentials, while the nonlinearities $f$ and $g$ are
nonnegative Lipschitz continuous functions on each interval $[\varepsilon
,\infty)$ (with $ \varepsilon >0$). In particular, this framework includes the
sublinear case. In all the results we  establish in
this paper we study only positive solutions, especially because of the
physical meaning of the corresponding unknowns.
 One of our main purposes in this paper is to establish necessary and sufficient conditions for
the existence of large solutions.  The existence of bounded positive solutions is also
studied in this paper, provided that $f$ and $g$ are nondecreasing and the Green potential
of $p$ and $q$ are continuous and bounded in ${\mathbb{R}}^n$,
$n\geq 3$. We recall that the Green potential of a nonnegative
measurable function $\phi$ is defined on ${\mathbb{R}}^n$ by
$$V\phi(x)=c_n\displaystyle\int_{{\mathbb{R}}^n}\displaystyle\frac{\phi(y)}{|x-y|^{n-2}}\,dy\,,\qquad
\mbox{where}\ \  c_n=\displaystyle\frac{\Gamma
(\frac{n}{2}-1)}{4{\pi}^{\frac{n}{2}}}\,,\ \ n\geq 3\,.$$ Moreover, $V
\phi$ is a lower semicontinuous function.

\smallskip
 In order to discuss the
existence of positive radial solutions to this class of nonlinear systems, we are concerned with the following system of nonlinear
differential equations
$$(\mathcal{P})\left\{%
\begin{array}{ll}
\di\frac{1}{A}\,(Au')'-p(t)g(v)=0 ,\,t\in (0,\infty),\\
\di\frac{1}{B}\,(Bv')'-q(t)f(u)=0 ,\,t\in (0,\infty),\\
u(0)=a>0,\,v(0)=b>0,\\
\displaystyle\lim_{t\rightarrow0}A(t)u'(t)=0,
\displaystyle\lim_{t\rightarrow0}B(t)v'(t)=0,
\end{array}%
\right. $$ where the continuous functions $A,B:[0,\infty)\rightarrow
[0,\infty)$ are differentiable and positive on $(0,\infty)$
and satisfy the following growth hypotheses:
$$\int_{0}^{1}\frac{1}{A(t)}\left(\int_{0}^{t}A(s)ds\right)dt<\infty$$and
$$\int_{0}^{1}\frac{1}{B(t)}\left(\int_{0}^{t}B(s)ds\right)dt<\infty\,.$$
In particular, these assumptions are fulfilled  if $A$ and $B$ are
nondecreasing.

Define the operators $K$ and $S$ on
the set of nonnegative measurable functions on $[0,\infty)$ by
$$K\varphi(t)=\int_{0}^{t}\frac{1}{A(s)}\left(\int_{0}^{s}A(r)p(r)\varphi(r)dr\right)ds$$
and
$$S\varphi(t)=\int_{0}^{t}\frac{1}{B(s)}\left(\int_{0}^{s}B(r)q(r)\varphi(r)dr\right)ds.$$
By induction, it follows that for all $t\geq 0$ and $m\in
{\mathbb{N}}$,
\begin{eqnarray}\label{eq1.1}
K^{m}{1}(t)\leq \displaystyle\frac{[K{1}(t)]^{m}}{m!} \quad \mbox{
and} \qquad \;S^{m}{1}(t)\leq \displaystyle\frac{[S{1}(t)]^{m}}{m!},
\end{eqnarray}
where    $K^{j}:=K^{j-1}\circ K$ and
$S^{j}:=S^{j-1}\circ S$, for any integer $j\geq 2$. Indeed, put
$$h(t)=K1(t)=\displaystyle\int_0^t\left(\frac{1}{A(s)}\int_0^sA(r)p(r)\,dr\right)ds$$
and assume that (\ref{eq1.1}) holds for some integer $m$. Then
\begin{eqnarray*}
K^{m+1}1(t)=K(K^m1)(t)\leq \frac{1}{m!}K([h]^m)(t)&=&\frac{1}{m!}
\int_{0}^{t}\frac{1}{A(s)}\left(\int_{0}^{s}A(r)p(r)[h(r)]^m\,dr\right)ds
\cr \cr  &\leq& \frac{1}{m!}
\int_{0}^{t}[h(s)]^m\left(\frac{1}{A(s)}\int_{0}^{s}A(r)p(r)\,dr\right)ds
\cr \cr &=& \frac{1}{m!} \int_{0}^{t}[h(s)]^m \, h'(s)\,ds =
\frac{1}{(m+1)!} [h(t)]^{m+1}.
\end{eqnarray*}
Now, since  $K1$ and $S1$ are nondecreasing, it follows
that
$$
(K\circ S)1\,(t) = K (S1)(t)=
\int_{0}^{t}\frac{1}{A(s)}\left(\int_{0}^{s}A(r)p(r)S1(r)dr\right)ds
 \leq S1(t)K1(t)
$$
and similarly
\begin{eqnarray*}
(S\circ K)1\,(t) &\leq& K1(t)S1(t).
\end{eqnarray*}
Hence, by induction we obtain  for each $t\geq0$ and
$m\in\mathbb{N}$,
\begin{eqnarray} \label{eq1.2}
(K\circ S)^{m}1\,(t)\leq \displaystyle\frac{[K
{1}(t)]^{m}}{m!}\,\displaystyle\frac{[S{1}(t)]^{m}}{m!}\quad \;
\mbox{ and }\quad \;(S\circ K)^{m}1\,(t)\leq
\displaystyle\frac{[K{1}(t)]^{m}}{m!}\,\displaystyle\frac{[S{1}(t)]^{m}}{m!}.
\end{eqnarray}

