\documentclass[twoside]{article}
\usepackage{amssymb} % used for R in Real numbers
\pagestyle{myheadings}
\markboth{\hfil Existence of continuous and singular ground states
 \hfil EJDE--1998/01}%
{EJDE--1998/01\hfil Cecilia S. Yarur \hfil}
\begin{document}
\title{\vspace{-1in}\parbox{\linewidth}{\footnotesize\noindent
{\sc  Electronic Journal of Differential Equations},
Vol.\ {\bf 1998}(1998), No.~01, pp. 1--27. \newline
ISSN: 1072-6691. URL: http://ejde.math.swt.edu or http://ejde.math.unt.edu
\newline ftp (login: ftp) 147.26.103.110 or 129.120.3.113}
\vspace{\bigskipamount} \\
  Existence of continuous and singular ground states for semilinear 
  elliptic systems 
\thanks{ {\em 1991 Mathematics Subject Classifications:} 35J60, 31A35.
\hfil\break\indent
{\em Key words and phrases:} Semilinear elliptic systems, ground states.
\hfil\break\indent
\copyright 1998 Southwest Texas State University  and University of
North Texas. \hfil\break\indent
Submitted September 3, 1997. Published January 16, 1998.
\hfil\break\indent Partially supported by FONDECYT grant 1961235 and by DICYT. 
} }
\date{}
\author{Cecilia S. Yarur}
\maketitle

\begin{abstract} 
We study existence results of a curve of continuous and singular ground states 
for the system
\begin{eqnarray*}
&-\Delta{u}  =  {\alpha(|x|)}f(v) &\\
&-\Delta{v}  =  \beta(|x|) g(u)\,,& 
\end{eqnarray*}  
where $x \in {\mathbb R}^N \setminus \{0\}$, the functions $f$ and $g$ are 
increasing Lipschitz continuous functions in ${\mathbb R}$,
 and $\alpha$ and $\beta$ are nonnegative continuous functions in 
${\mathbb R}^+$. We also study general systems of the form 
\begin{eqnarray*}
&\Delta u(x)+V(|x|)u+a(|x|)v^p= 0&\\
&\Delta v(x)+V(|x|)v+b(|x|)u^q= 0\,.&
\end{eqnarray*}
\end{abstract}

\catcode`\@=11
\@addtoreset{equation}{section}
\catcode`\@=12 
\def\theequation{\thesection.\arabic{equation}}
\newtheorem{theorem}{Theorem}[section]
\newtheorem{corollary}{Corollary}[section]
\newtheorem{lemma}{Lemma}[section]
\newtheorem{proposition}{Proposition}[section]
\newcommand{\be}{\begin{equation}}
\newcommand{\ee}{\end{equation}}
\section{Introduction} 
The purpose of this paper is to prove  existence of a curve of positive 
radially symmetric continuous ground states, and a curve of  singular  ground 
states with the singularity at zero,  for the system
\be \label{gmmy}
\begin{array}{c}
-\Delta{u}  =  {\alpha(|x|)}f(v) \\[5pt]
 -\Delta{v}  =  \beta(|x|) g(u) 
\end{array} \quad  x \in {\mathbb R}^N \setminus \{0\}\,. 
\ee
The existence of these curves depends on conditions on the functions $f$, 
$g$, $\alpha$ and $\beta$. For some functions there exists a 
curve of continuous ground states, and not a curve of singular ground states 
and vice versa.

We will assume throughout this paper that  $f\in C({\mathbb R},{\mathbb R}), $ $g\in 
C({\mathbb R},{\mathbb R})$,  $\alpha \in C^1({\mathbb R}^+,{\mathbb R}^+) $, and $\beta\in 
C^1({\mathbb R}^+,{\mathbb R}^+). $ Recall that no smoothness at zero for the function
 $(\alpha, \beta)$ is required. This leads to obtain existence of ground states
  for more general systems.   Moreover, we assume that $f$ and $g$ are 
increasing functions with $f(0)=0$ and $g(0)=0$.  Serrin and Zou \cite{sz},
 proved the existence of a curve of ground states for system (\ref{gmmy}) when 
 $\alpha= \beta \equiv 1$, $(f(v), g(u))=(v^p, u^q)$ and $(p, q)$  above the 
 critical hyperbola (see (\ref{hiperbola1}) below with 
 $(\alpha_o,\beta_o)=(0,0)$). P.L Lions \cite{li} proved existence of ground 
 states on the hyperbola by studying the scalar equation 
$$
-\Delta((-\Delta u)^{1/p})=u^q\,.
$$

In section 2, we give some preliminary results, such as existence results for the 
Cauchy problem, continuous dependence theorem and some properties for the 
function $(\alpha,\beta)$. Section 3 is devoted to prove the existence of a 
curve of regular (no classical) ground states. We assume that
 $(f, g)$ are Lipschitz continuous functions such that for all $u\ge 0$ and 
 $v\ge 0$,
\be \label{potencia1}
vf(v)\ge (p+1)F(v), \quad ug(u)\ge (q+1)G(u)\,,
\ee
and that the functions $\alpha$ and $\beta$ satisfy
\be \label{breli1}
\int^{\infty}r\alpha(r)dr= \infty, \quad \int^{\infty}r\beta(r)dr= \infty\,.
\ee
As shown in section 1, condition (\ref{breli1}) implies that positive  
radially symmetric solutions to (\ref{gmmy}) are ground states.
The existence of a solution, near zero, to the Cauchy problem needs
\be \label{cotaalfabetaen0s1}
\int_0s\alpha(s)ds< \infty, \quad \mbox{and} \quad \int_0s \beta(s)ds< \infty. 
\ee
We also assume 
 the existence of $\alpha_o$ and $\beta_o$ such that
\be \label{decrecimiento1}
r^{\alpha_o}\alpha(r), \quad \mbox{and} \quad r^{\beta_o}\beta(r) \quad 
\mbox{are non-increasing functions,}
\ee
and that
\be \label{hiperbola1}
\frac{N-\alpha_o}{p+1}+\frac{N-\beta_o}{q+1} \le N-2. 
\ee 
In section 4,  we prove two kinds of results of existence of singular ground 
states. The first, will be given as a limit of regular ground states 
constructed in section~3, and the second type will be given by those results 
of section 3 and the Kelvin transform. This last type, gives existence of a curve 
of singular ground states under the critical hyperbola (\ref{hiperbola1}) and 
above the \lq\lq first critical hyperbola". In section 5 we applied results of the 
previous sections to general systems of the form
\begin{eqnarray*}
\Delta u(x)+V(|x|)u+a(|x|)v^p&=& 0\,\nonumber\\
\Delta v(x)+V(|x|)v+b(|x|)u^q&=& 0\,
\end{eqnarray*}
with $x$ in  ${\mathbb R}^N\setminus \{0\}$, and $V$ not necessarily negative. 
In particular, if $V(r)=-dr^{-2}$,  
$d>-(N-2)^2/4$, and $\theta_{1,0}=(2-N\pm ((N-2)^2+4d)^{1/2})/2$ we get the  
following results. 
\medskip

\noindent {\bf Corollary \ref{d}} {\em Assume that there exists $(a_o, b_o)$ 
such that $r^{a_o}a(r)$ and $r^{b_o}b(r)$ are non-increasing functions for some 
$(a_o, b_o)$ satisfying
$$
 \frac{N-a_o}{p+1}+\frac{N-b_o}{q+1}\le N-2\,.
$$
Also assume that $a$ and $b$ satisfy $(\ref{cotaalfabetaen0vno0})$ and 
$(\ref{brelivno0})$.  
Then, there exists $g\in C({\mathbb R}^+, {\mathbb R}^+)$ strictly increasing and such 
that for any $c>0$ there exists a radially symmetric  solution $(u, v)$ to 
$(\ref{ralamenos2})$ such that
$$
\lim_{r\to 0}\frac{u(r)}{r^{\theta_1}}=g(c), \quad \mbox{and} \quad 
\lim_{r\to 0}\frac{v(r)}{r^{\theta_1}}=c\,.
$$}
\medskip

\noindent{\bf Corollary \ref{d1}} {\em Assume that there exists $(a_o, b_o)$ 
such that $r^{a_o}a(r)$ and $r^{b_o}b(r)$ are nondecreasing functions for some 
$(a_o, b_o)$ satisfying
$$
 \frac{N-a_o}{p+1}+\frac{N-b_o}{q+1}\ge N-2.
$$
Also assume that $a$ and $b$ satisfy $(\ref{condicioneskelvinvno0})$ and 
$(\ref{condicioneskelvin1vno0})$.
Then, there exists $\delta \in C({\mathbb R}^+, {\mathbb R}^+)$ strictly increasing and 
such that for any $c>0$ there exists a radially symmetric  solution $(u, v)$ to
 $(\ref{ralamenos2})$ such that}
$$
\lim_{r\to \infty}\frac{u(r)}{r^{\theta_0}}=\delta(c), \quad  
\lim_{r\to \infty}\frac{v(r)}{r^{\theta_0}}=c.
$$

After this work was completed, we learned of a paper by Serrin and Zou, 
\cite{sz13}, in which they prove existence of classical ground states for a 
general elliptic Hamiltonian system.

\section{Preliminary results}

This section is devoted to prove existence, continuous dependence results and 
some previous properties of nonnegative solutions to the problem
\begin{eqnarray} \label{pc}
&-u''(r) -\frac{N-1}{r}u'(r)= \alpha(r) f(v)\,& \nonumber\\
&-v''(r) -\frac{N-1}{r}v'(r)= \beta(r) g(u),&\\
&u(0)= c_1, \quad v(0)=c_2,&  \nonumber
\end{eqnarray}
where $c_1$ and $c_2$ are positive constants. In this section, the functions 
$\alpha$ and $\beta$ are $C^1$ functions defined in $(0, \infty)$, satisfying  
\be \label{breli}
\int^{\infty}r\alpha(r)dr= \infty, \quad \int^{\infty}r\beta(r)dr= \infty.
\ee
Condition (\ref{breli}) implies that positive  radial solutions to (\ref{gmmy})
 are ground states, as is shown below. Moreover,  
\be \label{cotaalfabetaen0}
\int_0s\alpha(s)ds< \infty, \quad\mbox{and} \quad \int_0s \beta(s)ds< \infty 
\ee
ensures the existence of nonnegative solutions to (\ref{pc}). Condition 
(\ref{cotaalfabetaen0}) appears usually on existence problems (\cite{as}, 
\cite{ni}), and is  the
 Kato class for radially symmetric functions.  

The solutions of (\ref{pc}) defined in $[0, \infty)$ are radially symmetric 
continuous solutions to (\ref{gmmy}), in the sense of distributions in 
${\mathbb R}^N$.
 We distinguish the radially symmetric continuous solutions to 
(\ref{gmmy}) from those which satisfy
\begin{eqnarray} \label{pc0}
-u''(r) -\frac{N-1}{r}u'(r)&=& \alpha(r)f(v), \nonumber\\
-v''(r) -\frac{N-1}{r}v'(r)&=& \beta(r)g(u),\\
u(0)= c_1, u'(0)=0, && v(0)=c_2, v'(0)=0, \nonumber
\end{eqnarray}
 by calling these last type {\bf classical solutions}. Our interest  in the 
 study of continuous solutions and not necessary classical solutions,  comes 
 from the existence of singular ground states treated in Section 4. 
The classical nature of  a solution to (\ref{pc}) with $c_1>0$ and $c_2>0$ only 
depends on the functions $\alpha$ and $\beta$ as is shown in the following.

\begin{proposition} \label{u'0} Assume that $(\alpha, \beta)$ satisfies 
$(\ref{cotaalfabetaen0})$. Let $(u, v)$ be a solution to $(\ref{pc})$ with 
$c_1>0$ and $c_2>0$. Then
\be \label{u'}
\lim_{r\to 0}ru'(r)=0, \quad \mbox{and} \quad \lim_{r\to 0}rv'(r)=0.
\ee
Moreover, if 
\be \label{cotaalfabetaen01}
\lim_{r\to 0}r^{1-N}\int_0^rs^{N-1}\alpha(s)ds=0, \quad \mbox{and} \quad 
\lim_{r\to 0}r^{1-N}\int_0^r s^{N-1}\beta(s)ds=0,
\ee
then, $u'(0)=0$, and $v'(0)=0$.
\end{proposition}

\noindent{\bf Proof.} Since $u$ is continuous at zero,  
$\lim_{r\to 0}r^{N-1}u'(r)=0$, and thus, since from the system 
 (\ref{pc}) $-r^{N-1}u'(r)$ is increasing for $r$ small, we have that $u$ is 
 decreasing for $r$ small enough. Similarly, $v$ is decreasing for all $r$ 
 small. Integrating from $0$ to $r$ the first equation of (\ref{pc}) and if 
 $C=\max_{v\in [c_2/2, c_2]}f(v)$, we get
\begin{eqnarray*}
-r^{N-1}u'(r)&=&\int_0^rs^{N-1}\alpha(s)f(v(s))\,ds \\
&\le&  C\int_0^rs^{N-1}
\alpha(s)ds  \le Cr^{N-2}\int_0^r s\alpha(s)\,ds\,,
\end{eqnarray*}
and thus (\ref{u'}) follows for $u$, and similarly for $v$. From the above 
estimate we also get $u'(0)=0$ assuming (\ref{cotaalfabetaen01}). \hfill$\diamondsuit$


The following is a nonexistence result of solutions to (\ref{pc}), without 
condition (\ref{cotaalfabetaen0}).  For a similar result in this direction, see 
\cite{bi}.

\begin{proposition} Assume that $(u, v)$ is a nonnegative solution to 
$(\ref{pc})$ with $c_1>0$ and $c_2>0$, then $(\alpha, \beta)$ satisfies 
$(\ref{cotaalfabetaen0})$. Moreover, if $(u, v)$ is a classical solution to 
$(\ref{pc})$, then $(\alpha, \beta)$ satisfies $(\ref{cotaalfabetaen01})$.
\end{proposition}

\noindent {\bf Proof.} Integrating twice (\ref{pc}) from $0$ to $r_0$ with $r_0$
 small enough such that $c_1\ge u(r)\ge c_1/2$ and $c_2\ge v(r)>c_2/2$, for all 
 $r \in [0, r_0]$ , we obtain
$$
c_1-\int_0^{r_0}s^{1-N}\left[\int_0^{s}t^{N-1}\alpha(t)f(v(t))dt\right]ds \ge 
c_1/2,
$$
and thus,
$$
m \int_0^{r_0}s^{1-N}\left[\int_0^{s}t^{N-1}\alpha(t)dt\right]ds \le 
\int_0^{r_0}s^{1-N}\left[\int_0^{s}t^{N-1}\alpha(t)f(v(t))dt\right]ds \le 
\frac{c_1}2\,,
$$
where $m=\min_{r\in [c_2/2, c_2]}\{f(v)\}$.  Therefore, 
$$
\int_0^{r_0}s^{1-N}\left[\int_0^{s}t^{N-1}\alpha(t)dt\right]ds < \infty,
$$
which is equivalent to $\int_0s\alpha(s)ds< \infty$. Moreover,
$$
-u'(r)=r^{1-N}\int_0^r s^{N-1}\alpha(s)f(v(s))ds \ge 
m r^{1-N}\int_0^r s^{N-1}\alpha(s)ds
$$
and thus if $u'(0)=0,$
$$
0=-u'(0)\ge m \lim_{r \to 0}r^{1-N}\int_0^r s^{N-1}\alpha(s)ds.
$$
 Similarly, it can be proven that
 $$\int_0s\beta(s)ds< \infty, \quad \mbox{ and} \quad 
 \lim_{r \to 0}r^{1-N}\int_0^r s^{N-1}\beta(s)ds=0\,.$$
  \hfill$\diamondsuit$ 

Condition (\ref{breli}) implies that nonnegative solutions to (\ref{pc})
defined in $(0, \infty)$ are ground states, as is proved in the following 
proposition.

\begin{proposition} \label{groundstate} Assume that $(\alpha, \beta)$ satisfies 
$(\ref{breli})$ and 
let $(u, v)$ be a nonnegative solution to $(\ref{pc})$ in $(0, \infty)$. Then,
$$
\lim_{r\to \infty}u(r)=0, \quad \lim_{r\to \infty}v(r)=0.
$$
\end{proposition}

\noindent{\bf Proof.} From \cite{blions}), we deduce the following: 

Let $0 \le h \in L_{\mbox{\scriptsize loc}}^1(B_1(0) \setminus \{0\})$, and 
$w \in L_{\mbox{\scriptsize loc}}^1(B_1(0) \setminus \{0\})$ be a nonnegative function, such that 
$ -\Delta w \in L_{\mbox{\scriptsize loc}}^1(B_1(0) \setminus \{0\})$  in the sense of 
distributions in $B_1(0) \setminus \{0\}$, and $ -\Delta w =h$.  Then, 
$h \in L_{\mbox{\scriptsize loc}}^1(B_1(0))$.  Moreover, for a   radially symmetric $w$; 
$\lim_{r\to 0}r^{N-2}w(r)$ exists and it is finite.

Next, we prove that $v$ tends to zero as $r$ tends to infinity. Consider $w$, 
be the Kelvin transform of the function $u$ (see \cite{gt}), that is 
$$
w(r)=r^{2-N}u(1/r).
$$
We easily get that 
$$-\Delta w = h, \quad \mbox{where}  \quad 
h(x)=|x|^{-(N+2)}\alpha(1/{|x|})f(v(1/{|x|})),
$$
and thus $h \in L_{\mbox{\scriptsize loc}}^1(B_1(0))$, and $\lim_{r \to \infty}u(r)= 
\lim_{r \to 0}r^{N-2}w(r)$ exists and is finite. 
 Similarly, there exists $\lim_{r \to \infty}v(r)= c$.  Now, $h \in 
 L_{\mbox{\scriptsize loc}}^1(B_1(0))$, 
 is equivalent to 
$$
\int^{\infty}s\alpha(s)f(v(s))< \infty.
$$
Therefore, from (\ref{breli}), we obtain  $c=0$.   \hfill$\diamondsuit$
\medskip

The next results will be needed in the following sections.

\begin{proposition} \label{r2alfa} Let $\alpha$ be any nonnegative function, 
and $\alpha_o \in {\mathbb R}$, be such that 
$\int\limits_{0}s\alpha(s)ds< \infty$ and $r^{\alpha_o}\alpha(r)$
is non-increasing.
Then, $\alpha_o<2$, and \newline $\lim_{r\to 0}r^2\alpha(r)=0$.
\end{proposition}

\noindent{\bf Proof.} Let $0<r<r_0$, with $r_0$ small, then 
$$
\int_0^r s\alpha(s)ds \ge \int_{r/2}^r s^{\alpha_o}\alpha(s)s^{1-\alpha_o}ds 
\ge Cr^{2}\alpha(r)\ge Kr^{2-\alpha_o},
$$
and the conclusion follows.  \hfill$\diamondsuit$
\medskip

For $(\alpha_o, \beta_o)\in {\mathbb R}^2$ 
 and $ pq>1$ we define,
\begin{equation} \label{defgammas}
\gamma_1(\alpha_o, \beta_o)=\frac{\alpha_o-2+ (\beta_o-2)p}{pq-1},\quad
\gamma_2(\alpha_o, \beta_o)=\frac{\beta_o-2+ (\alpha_o-2)p}{pq-1}\,.
\end{equation}
We have the following

\begin{proposition}\label{propiedades} 
\begin{enumerate}
\item[$(i)$] If $pq>1$, then conditions $(\ref{hiperbola1})$ and \newline 
$\gamma_1(\alpha_o, \beta_o)+\gamma_2(\alpha_o, \beta_o)+N-2\ge 0$
are equivalent.
\item[$(ii)$] Assume that $pq>1$ and assume that there exist 
$(\alpha_o, \beta_o)$ such that   $r^{\alpha_o}\alpha(r)$, 
$r^{\beta_o}\beta(r)$ are non-increasing functions, and  condition 
$(\ref{cotaalfabetaen0})$ is satisfied. Then,
$\gamma_i(\alpha_o, \beta_o)<0,\quad \mbox{for } i=1,2$.
\item[$(iii)$] Assume that $(\alpha, \beta, p, q)$ satisfy the hypothesis given 
on  (ii) and $(\ref{hiperbola1})$. Then $
2-\alpha_o-(N-2)p<0$ and $2-\beta_o-(N-2)q<0$.
\end{enumerate}
\end{proposition}
\noindent {\bf Proof.} The proof of  (i) is a consequence of  the definition of 
$\gamma_i$ for $i=1,2$.
 Item (ii) follows from Proposition \ref{r2alfa}. From that proposition 
 $2-\alpha_o>0$ and $2-\beta_o>0$, and thus $N-\alpha_o>0$ and $N-\beta_o>0$.  
 Therefore, if we are in the assumption of (iii), we get 
$$
\frac{N-\alpha_o}{p+1} \le N-2-(\frac{N-\beta_o}{q+1})< N-2,
$$
and the conclusion follows. \hfill$\diamondsuit$ 

\begin{theorem} \label{existencepc} Assume that $(\alpha, \beta)$ satisfies
$(\ref{cotaalfabetaen0})$ and $f$ and $g$ are continuous nonnegative functions 
defined in ${\mathbb R}^+\cup \{0\}$. Then, for any $(c_1, c_2) \in  {\mathbb R}^+\times 
{\mathbb R}^+$, there exists a solution $(u, v)$ to $(\ref{pc})$ defined in some 
interval $[0, k)$.  Moreover, the solution $(u, v)$ is a classical solution if 
$(\alpha, \beta)$ satisfies $(\ref{cotaalfabetaen01})$.
\end{theorem}

\noindent{\bf Proof.} Let
$$
S:= \{(u, v) \in C([0, \epsilon])\times C([0, \epsilon]) |\quad  0 \le u 
\le c_1, 0 \le v \le c_2 \},
$$
where $\epsilon>0$ is small enough. In $S$ we define $T(u,v)=(T_1(u, v), 
T_2(u, v))$ as follows
\begin{eqnarray*}
T_1(u, v)(r)&=& c_1-\int_0^r s^{1-N}\left[\int_0^s
t^{N-1}\alpha(t)f(v(t))dt\right]ds \\
T_2(u, v)(r)&=& c_2-\int_0^r s^{1-N}\left[\int_0^s
t^{N-1}\beta(t)g(u(t))dt\right]ds. \\
\end{eqnarray*}
Next, we will show that for $\epsilon$ small $T(S) \subset S$.   Since, 
$v \ge 0$, from the definition of $T_1$ it follows that $T_1(u, v) \le c_1$. We 
will prove that $T_1(u, v)\ge 0$.  Let $M \ge \max\{f(v):v \in [0, c_2]\}$, then 
if $(u, v) \in S$, we have
\begin{eqnarray*}
0 \le \int_0^{\epsilon} s^{1-N}\left[\int_0^s t^{N-1}\alpha(t)f(v(t))dt\right]ds &
\le & M \int_0^{\epsilon} s^{1-N}\left[\int_0^s t^{N-1}\alpha(t)dt\right]ds \\ 
&\le& \frac{M}{N-2}\int_0^{\epsilon} s\alpha(s)ds, 
\end{eqnarray*}
and thus, if $\frac{M}{N-2}\int_0^{\epsilon} s\alpha(s)ds\le c_1$, we obtain 
that $T_1(u, v)\ge 0$.  A similar argument  can be used to prove 
$0 \le T_2(u,v)\le c_2$.  It is not difficult to prove that $T$ is a continuous 
and compact operator in $S$, with respect to the uniform convergence, and thus 
it has a fixed point, which corresponds to a nonnegative $(u, v)$ to (\ref{pc}).
 \hfill$\diamondsuit$

\begin{lemma} \label{depen} Let $\alpha$ and $\beta$ be two nonnegative 
functions defined in $(0, \infty)$, satisfying $(\ref{cotaalfabetaen0})$. Let $f$ 
and $g$ be locally Lipschitz continuous functions, and nonnegative near zero. Let 
$(u_1, v_1)$, $(u_2, v_2)$ be nonnegative solutions to $(\ref{pc})$, with 
$(u_1(0), v_1(0))=(c_1, c_2)$,  $(u_2(0), v_2(0))=(d_1, d_2)$, and $c_1, c_2, 
d_1, d_2$ positive numbers.  Consider $I_1:=[0,r_1]$ to be any  interval where 
the functions $u_i$, and  $v_i$, for $i=1,2$,  are defined. Then, there exists 
a positive constant $C$ such that for all $r\in I_1$ 
\begin{eqnarray} 
|u_1(r)-u_2(r)|&\le& C\max\{|c_1-d_1|, |c_2-d_2|\}, \nonumber\\
|v_1(r)-v_2(r)|&\le& C\max\{|c_1-d_1|, |c_2-d_2|\},\end{eqnarray}
and
\begin{eqnarray}
r|u'_1(r)-u'_2(r)|&\le& C\max\{|c_1-d_1|, |c_2-d_2|\}, \nonumber\\
r|v'_1(r)-v'_2(r)|&\le& C\max\{|c_1-d_1|. |c_2-d_2|\}.
\end{eqnarray}
\end{lemma}