\section{Main results}
We are first concerned with the existence and the
uniqueness of a nonnegative solution of the system $(\mathcal{P})$.
For this purpose, we assume that $p,\,q,\,f$, and $g$ satisfy the following hypotheses.

\begin{description}
\item[$\mathbf{(H_{1})}$] The functions $ p,q,f,g\,:[0,\infty)\rightarrow  [0,\infty)$ are
continuous.
\item[$\mathbf{(H_2)}$] For all $c>0$, there exits $\beta>0$ such
that for all $ x,y\in[c,\infty)$$$|f(x)-f(y)|\leq\beta|x-y|$$ and$$
|g(x)-g(y)|\leq\beta|x-y| . $$
\end{description}
\begin{rem} Under the hypotheses $\mathbf{(H_1)}$ and $\mathbf{(H_2)}$
there exist $\lambda,\,\mu>0$ such that for each $x\geq 0$ we have
$$0\leq f(x)\leq\lambda x+\mu,$$ and
$$0\leq g(x)\leq\lambda x+\mu.$$
\end{rem}

Our first existence result is the following.

\begin{theo}\label{theo1}
Under the hypotheses $\mathbf{(H_{1})}$ and $\mathbf{(H_{2})}$, the
problem $(\mathcal{P})$ has a unique solution $(u,v)$ satisfying
$u,v \in C([0,\infty))\cap C^{1}((0,\infty))$ and $u,v>0$. Moreover
$$\lambda a+\mu\leq\lambda u(t)+\mu\leq (\lambda a+\mu)(1+\lambda
K1\,(t)) \exp (\lambda^{2}(K\circ S)1\,(t))$$ and $$\lambda
b+\mu\leq\lambda v(t)+\mu \leq (\lambda b+\mu)(1+\lambda S{1}(t))
\exp (\lambda^{2}(S\circ K)1\,(t)).$$
\end{theo}

Next, we are concerned with the existence and
the nonexistence of large or bounded solutions to the following system
of nonlinear elliptic equations
$$(\mathcal{Q})\left\{
\begin{array}{ll}
\Delta u=p(x)g(v), \qquad x \in {\mathbb{R}}^n\;\;\; (n\geq 3)\\
\Delta v=q(x)f(u),\qquad x \in {\mathbb{R}}^n \,.
\end{array}%
\right.$$


\begin{Defi}
Let $u,v:{\mathbb{R}}^n\rightarrow {\mathbb{R}}_+$ be  continuous
functions. We say that $(u,v)$ is a large solution of the problem
$(\mathcal{Q})$ if $(u,v)$ satisfies $(\mathcal{Q})$ in the sense of distributions  and
$\displaystyle\lim_{|x|\rightarrow\infty}[u(x)+v(x)]=\infty.$
\end{Defi}

We establish the following necessary and sufficient condition for the existence of large solutions, under the following additional hypothesis:

\smallskip
\noindent ${\mathbf{(H_{3})}}\;\; \displaystyle\inf_{t\geq a}f(t)\geq
\delta>0$ and
$\displaystyle\inf_{t\geq b}g(t)\geq \delta>0$.

\begin{theo}\label{theo2}
Let  $p$ and $q$ be radial functions and assume that the hypotheses
$\mathbf{(H_1)-(H_3)}$ are satisfied. Then the problem
$(\mathcal{Q})$ has a large solution on ${\mathbb{R}}^n$ if and only
if
\begin{equation}\label{eq3.1}
\int_0^\infty r(p(r)+q(r))dr=\infty.
\end{equation}
\end{theo}

We point out that condition \eq{eq3.1} is closely related with assumption (2) in \cite{c3}.