\noindent{\bf Proof.}
 Consider  $I_1$  as in the hypothesis of this lemma. Define   
 $b:=\max\{|c_1-d_1|, |c_2-d_2|\}$,  then integrating twice  (\ref{pc}), and 
 since from proposition \ref{u'0}, $\lim_{r\to 0}ru'(r)=0, $  we get
\be \label{cotau'}
|u'_1(r)-u'_2(r)|\le  r^{1-N}\int_0^r s^{N-1} \alpha(s)|f(v_1(s))-f(v_2(s))| ds, 
\ee
and
\be \label{dependencia}
|u_1(r)-u_2(r)|\le b + H,
\ee
where 
$$
H:=\int_0^r t^{1-N}\left[\int_0^t s^{N-1} \alpha(s)|f(v_1(s))-f(v_2(s))| ds
\right]dt.
$$
Since 
$$
|f(v_1(r))-f(v_2(r))|\le m|v_1(r)-v_2(r)| \quad \mbox{for any } \quad 
r\in [0, r_1], $$
where we take $ m$  as a Lipschitz constant for $f$ and $g$, 
from  (\ref{dependencia}) we get 
\be \label{dependencia1}
|u_1(r)-u_2(r)|\le b+ m\int_0^r t^{1-N}\left[\int_0^t s^{N-1}\alpha(s)
|v_1(s)-v_2(s)| ds\right]dt. \ee
Arguing as above, but  now with the second equation of (\ref{pc}), we obtain 
\be \label{dependencia2}
|v_1(r)-v_2(r)|\le b + m\int_0^r t^{1-N}\left[\int_0^t s^{N-1}\beta(s)|u_1(s)-
u_2(s)| ds\right]dt. 
\ee
Define 
\begin{eqnarray}
X(r)&:=&b + m\int_0^r t^{1-N}\left[\int_0^t s^{N-1}\alpha(s)|v_1(s)-v_2(s)| ds
\right]\,dt\,,\\
Y(r)&:=& b + m\int_0^r t^{1-N}\left[\int_0^t s^{N-1}\beta(s)|u_1(s)-u_2(s)| ds
\right]\,dt\,. 
\end{eqnarray}
The functions $X$ and $Y$ are nondecreasing functions such that $X(0)= Y(0)=b$,
and
\be \label{sistemaXY}
\begin{array}{rll}
(r^{N-1}Y')'(r)&=&  mr^{N-1}\beta(r)|u_1(r)-u_2(r)|\le mr^{N-1}\beta(r)X(r),\\[5pt]
(r^{N-1}X')'(r)&=&  mr^{N-1}\alpha(r)|v_1(r)-v_2(r)|\le mr^{N-1}\alpha(r)Y(r), \\
\end{array}
\ee
and thus using that $Y$ is increasing in the second inequality of 
(\ref{sistemaXY}), we get
\be \label{cotaXY}
X(r)\le b+ mC_{\alpha} Y(r), 
\ee
where $C_{\alpha}:=(1/(N-2)\int_0^{r_1}s\alpha(s)ds.$
Using (\ref{cotaXY}) in  the first inequality of (\ref{sistemaXY}), we have that
\be \label{Y}
(r^{N-1}Y')'(r)\le mr^{N-1}\beta(r)\left(b+mC_{\alpha}Y(r)\right).
\ee
Integrating twice (\ref{Y}) from $0$ to $r$, we easily get
\be \label{gronwall}
Y(r)\le b(1+mC_{\beta})+ \frac{m^2C_{\alpha}}{N-2}\int_0^r t\beta(t)Y(t)dt,
\ee
where $C_{\beta}$ is define by replacing $\beta$ for $\alpha$ in the definition 
of $C_{\alpha}$.  From Gronwall's  inequality and (\ref{gronwall}), we deduce 
$$
Y(r)\le b(1+mC_{\beta})\exp\left\{ \frac{m^2C_{\alpha}}{N-2}\int_0^r
t\beta(t)dt\right\}
\le b(1+mC_{\beta})\exp\{ m^2C_{\alpha}C_{\beta}\},
$$
and similarly for $X$, and thus the conclusion follows from the above 
inequality, (\ref{dependencia1}) and (\ref{dependencia2}). For the bound of 
$u'$, we use (\ref{cotau'}), and the bound for $v$.   \hfill$\diamondsuit$

\section{Existence of grounds states}
In this section we prove the existence of a curve of ground states for the 
system (\ref{gmmy}), under the following conditions:
 the functions $f\in C({\mathbb R})$ and $g\in C({\mathbb R})$ are increasing functions, 
 Lipschitz continuous such that $f(0)=0$, $g(0)=0$. Moreover, we assume  the 
 existence of two positive constants $p$ and $q$ such that for $u\ge 0$ and 
 $v\ge 0$
\be \label{potencia}
vf(v)\ge (p+1)F(v), \quad ug(u)\ge (q+1)G(u),
\ee
where  $$F(v)=\int_0^vf(t)dt, \quad G(u)=\int_0^ug(t)\,dt\,.$$

For the functions $\alpha$ and $\beta$ we assume that they are nonnegative $C^1$
 functions defined in $(0, \infty)$ and such that there exist $\alpha_o$ and 
 $\beta_o$ such that
\be \label{decrecimiento}
r^{\alpha_o}\alpha(r), \quad \mbox{and} \quad r^{\beta_o}\beta(r) \quad 
\mbox{are non-increasing functions,}
\ee
and
\be \label{hiperbola}
\frac{N-\alpha_o}{p+1}+\frac{N-\beta_o}{q+1} \le N-2. 
\ee 
Moreover, we assume that (\ref{breli}) and (\ref{cotaalfabetaen0}) are 
satisfied.

Nonexistence results can be found in \cite{mi1}, \cite{sz1} and the references 
therein.

Consider the system
\begin{eqnarray} \label{pc1}
-u''(r) -\frac{N-1}{r}u'(r)&=& \alpha f(v),\\
-v''(r) -\frac{N-1}{r}v'(r)&=& \beta g(u),\nonumber
\end{eqnarray}
and let $=(c_1, c_2)$ be such that $c_1>0$ and $c_2>0$, and $(u, v)$ be the 
solution to (\ref{pc1}) such 
that $(u(0), v(0))= (c_1, c_2)$.  We define the subsets 
 \begin{eqnarray*}
&&{\cal U}_0:=\{(c_1, c_2) \in {\mathbb R}^+\times {\mathbb R}^+ | \quad  \mbox{such that}
\quad u(\tau)=0 \quad \mbox{for some} \quad \tau>0\}, \\
&&{\cal V}_0:=\{(c_1, c_2) \in {\mathbb R}^+\times {\mathbb R}^+ | \quad  \mbox{such that}
\quad v(\tau)=0 \quad \mbox{for some} \quad \tau>0\},\\
&&{\cal G}:=\{(c_1, c_2)\in {\mathbb R}^+\times {\mathbb R}^+|\quad (u, v) \quad 
\mbox{ is a ground state}\}.
\end{eqnarray*}

  
Our main result of this section is the following.

\begin{theorem}\label{existenciadegroundstates} Assume that conditions 
$(\ref{breli})$, $(\ref{cotaalfabetaen0})$, $(\ref{potencia})$, 
$(\ref{decrecimiento})$ and $(\ref{hiperbola})$
 are satisfied. Then, there exists a function $h\in C({\mathbb R}^+, {\mathbb R}^+)$, 
 strictly increasing such that
\begin{enumerate}
\item[$1$.] For any $c_2>0$, the solution to $(\ref{pc})$ with $c_1=h(c_2)$, is a 
ground state.
\item[$2$.] ${\cal U}_0=\{(c_1, c_2) | \quad c_1<h(c_2)\}.$
\item[$3$.] ${\cal V}_0=\{(c_1, c_2) | \quad h(c_2)< c_1\}.$
\end{enumerate}
\end{theorem}
Before proving this theorem, we need some 
preliminary results. The following is a Pohozaev-Pucci-Serrin identity. The proof 
can be obtained by a direct computation.

\begin{proposition} \label{pohozaev} Let $(u, v)$ be a solution to $(\ref{pc1})$ 
in some interval $I$, and let  $a$ and $b$ be two  constants. For any $r\in I$ 
define 
\begin{eqnarray}\label{energia1}
 E_{ab}(r)&=& r^N\left[u'(r)v'(r) +ar^{-1}u'(r)v(r)\right] \\
&&+ r^N\left[br^{-1}v'(r)u(r) +\alpha F(v) + \beta G(u) \right].\nonumber 
\end{eqnarray}
Then, the derivative with respect to $r$ of $E_{ab}$ satisfies
\begin{eqnarray}
E'_{ab}(r)&=& (a+b+2-N)r^{N-1}u'(r)v'(r) + (r^N \alpha)'F(v)  \\
&& -ar^{N-1}\alpha vf(v)+(r^N \beta)'G(u)-br^{N-1}\beta ug(u). \nonumber 
\end{eqnarray}
\end{proposition} 

Next, we study the sets ${\cal U}_0$ and ${\cal V}_0.$

\begin{theorem} \label{g} Assume that conditions in theorem 
\ref{existenciadegroundstates} are satisfied.   We have the following: 
\begin{enumerate}
\item[$(i)$] If $(c_1, c_2)\in {\cal U}_0$ and  $(u, v)$ is the solution to 
$(\ref{pc})$ defined on his maximal right hand side  interval $I=[0, k)$, then 
there exists $\tau \in I$ such that
\begin{eqnarray} \label{A}
u(\tau)=0, && u(r)(\tau-r)>0 \quad r \in I\setminus \{\tau\} \nonumber \\
v(r)>0 \quad r\in I, && v'(r)<0 \in (0, \tau) \\
\lim\limits_{r\to k}u(r)=-\infty , && \lim\limits_{r\to k}v(r)=\infty. \nonumber
\end{eqnarray}
 
\item[$(ii)$]If $(c_1, c_2)\in {\cal V}_0$ and  $(u, v)$ is the solution to 
$(\ref{pc})$ defined on his maximal right hand side  interval $I=[0, k)$, then 
there exists $\tau \in I$ such that
\begin{eqnarray*}
v(\tau)=0, && v(r)(\tau-r)>0 \quad r \in I\setminus \{\tau\} \nonumber \\
u(r)>0 \quad r\in I, && u'(r)<0 \in (0, \tau) \\
\lim\limits_{r\to k}u(r)=\infty , && \lim\limits_{r\to k}v(r)=-\infty. \nonumber
\end{eqnarray*}

\item [$(iii)$] ${\mathbb R}^+\times {\mathbb R}^+= {\cal G} \cup {\cal U}_0 \cup{\cal V}_0$.  
\end{enumerate}
\end{theorem} 

\noindent {\bf Proof.} If $(u, v)$ is not a ground state, from proposition 
\ref{groundstate} either $u$ or $v$  has a zero at some point $\tau<\infty$, 
and thus (iii) is proved.