The following result deals with the existence of a  positive
bounded solution for the problem $(\mathcal{Q})$. Assume that
$p,\,q\in L_{loc}^1({\mathbb{R}}^n)$ are
nonnegative functions  and
 denote by $Vp$ and $Vq$ their Green potentials.

\begin{theo}\label{theo3}
Let $f\,,\,g\,$ be  nonnegative nondecreasing continuous
functions in $[0,\infty)$. Assume that $Vp$ and $Vq$ are continuous,
bounded in ${\mathbb{R}}^n$ and the following hypothesis is
satisfied:
$${\mathbf{(H_4)}}\, \mbox{ there exist }\; a>0\,,\,b>0 \,\mbox{ such that }\;
a-g(b)||Vp||_{\infty}>0 \mbox{ and } \;
b-f(a)||Vq||_{\infty}>0. $$  Then the system
$(\mathcal{Q})$ has a positive bounded continuous solution $(u,v)$
satisfying for each $x\in {\mathbb{R}}^n$
$$a-g(b)Vp (x)\leq u(x)\leq a \; \mbox{ and } \; \;
 b-f(a)Vq(x)\leq v(x)\leq b.$$
\end{theo}

In the last part of this paper we establish a nonexistence result, provided that $f$ and $g$ satisfy

\smallskip
\noindent ${\mathbf{(H_{5})}}$ $\qquad \forall a>0,\  \exists c>0 \mbox{ such that } \forall
\xi \in[0,a] \mbox{ we have } f(\xi)\geq c\xi  \mbox{ and }
g(\xi)\geq c\xi.$

\begin{theo}\label{5}Assume that p and q are two nonnegative continuous functions on
${\mathbb{R}}^{n}$ satisfying $\displaystyle\int_{0}^{\infty}r
\displaystyle\min_{|x|=r} [p(x)+q(x)]\,dr=\infty$ and the functions
$f,\,g$ satisfy ${\mathbf{(H_{5})}}$.
 Then the system
$({\mathcal{Q}})$ has no positive bounded solution.
\end{theo}

\section{Proof of Theorem \ref{theo1}}
Let $(u_{m})_{m\geq0}$ and $(v_{m})_{m\geq0}$ be sequences of
positive continuous functions defined on $[0,\infty)$ by
$$\left\{%
\begin{array}{ll}
 u_{0}(t)=a, \\
 v_{m}(t)=b+\displaystyle\int_{0}^{t}\frac{1}{B(s)}\left(\displaystyle\int_{0}^{s}
B(r)q(r)f(u_{m}(r)) dr\right)ds=b+S(f\circ u_m)(t) ,\\
 u_{m+1}(t)=a+\displaystyle\int_{0}^{t}\frac{1}{A(s)}\left(\displaystyle\int_{0}^{s}
A(r)p(r)g(v_{m}(r))dr\right)ds= a+K(g\circ v_m)(t).    \\
\end{array}%
\right.$$

Thus, for all $ t\geq 0$ and $m\in
{\mathbb{N}}$,   $u_{m}(t)\geq a$ and $v_{m}(t)\geq b$. Now,
since $\min(a,b)>0$ then by $\mathbf{(H_2)}$ there exists $\beta>0$ such
that for
all $t\geq 0$
$$\left\{%
\begin{array}{ll}
|f(u_{m+1}(t))-f(u_{m}(t))|\leq \beta |u_{m+1}(t)-u_{m}(t)|, \\
|g(v_{m+1}(t))-g(v_{m}(t))|\leq \beta |v_{m+1}(t)-v_{m}(t)|. \\
\end{array}%
\right.$$ Let $c=\max(\lambda b+\mu,\lambda f(a))$.  Then, using
(\ref{eq1.1}) and (\ref{eq1.2}), we show by induction that
$$|u_{m+1}(t)-u_{m}(t)|\leq
c\beta^{2m}\left[\displaystyle\frac{[K{1}(t)]^{(m+1)}}{(m+1)!}\,\displaystyle\frac{[S{1}(t)]^{m}}{m!}
+\displaystyle\frac{[K{1}(t)]^{(m+1)}}{(m+1)!}\,\displaystyle\frac{[S{1}(t)]^{(m+1)}}{(m+1)!}\right],$$
and $$|v_{m+1}(t)-v_{m}(t)|\leq
c\beta^{2m+1}\left[\displaystyle\frac{[K{1}(t)]^{(m+1)}}{(m+1)!}\,\displaystyle\frac{[S{1}(t)]^{(m+1)}}
{(m+1)!}+\displaystyle\frac{[K{1}(t)]^{(m+1)}}{(m+1)!}\,\displaystyle\frac{[S{1}(t)]^{(m+2)}}{(m+2)!}\right].$$
 Therefore, the sequences $(u_{m})_{m\geq0}$ and
$(v_{m})_{m\geq0}$ converge locally uniformly  to
functions $u$ and $v$ that satisfy for each $t\geq0$,
$$u(t)=a+\int_{0}^{t}\frac{1}{A(s)}\left(\int_{0}^{s}A(r)p(r)g(v(r))dr\right)ds$$
$$v(t)=b+\int_{0}^{t}\frac{1}{B(s)}\left(\int_{0}^{s}B(r)q(r)f(u(r))dr\right)ds.$$Hence
$u,v\in C([0,\infty))\cap C^{1}((0,\infty))$ and $(u,v)$ is a
solution of $(\mathcal{P})$.