 Let $(u, v)$ be a solution to (\ref{pc}) defined on his maximal right hand 
 side interval $I= (0,k)$, (positive or not)  and such that $u(\tau)=0$ with $u$
  and $v$  positive functions in  $I'=(0, \tau)$.  

\noindent{\em Assertion.}  $r^{N-2}v$ is increasing on  $I$, and $u(r)<0$ for 
all $r\in (\tau, k).$

We prove the assertion in two steps.

\noindent{\em Step 1.} $r^{N-2}v$ is increasing on  $I'.$
 
The functions $u $ and $v$ are non-increasing functions on $I'$.  For $a$ 
and $b$ such that $b \ge (N-\beta_o)/(q+1)$, $a\ge (N-\alpha_o)/(p+1)$,
and $a+b\le N-2$; we obtain that $E'_{ab}(r)$ is non-positive for all 
$r\in I'$.  In that case  the energy is  a decreasing function of $r$ and thus, 
$E_{ab}(r) \le E_{ab}(0)$.  Next, we prove that $E_{ab}(0)=0$.   From 
Proposition~\ref{u'0}, the three first terms on $E'_{ab}(r)$, goes to zero 
as $r$ does. The convergence to zero of the other two terms, follows from 
Proposition~\ref{r2alfa}; Thus for all $r\in (0, \tau)$
\be \label{e-}
E_{ab}(r) \le 0.
\ee
For $r\in (0, \tau)$, let
\be \label{de}
d(r):= -\frac{ru'(r)}{u(r)} \quad \mbox{and} \quad e(r) := -\frac{rv'(r)}{v(r)}. 
\ee
 From (\ref{e-})  in  particular we obtain
\be \label{ed}
e(r)d(r) -ad(r) - be(r) <0.  
\ee
As a consequence of $u(\tau)=0$ and the concave nature of $w_1(s):=su(r)$, 
$s=r^{N-2}$,  there exists  $r_1<\tau$ satisfying  $d(r_1)=N-2$, and  for all 
$r\in (r_1, \tau)$, $d(r)>N-2$. Therefore,
 \be \label{emenor}
\frac{N-\alpha_o}{p+1}+\frac{N-\beta_o}{q+1} \le d(r_1) = N-2.
\ee
For this $d(r_1)$ we can choose $a, b$ such that $a+b= d(r_1)$ and 
$\frac{N-\alpha_o}{p+1} \le a$, $\frac{N-\beta_o}{q+1}\le b$. Returning to 
(\ref{ed}) we get $e(r_1) < d(r_1)= N-2$.  On the other hand, the function 
$w_2(s):=sv(r),\quad s= r^{N-2}$ is a concave function with $w_2(0)=0$  and 
thus, $e(r) < N-2$  for all $r \in [0, r_1]$.   Moreover, if for some $r_2>r_1$ 
we have that $e(r_2)=N-2$, then arguing as above we get that $d(r_2)<N-2$, 
which is a contradiction. Thus
 we conclude that $e(r)<N-2$ for all $r\in (0,\tau).$
Therefore, $r^{N-2}v(r)$ is an increasing function in $ (0,\tau)$.  In particular
 $v(\tau)>0.$

\noindent{\em Step 2.} 
 Next, we will prove that $u(r)<0$ for all $r\in (\tau,k)$, and $v(r)>0$ for all
  $r\in I$.  For it, let $w_2(s):=r^{N-2}v(r)$ with $s=r^{N-2}$ and 
  $s\in (\tau^{N-2}, k^{N-2})$.  Assume that  $u(r)<0$ for $r\in (\tau, x)$. 
  Since $g(u)\le 0 $, for $u\le 0$, it can be easily verified that $w_2$ is a 
  convex function for all $s\in(\tau^{N-2}, x^{N-2})$, and thus in such interval
$$
\frac{dw_2}{ds}(s)\ge \frac{dw_2}{ds}(\tau^{N-2})\ge 0, 
$$
which in turn implies that $w_2(s)\ge w_2(\tau^{N-2})>0$, for all 
$s\in(\tau^{N-2}, x^{N-2})$.  Now, consider the function $w_1(s):=r^{N-2}u(r)$.  
Using now, that $w_2(s)>0$ and thus $v(r)>0$ for all $r\in (0, x)$, it can be 
verified that the function $w_1$ is concave in $(0, x)$.  We have used that 
$f(v)\ge 0$, for $v\ge 0$.  Therefore, in particular
$$
\frac{dw_1}{ds}(s)\le \frac{dw_1}{ds}(\tau^{N-2})<0, 
$$
for all $s\in(\tau^{N-2}, x^{N-2})$, and thus $w_1(s)$ is decreasing for 
$s>\tau^{N-2}$ and it cannot be zero for $s>\tau^{N-2}$.  Also in the above 
argument we can take $x=k$ to conclude that $v$ is  positive and $r^{N-2}v(r)$ 
is increasing on $I$. Moreover, from (\ref{pc}) and since $v$ is positive we 
obtain that $u$ is decreasing on $I$.  Thus the assertion follows. 

By changing the role between $u$ and $v$ in the above assertion, 
we obtain that ${\cal U}_0\cap{\cal V}_0=\emptyset.$
Next, we will show that 
$$
\lim_{r\to k}u(r)= -\infty, \quad \lim_{r\to k}v(r)= \infty.
$$
 Assume first that  $k<\infty$.  Since  $r^{N-2}v(r)$ is increasing, we obtain  
 the existence of $\lim_{r\to k}v(r)=l$.  Assume, by contradiction that 
 $l<\infty$, and thus  $v$ is bounded. Integrating the first equation of 
 (\ref{pc}) and using the facts that  $v$  bounded and $k<\infty$, it can be 
 easily verified that $u$ is also bounded. Therefore, $(u, v)$  can be extended 
 to the right of $k$, contradicting the definition of $k$.  The proof that 
 $\lim_{r\to k}u(r)=-\infty$ is analogous to the above.

Now, for $k=\infty$,  we prove first that 
$l=\lim_{r \to \infty }v(r)= \infty$.  Integrating twice the first 
equation of (\ref{pc}) we obtain 
 for any $r>2\tau$,
\begin{eqnarray}\label{vsevaainfinito}
 v(r)-v(2\tau)&=&-\int_{2\tau}^r s^{1-N}\left[\int_0^s t^{N-1}\beta(t)g(u(t))dt
 \right]ds  \nonumber \\
&=& \frac{1}{N-2}[r^{2-N}-{(2\tau)}^{2-N}]
\left[\int_0^{2\tau} t^{N-1}\beta(t)g(u(t))dt\right] \nonumber \\   
&&-\int_{2\tau}^rs^{1-N}\left[\int_{2\tau}^st^{N-1}\beta(t)g(u(t))dt\right]ds.
\end{eqnarray}
From (\ref{vsevaainfinito}) and since $|g(u(s))|=-g(u(s))\ge |g(u(2\tau))|$, 
for all $s\ge 2\tau$,  $l=\infty$ is implied by
$$
\lim_{r\to
\infty}\int_{2\tau}^rs^{1-N}\left[\int_{2\tau}^st^{N-1}\beta(t)dt\right]ds=
\infty.
$$
If $r>4\tau$, we get
\begin{eqnarray}\label{betadoble}
\int_{2\tau}^rs^{1-N}\left[\int_{2\tau}^st^{N-1}\beta(t)dt\right]ds &\ge & 
\int_{4\tau}^rs^{1-N}\left[\int_{2\tau}^st^{N-1}\beta(t)dt\right]ds  \nonumber \\
&\ge& \int_{4\tau}^rs^{1-N}\left[\int_{s/2}^st^{N-1-\beta_o}t^{\beta_o}
\beta(t)dt\right]ds \nonumber \\
&\ge& C\int_{4\tau}^rs\beta(s)\,ds\,, 
\end{eqnarray}
in the last inequality of (\ref{betadoble}) we have used that 
$t^{\beta_o}\beta(t)$ is non-increasing. Thus, from (\ref{betadoble}) and 
(\ref{breli}) we get that $l=\infty$.  A similar argument with 
\begin{eqnarray*}
-u(r)&=& \int_{\tau}^rs^{1-N}\left[\int_0^s t^{N-1}\alpha(t)f(v(t))dt\right]ds\\
&\ge & f(c)\int_{\tau}^rs^{1-N}\left[\int_0^s t^{N-1}\alpha(t)dt\right]ds,
\end{eqnarray*}
shows that $\lim_{r\to \infty}u(r)=\infty$.  \hfill$\diamondsuit$

As a consequence of the above result we get the following for the particular 
system 
\begin{eqnarray} \label{iguales}
-\Delta u&=&\alpha f(v),\\
-\Delta v&=&\alpha f(u).\nonumber
\end{eqnarray}
\begin{corollary} Assume that $f=g$ and $\alpha=\beta$ satisfy conditions in 
theorem~\ref{existenciadegroundstates}. Then, 
$$
{\cal G}=\{(c, c)|\quad c>0\}.
$$
\end{corollary}
\noindent{\bf Proof.} Let $c>0$, and let $u$ be a nonnegative solution to 
$$
-\Delta u=\alpha f(u),
$$
defined near zero and such that $u(0)=c$.  Thus, since $(u, u)$ is a solution to 
(\ref{iguales}) with $u=v$, we get that $(c, c)\notin {\cal U}_0\cup {\cal V}_0$
 (solutions with initial data in ${\cal U}_0\cup {\cal V}_0$ satisfy that only
  one between $u$ and $v$ have a zero). Therefore, $(c, c)\in {\cal G}.$ \hfill$\diamondsuit$
\medskip

With some extra conditions, it can be proven that the ground states are the 
unique solutions to (\ref{pc}) defined in $[0, \infty)$.  

\begin{theorem} Assume that conditions in theorem \ref{g} are satisfied. 
Moreover, assume that $f$ and $g$ are odd functions. If  there exist constants 
$\alpha_1$, $\beta_1$ and a positive constant $C$ such that 
$$
\alpha(r)\ge Cr^{-\alpha_1}, \quad \beta(r)\ge Cr^{-\beta_1}, \quad 
\mbox{for all $r$ large}
$$
with 
\be \label{gamas}
\min \left\{ \gamma_1(\alpha_1, \beta_1),\gamma_2(\alpha_1,\; \beta_1) 
\right\}\le 0,
\ee 
then for any $(c_1, c_2)$ with $c_1>0$ and $c_2>0$, we have that the 
corresponding solution $(u, v)$ is either a ground state or the maximal right 
hand side interval is  $(0, k)$ with $k<\infty.$
\end{theorem}
\noindent {\bf Proof.} Let $(u, v)$ be a solution which is not a ground state, 
and thus from theorem \ref{g}, we can assume the existence of $\tau$ such that 
$u(\tau)=0$, $u(r)<0$ for all $r\in (\tau, k)$, and $v(r)>0$ for all 
$r\in (0, \tau)$.  If $k=\infty$, let $u_1(r)=-u(r)$, thus  $(u_1, v)$ is a 
positive solution in an exterior domain to 
\be \label{cy}
\begin{array}{rlll}
\Delta{u_1} & \ge & C|x|^{-\alpha_1} f(v) \cr
 \Delta{v} & \ge & C|x|^{-\beta_1} g(u_1).\cr
\end{array}
\ee
 Now, since $u_1(r)$ and $v(r)$ tend to infinity as $r$ does, we have 
 $f(v)\ge Cv^p$ and $g(u_1)\ge Cu_1^q$, and thus $(u_1, v)$ is a solution to 
$$
\begin{array}{rlll}
\Delta{u_1} & \ge & C|x|^{-\alpha_1} v^p \cr
 \Delta{v} & \ge & C|x|^{-\beta_1} u_1^q.\cr
\end{array}
$$
From the hypothesis (\ref{gamas}) and  Theorem 3.1 or Theorem 3.2 in \cite{y}, 
we get that either $u_1$  or $v$ are bounded, and the contradiction follows. 
Therefore, $k<\infty$.   \hfill$\diamondsuit$
\medskip
 
The following lemmas allow us to define the function $h$ given in theorem 
\ref{existenciadegroundstates}.

\begin{lemma} \label{alomas}Assume that conditions on theorem 
\ref{existenciadegroundstates} are satisfied. 
 Then, for any $c_1>0$ (respectively $c_2>0$) there exists at most one $c_2>0$ 
 (respectively $c_1>0$) such that $(c_1, c_2) \in {\cal G}.$
\end{lemma}
\noindent{\bf Proof.}  Let $c_1>0$.  Assume that there exists $c_2$ such that  
$(u_1, v_1)$ is a ground state with $u_1(0)=c_1$ and $v_1(0)=c_2$.  Let us prove 
that solutions $(u, v)$ of (\ref{pc}) corresponding to $u(0)=c_1$ and 
$v(0)=  c_2-\delta$, with $0<\delta<c_2$, are not ground states. Assume by contradiction that some of the above $(u, v)$ is also a ground state. Consider 
$[0, r_1)$  be the maximal right hand side interval where $v_1(r)>v(r)>0$,  for 
all $r\in[0, r_1)$.  Integrating twice (\ref{pc}) we get for all $r\in[0, r_1)$
$$
u(r)-u_1(r)=\int_0^r s^{1-N}\left[\int_0^s t^{N-1} 
\alpha(t)(f(v_1(t))-f(v(t)))dt\right]ds >0,
$$
and thus, $u(r)>u_1(r)$, for all $r\in[0, r_1)$.  Moreover, in such interval,
$$
-r^{N-1}(v'(r)-v'_1(r))=\int_0^r t^{N-1} \beta(t)(g(u(t))-g(u_1(t)))dt]ds >0,
$$
 implying that $v_1(r)-v(r)$ is increasing in $[0, r_1)$, and thus 
\be \label{delta}
v_1(r)-v(r)> \delta.
\ee
If $r_1<\infty$, then from the definition of $r_1$ and  (\ref{delta}) we get 
$v(r_1)=0$, and thus $(u, v)$ is not a ground state. If $r_1=\infty$, the 
contradiction follows from (\ref{delta}) and proposition \ref{groundstate}. 
Therefore, if  $(c_1, c_2)$ corresponds to a ground state, then  
$(c_1, c_2-\delta)$ does not correspond to a ground state, and thus 
$(c_1, c_2+\delta)$ also cannot correspond to a ground state (if it does, 
$(c_1, c_2)=(c_1, c_2+\delta-\delta)$ does not). \hfill$\diamondsuit$ 
\medskip

Using similar arguments to those used in the proof of the above lemma, we have.

\begin{lemma} \label{crecimiento} Assume that conditions on theorem 
\ref{existenciadegroundstates} are satisfied. Then,
\begin{enumerate}
\item[$(i)$] If $(c_1, c_2)\in {\cal  V}_0$, then  $[ c_1, 
\infty)\times(0, c_2)\subset {\cal  V}_0$.  
\item[$(ii)$] If $(c_1, c_2)\in 
{\cal U}_0$, then $(0, c_1)\times [c_2, \infty)\in {\cal U}_0$.  
\end{enumerate}
\end{lemma}

 \begin{lemma}\label{novacios} Assume that conditions on theorem 
 \ref{existenciadegroundstates} are satisfied. 
 Then, 
\begin{enumerate}
\item[$(i)$] For any $c_1>0$ there exists a  $\bar c_2>0$  such that 
$\{c_1\}\times(0, \bar c_2) \subset {\cal V}_0.$
\item[$(ii)$] For any $c_2>0$ there exists a  $\bar c_1>0$  such that 
$(0, \bar c_1)\times\{ c_2\} \subset {\cal U}_0.$
\end{enumerate}
\end{lemma}
\noindent{\bf Proof.}
We prove (i). Assume by contradiction that for some  $c_1>0$ there exists no 
$\bar c_2$ such that $\{c_1\}\times(0, \bar c_2)\subset  {\cal  V}_0$. In that 
case, we choose a positive sequence $\{c_{2n}\} $ decreasing to zero such that 
 $(c_1, c_{2n}) \in {\cal U}_0$.  We call $(u_n, v_n)$, the solution to 
 (\ref{pc}) corresponding to $u_n(0)=c_1$ and $v_n(0)=c_{2n}$.  If $\tau_n$ is 
 the point where $u_n(\tau_n)=0$, we obtain
 $0\le v_n(r)\le c_{2n} $ for all  $r\in (0, \tau_n)$  and
$$
c_1 = \int_0^{\tau_n}s^{1-N}\left[\int_0^s t^{N-1}\alpha(t)f(v(t))dt\right]ds \le 
\frac{f(c_{2n})}{N-2}\int_0^{\tau_n}s\alpha(s)ds,
$$
and thus $\tau_n$ tends to infinity as $n $ does. 

Next, we will prove that for any $a>0$,  $(\{u_n\}_{n}, \{v_n\}_{n}) $ has a 
subsequence which converges uniformly in $[0,a]$ to a solution  $(u, v)$ of 
(\ref{pc}). Let $n$ be large enough such that $\tau_n>a$, and thus $0\le v_n(r)
\le c_{2n}$, for all $r\in (0, a)$.  Since  $\{c_{2n}\}_{n}$ tends to zero as 
$n$ tends to infinity, we get that  that $v\equiv 0$, which is a contradiction 
to $u(0)=c_1$.   We prove the existence of such a converging subsequence by 
showing  that $(\{u_n\}_{n}, \{v_n\}_{n}) $ is equicontinuous and bounded in 
$[0, a]$.  We  have
\be \label{vnuv}
0 \le v_n(r) \le c_{2n},  \quad 0\le u_n(r)\le c_1 \quad r\in (0, a),
\ee
and thus integrating twice (\ref{pc}) we get
\be \label{un-u}
|u_n(r)-u_n(x)|\le \int_r^x s^{1-N}\left[\int_0^s t^{N-1}\alpha(t)f(v_n(t))dt
\right]ds,
\ee
Therefore from (\ref{un-u}) and (\ref{vnuv})
\be \label{un-u2}
|u_n(r)-u_n(x)|\le f(c_{2n})\int_r^x s^{1-N}\left[\int_0^s t^{N-1}\alpha(t)dt
\right]ds,
\ee
and thus $\{u_n\}$ is equicontinuous.
Similarly, we obtain
$$
|v_n(r)-v_n(x)|\le  g(c_{1})\int_r^x s^{1-N}\left[\int_0^s t^{N-1}\beta(t)dt
\right]ds.
$$
The proof of (ii) follows the same ideas given on (i). \hfill$\diamondsuit$

 \begin{lemma} \label{supremo}Assume that conditions on theorem 
 \ref{existenciadegroundstates} are satisfied. 
 Then, 
\begin{enumerate}
\item[$(i)$] For any $c_2>0$ there exists $c_1^+$, such that $( c_1^+, \infty)
\times\{c_2\}\subset {\cal  V}_0.$
\item[$(ii)$] For any $c_1>0$ there exists $c_2^+$, such that  $\{c_1\}\times 
(c_2^+, \infty)\subset {\cal U}_0.$
\end{enumerate}
\end{lemma}

\noindent {\bf Proof.} Assume by contradiction the existence of $c_2>0$ and a 
sequence $\{ c_{1n}\}$ increasing to infinity such that $(c_{1n}, c_2)\in 
{\cal U}_0$. We consider $r_n$ to be such that $u_n(r_n)=c_{1n}/2$, where 
$(u_n, v_n)$ is the solution to (\ref{pc}) with 
$(u_n(0), v_n(0))=(c_{1n}, c_2)$.  Therefore, since $v_n$ is decreasing before 
the zero of $u_n$, we get
\be \label{c1grande}
\begin{array}{rll}
\frac{c_{1n}}{2}&=& u_n(0)-u_n(r_n)=\int_0^{r_n}s^{1-N}[\int_0^s t^{N-1}
\alpha(t)f(v_n(t))dt]ds\\
&\le & f(c_2) \int_0^{r_n}s^{1-N}[\int_0^s t^{N-1}\alpha(t)dt]ds,
\end{array}
\ee
and thus $r_n$ tends to infinity as $n$ does. On the other hand, $v_n(r_n)>0$ 
implies
\begin{eqnarray*}
c_2&>&\int_0^{r_n}s^{1-N}\left[\int_0^s t^{N-1}\beta(t)g(u_n(t))dt\right]ds\\ &\ge & 
g(c_{1n}/2)\int_0^{r_n}s^{1-N}\left[\int_0^s t^{N-1}\beta(t)dt\right]ds,
\end{eqnarray*}
which is a contradiction to (\ref{c1grande}). \hfill$\diamondsuit$
 
 \begin{lemma} \label{abiertos} Assume that conditions on theorem 
 \ref{existenciadegroundstates} are satisfied. 
 Then, ${\cal U}_0$ and ${\cal V}_0$ are open subsets of ${\mathbb R}^+
 \times{\mathbb R}^+.$

\end{lemma}
\noindent{\bf Proof.} Let $(d_1, d_2)\in {\cal U}_0, $ then $u(\tau_d)=0$, for 
some $\tau_d>0$,  and thus
\begin{eqnarray} \label{definidasenunintervalo}
d_1&=&\int_0^{\tau_d}s^{1-N}[\int_0^s t^{N-1}\alpha(t)f(v(t))dt]ds \nonumber \\ 
    &\le & f(d_2)\int_0^{\tau_d}s^{1-N}[\int_0^s t^{N-1}\alpha(t)dt]ds  \\
&\le & \frac{f(d_2)}{N-2}\int_0^{\tau_d}s\alpha(s)ds.\nonumber
\end{eqnarray}
Similarly if $(d_1, d_2)\in {\cal V}_0, $ we get 
\be \label{definidasenunintervalo2}
d_2 \le  \frac{g(d_1)}{N-2}\int_0^{\tau_d}s\beta(s)ds.
\ee
Now, let ${\bf c}:=(c_1, c_2)\in {\cal U}_0$, and let ${\bf d}:=(d_1, d_2)\in 
{\mathbb R}^+\times {\mathbb R}^+$, with $|c_i-d_i|<\epsilon$, for $i=1,2$.  Consider 
$(u_1, v_1)$ to be the solution to (\ref{pc}) associated to ${\bf c}$ and 
$(u, v)$ the corresponding ${\bf d}$.  Then, if $\epsilon<\max\{c_1/2, c_2/2\}$,
  from  (\ref{definidasenunintervalo}) and (\ref{definidasenunintervalo2})we 
  get that the solution $(u, v)$ is at least defined on $I_1=[0, \tau_1]$, 
  where $\tau_1$ satisfies
$$
\frac{c_1(N-2) }{2f(3c_2/2)}\le  \int_0^{\tau_1}s\alpha(s)ds, \quad \mbox{and} 
\quad\frac{c_2(N-2)}{2g(3c_1/2)} \le  \int_0^{\tau_1}s\beta(s)ds.
$$
Therefore, from Lemma \ref{depen} for the above interval $I_1$ we get that 
$|u_1(\tau_1)-u(\tau_1)|<\epsilon_0$, $|u'_1(\tau_1)-u'(\tau_1)|<\epsilon_0$, 
$|v_1(\tau_1)-v(\tau_1)|<\epsilon_0$, and $|v'_1(\tau_1)-v'(\tau_1)|<\epsilon_0,
$ where $\epsilon_0$ is  small. Thus, from classical continuous dependence 
results for  $(u_1, v_1)$ defined in $[\tau_1, k_0]$, for any $k_0<k$, we get 
that the solution $(u, v)$  is also defined in $[\tau_1, k_0]$, and it is near 
to $(u_1, v_1)$.   Since, $u_1$ is negative at some point, we  conclude that $u$
 must be negative at that point. \hfill$\diamondsuit$
 \bigskip

\noindent {\bf Proof of theorem \ref{existenciadegroundstates}.}
\medskip
\noindent We first construct the function $h$.
Let $c_2>0$, thus from lemma \ref{novacios} and lemma \ref{supremo} we obtain
$$
0 < h(c_2):= sup\left \{c_1>0 \quad | \quad (0, c_1)\times \{c_2\}\subset 
{\cal U}_0 \right \}<\infty.
$$
Next, we will prove that $(h(c_2), c_2)$ corresponds to a ground state.
Since by lemma \ref{abiertos} ${\cal U}_0$ is open and $h(c_2)$ is a supremum, 
we get that $(h(c_2), c_2)\notin {\cal U}_0$. If  $(h(c_2), c_2)\in 
{\cal  V}_0$, and since ${\cal  V}_0$ is open $(h(c_2)-\epsilon, c_2)\in 
{\cal  V}_0$, for some $\epsilon$ small, which  is a contradiction to the 
definition of $h(c_2)$, and thus $(h(c_2), c_2)$ correspond to a ground state.
 