Now, we prove the uniqueness of the solution.
Indeed, let $(u,v)$ and $(\widetilde{u},\widetilde{v})$ be two
solutions of the problem $(\mathcal{P})$. Then for each $R\in
(0,\infty)$ and $t\in [0,R]$, we have $$|u(t)-\widetilde{u}(t)|\leq
\beta K(|v-\widetilde{v}|)(t)\qquad \mbox{and}\qquad |v(t)-\widetilde{v}(t)|\leq
\beta S(|u-\widetilde{u}|)(t)\,.$$
By induction, we deduce that for each $m\geq
0$,$$|u(t)-\widetilde{u}(t)|\leq \beta^{2m} (K\circ
S)^{m}(|u-\widetilde{u}|)(t)$$  Since $K\circ S$ is a nondecreasing
operator, it follows from (\ref{eq1.2}) that for each $m\geq0$
$$\begin{array}{ll}
\di |u(t)-\widetilde{u}(t)|&\di\leq \beta^{2m}(K\circ S)^{m}1(R))\displaystyle\sup_
{r\in[0,R]}(|u(r)-\widetilde{u}(r)|\\
&\di\leq
\frac{[\beta^{2}K{1}(R)S{1}(R)]^{m}}{(m!)^{2}}\displaystyle\sup_
{r\in[0,R]}(|u(r)-\widetilde{u}(r)|)\,.\end{array}$$
Now,  letting $m\ri\infty$,
 we deduce that $|u(t)-\widetilde{u}(t)|=0,$ for all $t\in [0,R]$.
So $u=\widetilde{u}$ on $[0,\infty)$ and $v=\widetilde{v}$ on
$[0,\infty)$.

 Finally, we obtain for each $t>0$
\begin{eqnarray*}
  u'(t)&=& \frac{1}{A(t)}\int_{0}^{t}A(r)p(r)g(v(r))dr \\
  &\leq&\frac{1}{A(t)}\int_{0}^{t}A(r)p(r)(\lambda v(r)+\mu)dr \\
  &\leq&\frac{\lambda}{A(t)}\int_{0}^{t}A(r)p(r)(b+S\circ f (u(r)))dr+\mu(K{1})'(t) \\
  &\leq&\frac{\lambda(\lambda u(t) +\mu)}{A(t)}\int_{0}^{t}A(r)p(r)S{1}(r)dr+(\lambda b+\mu)(K{1})'(t) \\
  &\leq&\lambda^{2}u(t)\,K ( S{1})'(t)+\lambda\mu\,K(S{1})'(t)+(\lambda
  b+\mu)(K{1})'(t).
\end{eqnarray*}
Hence
$$u'(t)-\lambda^{2}u(t)\,K(S{1})'(t)\leq\lambda\mu\,K(S{1})'(t)+(\lambda
  b+\mu)(K{1})'(t).$$ For $t\geq0$, let $b(t)=\lambda^{2}\,K(S{1})'(t)$ and
$c(t)=\lambda\mu\, K(S{1})'(t)+(\lambda
  b+\mu)(K{1})'(t)$. Then
  $$u'(t)-b(t)u(t)\leq c(t).$$ It follows that
  $$\left[u(t)\exp(-\int_{0}^{t}b(s)ds)\right]'\leq
  c(t)\exp(-\int_{0}^{t}b(s)ds)$$ and consequently $$u(t)\exp\left(-\int_{0}^{t}b(s)ds\right)-a\leq
  \int_{0}^{t}c(r)\exp\left(-\int_{0}^{r}b(s)ds\right)dr.$$ Therefore,
\begin{eqnarray*}
u(t)&\leq& \left[a+
\int_{0}^{t}c(r)\exp\left(-\int_{0}^{r}b(s)ds\right)dr\right]\exp(\int_{0}^{t}b(s)ds) \\
&\leq &\left[a+\int_{0}^{t}(\lambda\mu(K(S{1})'(r)+(\lambda b
+\mu)(K{1})'(r)) \exp \left(-\lambda^{2}(K (S{1})(r)\right)dr\right]
\exp\left(\lambda^{2}K( S{1})(t) \right)\\&\leq&
\left[a+\frac{\mu}{\lambda}(1-\exp\left(-\lambda^{2}K(
S{1})(t)\right) )+(\lambda
b+\mu)K{1}(t)\right]\exp\left(\lambda^{2}\,K (S{1})(t)\right).
\end{eqnarray*}
Finally we obtain $$\lambda a+\mu\leq\lambda
  u(t)+\mu\leq\left[\lambda a+\mu+(\lambda^{2}b+\lambda\mu)K{1}(t)\right]
  \exp\left(\lambda^{2}\,K(S{1})(t)\right).$$ Similarly, we prove that
$$\lambda b+\mu\leq\lambda
  v(t)+\mu\leq\left[\lambda b+\mu+(\lambda^{2}a+\lambda\mu)S{1}(t)\right]
  \exp\left(\lambda^{2}\,S(K{1})(t)\right).$$
  This completes the proof of Theorem \ref{theo1}.\qed