The proof that  $h$ is increasing is as follows.
Let $c^*_2>c_2$.  From the definition of $h(c_2)$ we get that $(0, h(c_2))
\times \{c_2\}\subset {\cal U}_0$,  and thus from lemma \ref{crecimiento} 
$(0, h(c_2))\times [c_2, \infty)\subset {\cal U}_0$.     Therefore from the 
definition of $h(c^*_2)$ we get $h(c_2) \le h(c^*_2)$.  Now, since 
$(h(c_2), c_2)$ and  $(h(c^*_2), c^*_2)$ are ground states, from lemma 
\ref{alomas} $h(c_2) < h(c^*_2)$.

We end by showing item 1 and 2 of the theorem.
Let $(c_1, c_2)$ such that $c_1<h(c_2)$, from the definition of $h(c_2)$, we 
have that $(c_1, c_2)\in {\cal U}_0$.  Assume now that $c_1>h(c_2)$.  If 
$(c_1, c_2)\in {\cal U}_0$, from  lemma \ref{crecimiento}, $(h(c_2), c_2)$ also
 belongs to ${\cal U}_0$, which is a contradiction, and thus $(c_1, c_2)\in 
 {\cal  V}_0$, for any $c_1>h(c_2).$ \hfill$\diamondsuit$
\medskip

The following result has to do with  the particular system 
\begin{eqnarray} \label{potencias}
u''(r)+\frac{N-1}{r}u'(r)&=&-r^{-\alpha_o}v^p,\nonumber\\
v''(r)+\frac{N-1}{r}v'(r)&=&-r^{-\beta_o}u^q,\\
u(0)=c_1&& v(0)=c_2.\nonumber
\end{eqnarray}
\begin{corollary}Let $(f(v), g(u))=( v^p, u^q)$, with $p\ge 1$, $q\ge 1$.  Assume
 that $(\alpha(r), \beta(r))= (r^{-\alpha_o}, r^{-\beta_o}) $ with $\alpha_o<2$,
  $\beta_o<2$, and $(\alpha_o, \beta_o, p, q)$ satisfies $(\ref{hiperbola})$. 
  Then, there exists a positive constant $C_0$ such that $h$ given on theorem 
  \ref{existenciadegroundstates} is
$$
 h(x)=C_0x^{(\alpha_o-2+(\beta_o-2)p)/(\beta_o-2+(\alpha_o-2)q)}. $$
\end{corollary}

\noindent {\bf Remark.} The above result was proven by Serrin and Zou \cite{sz},
 when $\alpha_o=\beta_o=0$.  The behavior at infinity of the ground states when 
 (\ref{hiperbola}) is an equality an $\alpha_o=\beta_o=0$ is given in 
 \cite{hvv}. 
\medskip

 \noindent{\bf Proof.} Let $x$ positive, then $(h(x), x)$ is an initial value 
 for  a  ground state. If $(u, v)$ is such ground state, then for any positive 
 constants $a,$ $b$ and $k$, we define  $(w, y)$ by
$$
w(r)=au(kr), \quad y(r)=bv(kr).
$$
Thus, $(w, y)$ is a ground state to 
\begin{eqnarray*}
&&\Delta w+ab^{-p}k^{2-\alpha_o}r^{-\alpha_o}y^p=0, \\
&& \Delta y+ba^{-q}k^{2-\beta_o}r^{-\beta_o}w^q=0.
\end{eqnarray*}
If $ab^{-p}k^{2-\alpha_o}=1$,  and $ba^{-q}k^{2-\beta_o}=1$,
which implies that
$$
a=b^{\nu}    \quad \mbox{where} \quad \nu={\frac{\alpha_o-2+(\beta_o-2)p}
{\beta_o-2+(\alpha_o-2)q}}.
$$
In that case,  $(w, y)$ is a ground state to (\ref{potencias}) 
and $(w(0), y(0))=(b^{\nu}h(c), bc)$.  Let $x$ be positive 
and choose $b$ such that $x=bc$. Since the ground state corresponding to $x$ is
 unique we get that 
$$
h(x)=C_0x^{\nu},$$
where $C_0=h(c)c^{-\nu}$. \hfill$\diamondsuit$

\section{Existence of singular ground states}

In this section we study the problem of existence of singular ground states for 
(\ref{gmmy}). A singular ground state means a radially symmetric nonnegative 
solution $(u, v)$ to (\ref{gmmy}) such that either $\lim_{r\to 0}u(r)=
\infty$, or $\lim_{r\to 0}v(r)=\infty$.

Next, we will prove an existence result of singular radially symmetric ground 
state as a limit of ground states constructed in Section 3. 

\begin{theorem} \label{gssingulares} Assume that condition on Theorem 
\ref{existenciadegroundstates} are satisfied and $pq>1$.  Moreover, assume that 
$$
f(v)\ge Cv^p, \quad v\ge 0, \quad g(u)\ge Cu^q, \quad u\ge 0,
$$
and for any $k$ positive
\be \label{condfalpha}
\int_{0}t^{N-1}\alpha(t)f(kt^{\gamma_2})dt<\infty, \quad \int_{0}t^{N-1}\beta(t)
g(kt^{\gamma_1})dt<\infty,
\ee
where $\gamma_1$ and $\gamma_2$ are defined by $(\ref{defgammas})$.
Then, there exists a nonnegative singular ground state to $(\ref{gmmy})$. 
Moreover, 
$$\lim_{r\to 0}u(r)=\infty,\quad \mbox{and} \quad \lim_{r\to 0}v(r)=\infty.$$
\end{theorem}
\medskip

\noindent{\bf Remark.} If in the above theorem $f(v)\le D v^p$, and $g(u)\le Du^q$, and 
$\alpha(r)\le Dr^{\alpha_o}$,  $\beta(r)\le Dr^{\beta_o}$, for some positive 
constant $D$, then condition (\ref{condfalpha}) is satisfy, since
$$
\int_{0}t^{N-1}\alpha(s)f(kt^{\gamma_2})dt\le C\int_{0}t^{N-1-\alpha_o+
p\gamma_2}dt<\infty.
$$
The last integral is finite since $N-\alpha_o+p\gamma_2=N-2+\gamma_1>0$, and we
 are in the region where $N-2+\gamma_1+\gamma_2\ge 0$ and $\gamma_1<0$ and 
 $\gamma_2<0.$
\medskip

\noindent{\bf Proof.} As usual it can be proved  that for any $(u, v)$ 
nonnegative radially symmetric solution to (\ref{gmmy}) we have
\be \label{cotauv}
u(r)\ge Cv^p(r)r^2\alpha(r),
\ee
In the proof of  (\ref{cotauv}) we have used that $r^{N-2}u(r)$ is nondecreasing
 and $r^{\alpha_o}\alpha(r)$ is non-increasing. Similarly, we get
\be \label{cotavu}
v(r)\ge Cu^q(r))r^2\beta(r),
\ee
and thus from (\ref{cotauv}) and (\ref{cotavu}) we obtain
\begin{eqnarray} \label{acotacion}
u^{pq-1}(r)&\le& \frac{C}{r^{2(p+1)}\beta^p(r)\alpha(r)},\\
v^{pq-1}(r)&\le& \frac{C}{r^{2(q+1)}\alpha^q(r)\beta(r)},
\end{eqnarray}
Moreover, for any interval $[0,a]$, there exist a constant $A>0$, depending on 
$a$ such that $r^{\alpha_o}\alpha(r)\ge A$, and $r^{\beta_o}\beta(r)\ge A$, for
 all $r\in [0,a]$.  Thus, we obtain from (\ref{acotacion}) that for all 
 $r\in [0,a]$ 
\begin{eqnarray} \label{acotacion1}
u(r)&\le& Kr^{\gamma_1},\\
v(r)&\le& Kr^{\gamma_2},
\end{eqnarray}
where $K$ is some positive constant depending on $a.$
 Let $(h(c_n), c_n)$ with $c_n>0$ converging to $\infty$, and $(u_n, v_n)$ the 
 associated ground state. From (\ref{acotacion}) the sequence $\{u_n, v_n)\}$ 
 is uniformly bounded in any compact subset of $(0, \infty)$. We will show that
  the sequence $\{u_n, v_n)\}$ is equicontinuous  away from zero. Let $x>0$ be 
  fixed, we prove equicontinuity on $x$.  
From the definition of $(u_n, v_n)$, we have
\begin{eqnarray*}
|u_n(r)-u_n(x)|&\le &\int_{r}^{x}s^{1-N}\left[\int_{0}^st^{N-1}\alpha(s)
f(v_n(t))dt\right]ds \nonumber \\
&\le &
\int_{r}^{x}s^{1-N}\left[\int_{0}^st^{N-1}\alpha(s)f(Kt^{\gamma_2})dt\right]ds, \\
\end{eqnarray*}
and thus from (\ref{condfalpha}) we get the equicontinuity of the sequence 
$\{(u_n, v_n)\},$
in $x$.  Therefore, there exists a subsequence of $\{(u_n, v_n)\}$, which 
converges uniformly on any compact subset of $(0, \infty)$ to a continuous 
function $(u, v)$.  Moreover, $(u, v)$ is a solution of (\ref{gmmy}) in the 
sense of distribution in ${\mathbb R}^N \setminus \{0\}$.   \hfill$\diamondsuit$