\begin{coro}\label{coro2}
$1)$ Assume that
$$\int_{0}^{\infty}\left(\frac{1}{A(s)}\,\int_{0}^{s}A(r)p(r)dr\right)ds<\infty$$
and
$$\int_{0}^{\infty}\left(\frac{1}{B(s)}\,\int_{0}^{s}B(r)q(r)dr\right)ds<\infty\,.$$ Then
$u$ and $v$ are bounded.\vspace*{1mm}\\
$2)$ Assume, moreover, that condition
${\mathbf{(H_{3})}}$ holds.
Then for each $t\geq0$ we have   $u(t)\geq a+\delta K{1}(t)$ and
$v(t)\geq b+\delta S{1}(t)$.  In particular, $u$ and $v$ are
bounded if and only if
$$\int_{0}^{\infty}\left(\frac{1}{A(s)}\,\int_{0}^{s}A(r)p(r)dr\right)ds<\infty$$
and
$$\int_{0}^{\infty}\left(\frac{1}{B(s)}\,\int_{0}^{s}B(r)q(r)dr\right)ds<\infty.$$
\end{coro}

\begin{exa} For $\gamma\geq 0 \mbox{ and }\nu\geq 0$,
we consider the nonnegative functions $A$ and $B$ defined on
$[0,\infty)$ by
$$A(t)=t^{\gamma}\mbox{ and }B(t)=t^{\nu}.$$
Let $p,q:[0,\infty)\rightarrow [0,\infty)$ be two continuous
nonnegative functions and let $\delta \,,\,\theta \in [0,1]$. Then
the following problem
$$\left\{%
\begin{array}{ll}
\di u''(t)+\frac{\gamma}{t}u'(t)=p(t)v^{\delta}(t), \\
\di v''(t)+\frac{\nu}{t}v'(t)=q(t)u^{\theta}(t),\\
u(0)=a>0,v(0)=b>0, \\
u'(0)=v'(0)=0\\
\end{array}%
\right.$$  has a unique positive solution $(u,v)$ with $u,v\in
C([0,\infty))\cap C ^{2}((0,\infty))$. Moreover $u$ and $v$ are
bounded if and only if $ \gamma>1$, $\nu >1$,
$\int_{0}^{\infty}tp(t)dt<\infty$ and $
\int_{0}^{\infty}tq(t)dt<\infty$.
\end{exa}
\begin{coro}\label{coro3}
Let  $p$, $q$, $f$ and $g$ satisfying $\mathbf{(H_1)-(H_2)}$ and let
$\rho \,,\, \theta : (0,\infty) \rightarrow {\mathbb{R}}$ be two
continuous functions. Then, the following problem
$$(*)\qquad\left\{%
\begin{array}{ll}
\di\Delta u+\rho(|x|)\,x\cdot \nabla u=p(|x|)g(v), \qquad x \in {\mathbb{R}}^n\;\;\; (n\geq 3)\\
\di\Delta v + \theta(|x|)\,x\cdot \nabla v=q(|x|)f(u), \qquad x \in {\mathbb{R}}^n \,\\
\end{array}%
\right.$$ has infinitely many positive radial solutions $(u,v)$.
\end{coro}

{\bf{Proof.}} Let $u$ and $v$ be two radial functions. Then $(u,v)$
is a solution of $(*)$ if and only if
$$\left\{%
\begin{array}{ll}
\di u''(r)+ \left[\frac{n-1}{r}+r \rho(r)\right]u'(r)+=p(r)g(v(r)) \qquad r>0\\
\di v''(t)+\left[\frac{n-1}{r}+r \theta(r)\right]v'(r)=q(r)f(u(r))\qquad r>0
\end{array}%
\right.$$
or, equivalently,
$$\left\{%
\begin{array}{ll}
\displaystyle\frac{1}{A}\,(Au')'=p(r)g(v)\,,\; r>0 \\
\displaystyle\frac{1}{B}\,(Bv')'=q(r)f(u) \,,\; r>0,
\end{array}%
\right. $$ where $A(r)=r^{n-1}\exp{\left(\displaystyle\int_1^rs
\rho(s)ds\right)}$ and $B(r)=r^{n-1}
\exp{\left(\displaystyle\int_1^r s \theta(s)ds\right)}$. So the
result follows from Theorem \ref{theo1}.\qed

\section{Large and bounded solutions}

\subsection{Proof of Theorem \ref{theo2}} If $p$ and $q$ satisfy the condition
(\ref{eq3.1}), then using Theorem \ref{theo1} and  Corollary
\ref{coro2} for $A(t)=B(t)=t^{n-1},$ we obtain a large radial
solution for $(\mathcal{Q})$.

 Now, we assume that
$$\int_0^\infty rp(r)dr<\infty\quad\mbox{ and }\quad\int_0^\infty
rq(r)dr<\infty$$ and let $(u,v)$ be a solution for the problem
$(\mathcal{Q})$. We define the functions $\tilde{u}$ and $\tilde{v}$
on $(0,\infty)$ by
$$\tilde{u}(t)=\int_{S^{n-1}}u(tw)d\sigma(w)\quad\mbox{ and }\quad\tilde{v}
(t)=\int_{S^{n-1}}v(tw)d\sigma(w),$$ where $S^{n-1}$ is the united
sphere in ${\mathbb{R}}^n$ and $\sigma$ denotes the Lebesgue
measure on $S^{n-1}$. Then for all $t>0$,
\begin{eqnarray*}
\Delta\tilde{u}(t)&=& \frac{1}{t^{n-1}}(t^{n-1}\tilde{u}')' =
\int_{S^{n-1}}\Delta u(tw)d\sigma(w)=p(t)\int_{S^{n-1}}
g(v(tw))d\sigma(w)\cr \cr \Delta\tilde{v}(t)&=&
\frac{1}{t^{n-1}}(t^{n-1}\tilde{v}')' = \int_{S^{n-1}} \Delta
v(tw)d\sigma(w)=q(t)\int_{S^{n-1}}f(u(tw))d\sigma(w) \cr \cr
\tilde{u}'(t)&=&\frac{1}{t^{n-1}}\int_{B(0,t)}\Delta u(x)dx =
\frac{1}{t^{n-1}}\int_{B(0,t)}p(|x|)g(v(x))dx \geq 0\cr
\cr \cr \tilde{v}'(t)&=&\frac{1}{t^{n-1}}\int_{B(0,t)}\Delta
v(x)dx
   = \frac{1}{t^{n-1}}\int_{B(0,t)}q(|x|)f(u(x))dx\geq0.
\end{eqnarray*}
Thus, $\tilde{u}$ and $\tilde{v}$ are nondecreasing. Using now hypothesis $\mathbf{(H_2)}$ we obtain
\begin{eqnarray*}
\tilde{u}'(t)=\frac{1}{t^{n-1}}\int_{B(0,t)}\Delta u(x)dx &\leq&
[\lambda \tilde{v}(t)+\mu] \, \frac{1}{t^{n-1}}\int_0^tr^{n-1}p(r)dr
\cr \cr &\leq& [\lambda (\tilde{u}(t)+\tilde{v}(t))+\mu] \,
\frac{1}{t^{n-1}}\int_0^tr^{n-1}p(r)dr
\end{eqnarray*}
and
\begin{eqnarray*}
\tilde{v}'(t)=\frac{1}{t^{n-1}}\int_{B(0,t)}\Delta v(x)dx &\leq&
[\lambda \tilde{u}(t)+\mu] \, \frac{1}{t^{n-1}}\int_0^tr^{n-1}q(r)dr
\cr \cr &\leq& [\lambda (\tilde{u}(t)+\tilde{v}(t))+\mu] \,
\frac{1}{t^{n-1}}\int_0^tr^{n-1}q(r)dr
\end{eqnarray*}
Hence
$$\frac{ \tilde{u}'(t)+\tilde{v}'(t)}{\lambda (\tilde{u}(t)+\tilde{v}
(t))+\mu}\leq \frac{1}{t^{n-1}}\int_0^tr^{n-1}(p(r)+q(r))dr.$$
Consequently
$${\lambda (\tilde{u}(t)+\tilde{v}
(t))+\mu}\leq  [\lambda (\tilde{u}(0)+\tilde{v} (0))+\mu ]\exp
\left(\frac{\lambda}{n-2} \int_0^\infty r(p(r)+q(r))dr \right).$$
This implies that $\tilde{u}+\tilde{v}$ is bounded and consequently
$(u,v)$ is
not a large solution. This completes the proof.\qed

 \subsection{Proof of Theorem \ref{theo3}}
 We start with the following auxiliary result.

\begin{lem}\label{lem1}
Let $\phi$ and  $\psi$ be two  measurable functions such that $0\leq
\phi \leq \psi$. If $\,V \psi$ is continuous then $V\phi$ is also
continuous.
\end{lem}
{\bf{Proof.}} Let $\theta \geq 0$ such that $\psi=\phi +\theta $.
Since  $V\phi$ and $V\theta$ are lower semi-continuous and $V\phi +
V\theta =V \psi$ which is continuous, then $V\phi$ is an upper and
lower semi-continuous function. So, $V\phi$ is a continuous
function.\qed