Next, by using Kelvin transform on the results of Section 3,
we prove existence of singular ground states for the system 
\be \label{gmmy2}
\begin{array}{rlll}
-\Delta{u} & = & {\alpha^*(|x|)}v^p \cr
 -\Delta{v} & = & \beta^*(|x|) u^q \cr
\end{array} \quad  x \in {\mathbb R}^N \setminus \{0\}, 
\ee
where, $p\ge 1$, $q\ge 1$ and $pq>1$, under the following conditions
\be \label{condicioneskelvin}
\int_0s^{N-1-(N-2)p}\alpha^*(s)ds =\infty,  \quad 
\int_0s^{N-1-(N-2)q}\beta^*(s)ds =\infty, 
\ee 
\be\label{condicioneskelvin1}
\int^{\infty}s^{N-1-(N-2)p}\alpha^*(s)ds <\infty,  \quad
\int^{\infty} s^{N-1-(N-2)q}\beta^*(s)ds <\infty.
\ee
Moreover, we will assume the existence of $\alpha'_{0}$ and $\beta'_{0}$ such 
that
\be \label{bajohiperbola}
r^{\alpha'_o}\alpha^*(r)\quad \mbox{and} \quad r^{\beta'_o}\beta^*(r) \quad 
\mbox{are nondecreasing and }
\ee
\be\label{bajohiperbola1}
\frac{N-\alpha'_o}{p+1}+\frac{N-\beta'_o}{q+1} \ge N-2. 
\ee
\begin{theorem} \label{existenciabajolahiperbola} Assume that conditions 
$(\ref{condicioneskelvin})$, $(\ref{condicioneskelvin1})$, 
$(\ref{bajohiperbola})$ and $(\ref{bajohiperbola1})$ are satisfied. Then, there 
exists a function $h^*\in C({\mathbb R}^+, {\mathbb R}^+)$, strictly increasing such that
 for any $c_2>0$ there exists a positive ground state  $(u, v)$ such that
\be \label{behavior}
\lim_{r\to \infty }r^{N-2}u(r)=h^*(c_2), \quad \mbox{and} \quad \lim_{r\to 
\infty }r^{N-2}v(r)= c_2.
\ee
 Moreover, assume that either $r^{\alpha'_o}\alpha^*(r)$ or $r^{\beta'_o}
 \beta^*(r)$ are strictly increasing at some point or $(\ref{bajohiperbola1})$ 
 is not an equality, then 
\be \label{comportamientosingular}
\lim_{r\to 0}u(r)=\infty, \quad \mbox{and} \quad \lim_{r\to 0}v(r)=\infty. 
\ee
\end{theorem}

\noindent{\bf Proof.} The proof of this result is fundamentally based on Kelvin 
transform and on theorems of the previous section. If  $(u, v)$ is a radially 
symmetric nonnegative solution to (\ref{gmmy2}), we define 
$$u_1(r):=r^{2-N}u(r^{-1}), \quad v_1(r):=r^{2-N}v(r^{-1}),$$  the Kelvin 
transform of the function $u$ and $v$.  The pair $(u_1, v_1)$ is thus a 
solution to 
\be \label{gmmy3}
\begin{array}{rlll}
-\Delta{u_1} & = & {\alpha(|x|)}v_1^p \cr
 -\Delta{v_1} & = & \beta(|x|) u_1^q \cr
\end{array} \quad  x \in {\mathbb R}^N \setminus \{0\}, 
\ee
where the functions $\alpha$ and $\beta$ are given by
\begin{eqnarray} \label{cambio}
\alpha(r):&=&r^{(N-2)p-(N+2)}\alpha^*(r^{-1}) \nonumber\\
\beta(r):&=&r^{(N-2)q-(N+2)}\beta^*(r^{-1}).
\end{eqnarray}
Define 
$$
\alpha_o=N+2-(N-2)p-\alpha'_o, \quad \beta_o=N+2-(N-2)q-\beta'_o,
$$
and thus we easily get that $(\alpha, \beta, p, q)$ satisfies the hypothesis on 
Theorem \ref{existenciadegroundstates} and the existence of $h^*$ follows. 

Next, we will show that  those ground states satisfy 
(\ref{comportamientosingular}) if $r^{\alpha'_o}\alpha^*(r)$ or 
$r^{\beta'_o}\beta^*(r)$ are strictly increasing at some point or 
(\ref{bajohiperbola1}) is not an equality.
 
  Let $(u, v)$ be a radially symmetric ground state constructed as above. The 
  functions $u$ and $v$ are decreasing functions such that $r^{N-2}u(r)$ and 
  $r^{N-2}v(r)$ are increasing. Moreover,
\be \label{cotaen0}
\lim_{r\to 0}r^{N-2}u(r)=0, \quad \mbox{and} \quad \lim_{r\to 0}r^{N-2}v(r)=0,
\ee
as it follows from the behavior at infinity of $(u_1, v_1)$.   The energy 
function given by (see Proposition \ref{pohozaev})
\begin{eqnarray} \label{energia2}
E^*_{ab}(r)&=& r^N\left[u'(r)v'(r) +ar^{-1}u'(r)v(r) + br^{-1}v'(r)u(r)\right]  \cr 
&&+ r^N\left[ \frac{\alpha^*}{p+1} v^{p+1} + \frac{\beta^*}{q+1} u^{q+1} \right],
\end{eqnarray}
satisfies
\begin{eqnarray}
(E^*_{ab})'(r)&=& (a+b+2-N)r^{N-1}u'(r)v'(r) + 
(\frac{r^N\alpha^*}{p+1} )'v^{p+1} \\
&&-ar^{N-1}\alpha^* v^{p+1}+
(\frac{r^N\beta^*}{q+1} )'u^{q+1}-br^{N-1}\beta^*u^{q+1}. \nonumber 
\end{eqnarray}
Moreover, for $a=(N-\alpha'_o)/(p+1)$ and $b=(N-\beta'_o)/(q+1)$,  we 
get that $E^*_{ab}$ is a nondecreasing function. First, we will prove that 
$\lim_{r\to \infty}E^*_{ab}(r)=0$.  From (\ref{behavior}) we get  that 
the  first three terms in (\ref{energia2}) tend to zero as $r$ tends to 
infinity. Moreover, from Proposition \ref{r2alfa}, we get that 
$\lim_{r \to 0}r^2\alpha(r)=0$ and 
$\lim_{r \to 0}r^2\beta(r)=0$, where $(\alpha, \beta)$ is given by  
(\ref{cambio}). Therefore \newline
$\lim_{r \to \infty}r^{N-(N-2)p}\alpha^*(r)=0$
and $\lim_{r \to \infty}r^{N-(N-2)q}\beta^*(r)=0$, and thus the  last 
two terms in (\ref{energia2}) tend to zero as $r$ tends to infinity. 
\medskip

Assume by contradiction that $\lim_{r\to 0}u(r)=l<\infty.$

\noindent {\bf Assertion:} If $l<\infty$, then 
$\lim_{r\to 0}E^*_{ab}(r)=0$, and thus $(E^*_{ab})'(r)=0$, for all $r$.  

Assume that the assertion is true.  If either 
$r^{\alpha'_o}\alpha^*(r)$ or $r^{\beta'_o}\beta^*(r)$ are increasing functions 
at some point $x$, or the inequality in (\ref{bajohiperbola1}) is strictly, then
 $(E^*_{ab})'(x)=0$, implies that either $u(x)=0$, or $v(x)=0$, or $u'(x)=0$, or
  $v'(x)=0$, and the contradiction follows. 

Now, we prove the  assertion. If $u$ is bounded, the three first term in 
$E^*_{ab}(r)$, goes to zero as a consequence of (\ref{cotaen0}). For the fourth
 term, we argue as follows.
Since $r^{N-2}u(r)$ is nondecreasing, $v$ is non-increasing and $r^{\alpha'_0}
\alpha^*(r)$ is nondecreasing, we get 
\begin{eqnarray} \label{acotando}
r^{N-2}u(r)&\ge& C\int_{r/2}^r s^{N-1}\alpha^*(s)v^p(s) \nonumber \\ 
&\ge& C r^{N}\alpha^*(r/2)v^p(r),
\end{eqnarray}
and thus, using that $u$ is non-increasing and $r^{N-2}v(r)$ is  nondecreasing, 
we can change in (\ref{acotando})
$u(r)$ by $Cu(r/2)$ and $v^p(r)$ by $Cv^p(r/2)$ 
to get 
\be \label{acotando1}
u(r) 
\ge C r^{2}\alpha^*(r)v^p(r).
\ee

 Therefore, from (\ref{acotando1}) and since $l<\infty$ we get
$$
Cr^{N-2}v(r)\ge C'r^{N-2}u(r)v(r)\ge r^{N}\alpha^*(r)v^{p+1}(r),
$$
and thus $\lim_{r\to 0}r^{N}\alpha^*(r)v^{p+1}(r)=0.$
Now, from Proposition \ref{propiedades} we conclude that 
$r^N\beta^*(r)=r^{(N-2)q-2}\beta(r^{-1})$ goes to zero as $r$ does, and thus 
the last term in $E^*_{ab}(r)$ goes to zero as $r$ does, and the assertion 
follows. \hfill$\diamondsuit$
\medskip

\noindent{\bf Remark.} Assume that conditions on theorem 
\ref{existenciabajolahiperbola} are satisfied with $p=q=1$.   In this case 
(\ref{bajohiperbola1}) is equivalent to 
\be \label{linealhiperbola}
2-\alpha'_o+2-\beta'_o \ge 0.
\ee
The conclusion of the above theorem is in this case also valid.  If either 
$r^{\alpha'_o}\alpha^*(r)$  or  $r^{\beta'_o}\beta^*(r)$ is non constant, no 
changes on the proof of the above result. If  $\alpha^*(r)=C_1r^{-\alpha'_o}$ 
and  $\beta^*(r)=C_2r^{-\beta'_o}$, then from (\ref{condicioneskelvin1}) we 
get that (\ref{linealhiperbola}) is not an equality, and the proof follows that 
of the above theorem. 

On the other hand, it is known that the existence of radially symmetric 
positive solutions to 
$$
-\Delta u= c|x|^{-2}u,
$$
depends on the constant $c$.  For $4c> (N-2)^2$, those solutions do not exist.
\medskip

For system (\ref{potencias}) we have the following 
\begin{corollary} Assume that $\alpha'_o > N-(N-2)p$, $\beta'_o>N-(N-2)q$, and 
$(\alpha'_o, \beta'_o, p, q)$ satisfies $(\ref{bajohiperbola1})$. Then, there 
exists a positive constant $C^*$ such that
$$
 h^*(x)=C^*x^{(2-\alpha'_o+(2-\beta'_o)p)/(2-\beta'_o+(2-\alpha'_o)q)}. $$
Moreover, if 
$$(\alpha'_o, \beta'_o)=(0, 0), \quad \frac{N}{p+1}+\frac{N}{q+1} > N-2,$$ 
and if $(u, v)$ is a ground state  and $pq>1$, then there exist two 
positive constants $c$ and $d$ such that for all $r$ near zero we have
\begin{eqnarray*}
&dr^{(-2(p+1))/(pq-1)}\le u(r)\le cr^{(-2(p+1))/(pq-1)}\,,&\\  
&dr^{(-2(p+1))/(pq-1)}\le v(r)\le cr^{(-2(p+1))/(pq-1)}\,,&
\end{eqnarray*}
\end{corollary}
\noindent{ \bf Proof.} The proof of the behavior at zero of the ground states 
follows from Theorem 3.2 in \cite{gy}.  \hfill$\diamondsuit$
\medskip

From theorem \ref{gssingulares} and Kelvin transform, we obtain the following.

\begin{theorem} \label{gssingulares1} Assume that conditions on theorem 
\ref{existenciabajolahiperbola} are satisfied. Moreover, assume that 
\be \label{gammasnegativos}
\int^{\infty} s^{1+p\gamma_2}\alpha^*(s)ds<\infty\,, \quad \int^{\infty} 
s^{1+q\gamma_1}\beta^*(s)ds<\infty\,,
\ee
where $\gamma_{1,2}=\gamma_{1,2}(\alpha'_o, \beta'_o)$ are given by 
$(\ref{defgammas})$. Then, there exists $(u,v)$ a nonnegative  ground state to 
$(\ref{gmmy2})$, such that 
\be \label{eninfinito}
\lim_{r\to \infty}r^{N-2}u(r)=\infty\,,\quad \mbox{and} \quad 
\lim_{r\to \infty}r^{N-2}v(r)=\infty\,.
\ee
Moreover, if $(\ref{bajohiperbola1})$ is not an equality, then those ground states
 satisfy
$$\lim_{r\to 0}u(r)=\infty,\quad \mbox{and} \quad \lim_{r\to  0}v(r)=\infty.$$
\end{theorem}

\noindent{\bf Proof.} The existence of a positive solution $(u, v)$ of 
(\ref{gmmy2}) satisfying (\ref{eninfinito}) follows from Theorem 
\ref{gssingulares} and Kelvin transform. Next, we will prove that 
$(u(r), v(r))$ tends to zero as $r$ tends to infinity. Similarly to  
(\ref{acotando1}) we get 
\be \label{acotando2}
v(r) 
\ge C r^{2}\beta^*(r)u^q(r),
\ee
and thus  from (\ref{acotando1}),  (\ref{acotando2}) and since 
$r^{\alpha'_o}\alpha(r)$ and  $r^{\beta'_o}\beta(r)$ are nondecreasing 
functions we obtain the existence of  $r_0$ and a positive constant $C$ such 
that for all $r\ge r_0$ 
\begin{eqnarray} \label{cotasuvgammas}
u(r)&\le& Cr^{\gamma_1(\alpha'_o, \beta'_o)}, \\ 
v(r)&\le& Cr^{\gamma_2(\alpha'_o, \beta'_o)}. \nonumber
\end{eqnarray}
Moreover, from (\ref{gammasnegativos}) we obtain that $2-\alpha'_o+
\gamma_2(\alpha'_o, \beta'_o)p<0$, but it can be easily verified that 
$\gamma_1(\alpha'_o, \beta'_o)=2-\alpha'_o+\gamma_2(\alpha'_o, \beta'_o)p$  and 
thus $\gamma_1(\alpha'_o, \beta'_o)<0$.  Similarly $\gamma_2(\alpha'_o, \beta'_o)
<0$.  Therefore, the solution $(u, v)$ is a ground state. 