\medskip We
consider the sequences $(u_{k})_{k\geq0}$ and $(v_{k})_{k\geq0}$
defined by
$$ \left\{%
\begin{array}{ll} u_{0}=a-V(pg(b)), \\ u_{k+1}=a-V(p\,g(v_{k})), \\ v_{k}=b-V(q\,f(u_{k}))\,.
\end{array}%
\right.
$$

We claim  that for each $k\geq 0$,
$$a-g(b)Vp\leq u_{k}\leq u_{k+1}\leq a\; \mbox{ and }\; b-f(a)Vq\leq v_{k+1}
\leq v_{k}\leq b .$$ Indeed, from hypothesis $\mathbf{(H_4)}$ we
have $0<u_{0}=a-g(b)Vp\leq a$. So $$b\geq v_{0}=b-V(qf(u_{0}))\geq
b-V(qf(a))>0$$  $$u_{1}-u_{0}=-V[p(g(v_{0})-g(b))]\geq0$$ and
$$v_{1}-v_{0}=-V[q(f(u_{1})-f(u_0))]\leq0\,.$$
Let us assume that the
claim holds for some $k\in\mathbb{N}$, then we have
$$v_{k+2}-v_{k+1}=-V[q(f(u_{k+1})-f(u_{k}))]\leq0$$ and
$$u_{k+2}-u_{k+1}=-V[p(g(v_{k+1})-g(v_{k}))]\geq0.$$ Moreover, for all integer $k$
we have $u_{k}\leq a$ , $v_{k}\leq b$, $u_{k+1}=a-V(p g(v_{k}))\geq
a-g(b)Vp$ and $v_{k+1}=b-V(qf(u_{k+1}))\geq b-f(a)Vq$. This
completes the proof of the claim. Therefore, the sequences
$(u_{k})_{k\geq0}$ and $(v_{k})_{k\geq0}$ converge respectively to
two functions $u$ and $v$ satisfying  $0<a-g(b)Vp\leq u\leq a$ and
$0< b-f(a)Vq\leq v\leq b$. By the dominated convergence theorem we
deduce that $u$ and $v$ satisfies
\begin{eqnarray}\label{eq3.2}
\left\{%
\begin{array}{ll}
 u=a-V(p\,g(v)) \\
 v=b- V(q\,f(u)).
\end{array}%
\right.
\end{eqnarray}
Now, using the fact that  $Vp$ and $Vq$ are continuous and
$V(pg(v))\leq g(b)Vp\,,\;V(qf(u))\leq f(a) Vq$, we deduce from Lemma
\ref{lem1} that $u$ and $v$ are continuous. Finally, we deduce from relation (\ref{eq3.2}) that $(u,v)$ is a
weak bounded positive solution  of problem
$(\mathcal{Q})$.
This completes the proof of Theorem \ref{theo3}.

\begin{rem}
The condition ${\mathbf{(H_4)}}$ is satisfied in the particular case
where $f(t)=t^{\alpha}$, $g(t)=t^{\beta}$ with $0\leq
\alpha\;,\;\beta$ and $\alpha \beta \neq 1$. Hence, Theorem
$\ref{theo3}$ generalizes Theorem $1$ in \cite{w1}.
\end{rem}

\begin{rem}
Let $p$ be a nonnegative function in $L^{m}({\mathbb{R}}^{n})\cap
L^{1}({\mathbb{R}}^{n})$ with $m>\frac{n}{2}\geq \frac{3}{2}$. Then
it follows from \cite[p.64-66]{c2} that  $Vp$ is continuous in
${\mathbb{R}}^n$ and tends to zero at infinity.
\end{rem}

For the next result we fix two nonnegative functions $p,q\in
L^{m}({\mathbb{R}}^{n})\bigcap L^{1}({\mathbb{R}}^{n})$ with
\\ $m>\frac{n}{2}\geq \frac{3}{2}$ and $a>0$, $b>0$. Put
$\lambda_0=\displaystyle\frac{a}{g(b){\|Vp\|}_\infty}$ and
$\mu_0=\displaystyle\frac{b}{f(a){\|Vq\|}_\infty}$.

\begin{coro}
Let f, g be two nonnegative nondecreasing continuous functions and
$a\,,\,b>0$. Then for each
$\lambda\in [0,\lambda_{0})$ and $\mu\in[0,\mu_{0})$, the nonlinear elliptic system $$\left\{%
\begin{array}{llll}
\Delta u=\lambda p(x)g(v), \\
\Delta v=\mu q(x)f(u),\\
\displaystyle\lim_{x\rightarrow \infty} u(x)=a, \\
\displaystyle\lim_{x\rightarrow \infty}v (x)=b
\end{array}%
\right. $$ has a positive bounded continuous solution $(u,v)$
satisfying
$$a\left(1-\frac{\lambda}{\lambda_{0}}\right)\leq u\leq a \;\mbox{ and }\;\; b\left(1-
\frac{\mu}{\mu_0}\right)\leq v\leq b.$$
\end{coro}
\begin{exa}
Let $p,q$ be two  nonnegative functions in
$L^{m}({\mathbb{R}}^{n})\bigcap L^{1}({\mathbb{R}}^{n})$ with
$m>\frac{n}{2}\geq \frac{3}{2}$ and $a>0$, $b>0$. Let $\alpha\geq 0$
and $\beta \geq 0$. Then there exist $\lambda_0>0$ and $\mu_0>0$
such that for each $\lambda\in [0,\lambda_{0})$ and $\mu \in
[0,\mu_{0})$, the nonlinear elliptic system
$$\left\{%
\begin{array}{llll}
\Delta u=\lambda\, p(x)\,v^{\beta}, \\
\Delta v=\mu q(x)\,u^{\alpha},\\
\displaystyle\lim_{x\rightarrow \infty} u(x)=a, \\
\displaystyle\lim_{x\rightarrow \infty}v (x)=b
\end{array}%
\right. $$ has a positive bounded solution $(u,v)$.
\end{exa}

\section{A nonexistence result}
In this section we are concerned with the proof of
 Theorem \ref{5}.

 Let $M>0$ and  assume that $(u,v)$ is a solution of
$({\mathcal{Q}})$ with $0< u\leq M$
and $0< v\leq M$. Let $\tilde{u}$ and $\tilde{v}$ be as in the proof
of Theorem \ref{theo2}, then for each $t>0$ we have
\begin{eqnarray} \label{eq3.3}\begin{array}{ll}
\di\tilde{u}'(t)&\di=\frac{1}{t^{n-1}}\int_{B(0,t)}p(x)g(v(x))dx\\ &\di=
\frac{1}{t^{n-1}}\int^{t}_{0}s^{n-1}\int_{S^{n-1}}p(sw)g(v(sw))d\sigma(w)ds\geq0\end{array}
\end{eqnarray}
and
\begin{eqnarray}\label{eq3.4}\begin{array}{ll}
\di
\tilde{v}'(t)&\di =\frac{1}{t^{n-1}}\int_{B(0,t)}q(x)f(u(x))dx\\ &\di=\frac{1}{t^{n-1}}
\int^{t}_{0}s^{n-1}\int_{S^{n-1}}q(sw)f(u(sw))d\sigma(w)ds\geq 0.\end{array}
\end{eqnarray}
Since $\tilde{u}$ and $\tilde{v}$ are nondecreasing, there exist
$R>0$ and $\varepsilon
> 0$ such that $\tilde{u(t)}\geq \varepsilon$ and  $\tilde{v(t)}\geq \varepsilon$
$(\forall t\geq R)$. Using relations (\ref{eq3.3}), (\ref{eq3.4})  and
hypothesis $\mathbf{(H_5)}$, we obtain for all $t\geq R$,
\begin{eqnarray*}
M\geq \tilde{u}(t)&=&
\tilde{u}(0)+\int_{0}^{t}r^{1-n}\int^{r}_{0}s^{n-1}
\int_{S^{n-1}}p(sw)g(v(sw))d\sigma(w)\,ds dr\\
&\geq&\tilde{u}(0)+
\int_{0}^{t}r^{1-n}\int^{r}_{0}s^{n-1}(\displaystyle\min_{|x|=s}
p(x))\int_{S^{n-1}}g(v(sw))d\sigma(w)ds dr\\
&\geq&
\tilde{u}(0)+c\int_{0}^{t}\int_{0}^{r}s^{n-1}(\displaystyle\min_{|x|=s}
p(x))\tilde{v}(s)\,dsdr\\
&\geq& \tilde{u}(0)+ c\varepsilon\int_{R}^{t}r^{1-n}
\int_{R}^{r}s^{n-1}(\displaystyle\min_{|x|=s}p(x))\,dsdr\\
&\geq&\tilde{u}(0)+
c\varepsilon\int_{R}^{t}s^{n-1}(\displaystyle\min_{|x|=s}p(x))(\int_{s}^{t}r^{1-n}dr)ds.
\end{eqnarray*}
Similarly, we prove that
\begin{eqnarray*}
M\geq \tilde{v}(t) \geq\tilde{v}(0)+
c\varepsilon\int_{R}^{t}s^{n-1}(\displaystyle\min_{|x|=s}q(x))(\int_{s}^{t}r^{1-n}dr)ds.
\end{eqnarray*}
Consequently
\begin{eqnarray*}
2M\geq \tilde{u}(t)+\tilde{v}(t) \geq \tilde{u}(0)+ \tilde{v}(0)+
c\varepsilon\int_{R}^{t}s^{n-1}\,\displaystyle\min_{|x|=s}[p(x)+q(x)]\left(\int_{s}^{t}r^{1-n}dr\right)ds.
\end{eqnarray*}
Or, from the hypothesis on $p$ and $q$ the right-hand side of this
inequality tends to infinity as $t\rightarrow \infty$. This yields
to a contradiction and achieves the proof.\qed

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