Assume now that (\ref{bajohiperbola1}) is a strict inequality. To prove that 
$u$ and $v$ are singular at zero we use the same argument used on the proof of 
theorem \ref{existenciabajolahiperbola}. The unique difference with that  proof
  will be the proof of $\lim_{r\to \infty}E^*_{ab}(r)=0$, which is as 
  follows:
From (\ref{acotando1}), (\ref{acotando2}) and (\ref{cotasuvgammas})
we obtain that
\begin{eqnarray} \label{acotando3}
r^{N-2+\gamma_1+\gamma_2}&\ge& Cr^{N-2}u(r)v(r)\ge r^{N}\beta^*(r)u^{q+1}(r),\\ 
\nonumber
r^{N-2+\gamma_1+\gamma_2}&\ge& Cr^{N-2}u(r)v(r)\ge r^{N}\alpha^*(r)v^{p+1}(r).
 \nonumber
\end{eqnarray}
Now since $r^{N-2}u$ and $r^{N-2}v$ are nondecreasing we also get that 
\begin{eqnarray} \label{acotando4}
r^{\gamma_1-1}&\ge& C|u'(r)|,\\ \nonumber
r^{\gamma_2-1}&\ge& C|v'(r)|. \nonumber
\end{eqnarray}
On the other hand, since condition (\ref{bajohiperbola1}) (without equality) is
 equivalent to $\gamma_1+\gamma_2+N-2<0$, from (\ref{acotando3}) and  
 (\ref{acotando4}) the conclusion follows. \hfill$\diamondsuit$

\section{Applications}
In this section we will apply the results of the above sections to the more 
general system 
\begin{eqnarray} \label{general}
\Delta u(x)+V(|x|)u+a(|x|)v^p&=& 0,\nonumber\\
\Delta v(x)+V(|x|)v+b(|x|)u^q&=& 0, \quad \mbox{in} \quad {\mathbb R}^N\setminus 
\{0\}.
\end{eqnarray}
where $a$ and $b$ are nonnegative functions, $p\ge1$, $q\ge 1$.  The \lq\lq 
potential" $V$ is not necessarily non-positive. 

Let $V\in L_{\mbox{\scriptsize loc}}^\infty(0, \infty)$, such that the equation
\be \label{lineal}
h''(r)+\frac{N-1}{r}h'(r)+V(r)h(r)=0, \quad r\in(0, \infty)
\ee
is disconjugate in $(0, \infty)$; i.e., there exists a positive solution $h_0$ 
of (\ref{lineal}) such that
$$
\int^{\infty}r^{1-N}h_0^{-2}(r)dr=\infty.
$$
( See, e.g.,\cite{ha}, \cite{wi} for the definition and properties of 
disconjugacy.) Note that for $V=0$, (\ref{lineal}) is disconjugate, and in 
this case $h_0=r^{2-N}$.  We define as in \cite{bly}, 
$h_1(r)=h_0(r)\int_0^rt^{1-N}h_0^{-2}(t)dt$ if 
$$
D\equiv \int_0t^{1-N}h_0^{-2}(t)dt<\infty,
$$ 
and $h_1(r)=h_0(r)\int_R^rt^{1-N}h_0^{-2}(t)dt$ if $D=\infty$,  where $R>0$ 
 is fixed. In any case $h_0, h_1$ are two linearly independent solutions to 
 (\ref{lineal}). 
In the sequel we assume that $D<\infty$.  
If $(u,v)$ is a radially symmetric solution to (\ref{general}), then 
$(u_1, v_1)$ given by 
\begin{equation}
u_1(s)=\frac{u(r)}{h_1(r)}, \quad v_1(s)=\frac{v(r)}{h_1(r)}\,,
\end{equation}
where $s=\frac{h_1}{h_0}$, and $v$ is a solution for all $s>0$ to
\begin{eqnarray} \label{sistematransformado}
&& u''_1(s)+\frac{2}{s}u'_1(s)+\alpha(s)v_1^p=0, \nonumber \\
&& v''_1(s)+\frac{2}{s}v'_1(s)+\beta(s)u_1^q =0,
\end{eqnarray}
where
$$
\alpha(s)=a(r)h_1^{p-1}(r)h_0^4(r)r^{2(N-1)},\quad
\beta(s)=b(r)h_1^{q-1}(r)h_0^4(r)r^{2(N-1)},
$$
and thus with the appropriate conditions on $\alpha$ and $\beta$, the results 
of the above sections give existence of positive solutions to (\ref{general}). 
The function $(\alpha(s), \beta(s))$ satisfies (\ref{cotaalfabetaen0}) if and 
only if $(a(r), b(r))$ satisfies
\be \label{cotaalfabetaen0vno0}
\int_{0}r^{N-1}h_0h_1^{p}a(r)dr<\infty, \quad  \int_{0}r^{N-1}h_0h_1^{q}b(r)dr
<\infty,
\ee 
and  $(\alpha(s), \beta(s))$ satisfies (\ref{breli}) if and only if $(a(r), 
b(r))$ satisfies
\be \label{brelivno0}
\int^{\infty}r^{N-1}h_0h_1^{p}a(r)dr=\infty, \quad \int^{\infty}r^{N-1}
h_0h_1^{q}b(r)dr=\infty.
\ee 
We have the following 
\begin{theorem}Assume that $a$ and $b$ satisfy $(\ref{cotaalfabetaen0vno0})$ and 
$(\ref{brelivno0})$.  
Assume that there exists $(\alpha_o, \beta_o)$ such that 
\be
 \frac{3-\alpha_o}{p+1}+\frac{3-\beta_o}{q+1}\le 1,
\ee
and 
$$r^{2(N-1)}h_1^{p-1+\alpha_o}h_0^{4-\alpha_o}a(r), \quad r^{2(N-1)}h_1^{q-1+
\beta_o}h_0^{4-\beta_o}b(r)$$
 are non-increasing functions.
Then, there exists $g\in C({\mathbb R}^+, {\mathbb R}^+)$ strictly increasing and such 
that for any $c>0$ there exists a radially symmetric  solution $(u, v)$ to 
$(\ref{general})$ such that
$$
\lim_{r\to 0}\frac{u(r)}{h_1(r)}=g(c), \quad \mbox{and} \quad \lim_{r\to 0}
\frac{v(r)}{h_1(r)}=c.
$$
\end{theorem}
\medskip

For instance, if 
$V(r)=-dr^{-2}$, with $d>-(N-2)^2/4,$
the equation 
$$
h''(r)+\frac{N-1}{r}h'(r)-\frac{d}{r^2}h(r)=0, \quad r\in(0, \infty)
$$
has the solutions 
\begin{eqnarray*} 
&& h_0(r)=r^{\theta_0} \quad \mbox{and} \quad h_1(r)=r^{\theta_1}, \\
&& \theta_{1,0}=\frac{2-N\pm ((N-2)^2+4d)^{1/2}}{2},
\end{eqnarray*}
and for the system 
\begin{eqnarray} \label{ralamenos2}
\Delta u(x)-\frac{d}{|x|^2}u+a(|x|)v^p&=& 0,\nonumber\\
\Delta v(x)-\frac{d}{|x|^2}v+b(|x|)u^q&=& 0, \quad \mbox{in} \quad 
{\mathbb R}^N\setminus \{0\},
\end{eqnarray}
we have 
\begin{corollary}\label{d} Assume that there exists $(a_o, b_o)$ such that 
$r^{a_o}a(r)$ and $r^{b_o}b(r)$ are non-increasing functions for some 
$(a_o, b_o)$ satisfying
\be \label{hiperbola2}
 \frac{N-a_o}{p+1}+\frac{N-b_o}{q+1}\le N-2.
\ee
Moreover, assume that $a$ and $b$ satisfy $(\ref{cotaalfabetaen0vno0})$ and 
$(\ref{brelivno0})$.  
Then, there exists $g\in C({\mathbb R}^+, {\mathbb R}^+)$ strictly increasing and such 
that for any $c>0$ there exists a radially symmetric  solution $(u, v)$ to 
$(\ref{ralamenos2})$ such that
$$
\lim_{r\to 0}\frac{u(r)}{r^{\theta_1}}=g(c), \quad \mbox{and} \quad 
\lim_{r\to 0}\frac{v(r)}{r^{\theta_1}}=c.
$$
\end{corollary}
\noindent{\bf Remark.} If in the above corollary $d>0$, then for any $c>0$,  
the solution $(u, v)$ corresponding to $(g(c), c)$ satisfies that $u(0)=0$ and 
$v(0)=0$. On the other hand, if $d<0$, the solutions  are singular at zero. 
Moreover, if $(a(r), b(r))=(r^{-a_o}, r^{-b_o})$ with $(a_o, b_o)$ satisfying 
(\ref{hiperbola2}) and $$ a_o<2+(p-1)\theta_1, \quad b_o<2+(q-1)\theta_1$$
then  there exists a positive constant $D$ such that $g(x)=Dx^{k}$, where 
$$k=\frac{a_o-2+(b_o-2)p-\theta_1(pq-1)}{b_o-2+(a_o-2)q-\theta_1(pq-1)}.$$

Now, we will obtain existence of a curve of ground states of (\ref{general}) 
which behaves at infinity as $h_0$.   To this end, we will apply results of 
section 4 to the system (\ref{sistematransformado}). Condition 
(\ref{condicioneskelvin}) for $(\alpha(s), \beta(s))$ in terms of $(a(r), b(r))$
 is the following
\be \label{condicioneskelvinvno0}
\int_0r^{N-1}h_1h_0^pa(r)dr =\infty,  \quad \int_0r^{N-1}h_1h_0^qb(r)dr 
=\infty, 
\ee
and condition  (\ref{condicioneskelvin1}) for $(\alpha(s), \beta(s))$ reads for
 $(a(r), b(r))$ 
\be\label{condicioneskelvin1vno0}
\int^{\infty}r^{N-1}h_1h_0^pa(r)dr  <\infty,  \quad \int^{\infty}r^{N-1}
h_1h_0^qb(r)dr  <\infty.
\ee

\begin{theorem}Assume that $a$ and $b$ satisfy $(\ref{condicioneskelvinvno0})$ 
and $(\ref{condicioneskelvin1vno0})$.
Assume that there exists $(\alpha'_o, \beta'_o)$ such that 
\be
 \frac{3-\alpha'_o}{p+1}+\frac{3-\beta'_o}{q+1}\ge 1\,.
\ee
and
$$r^{2(N-1)}h_1^{p-1+\alpha'_o}h_0^{4-\alpha'_o}a(r),\quad r^{2(N-1)}h_1^{p-1+
\beta'_o}h_0^{4-\beta'_o}b(r),$$ 
are nondecreasing functions. Then, there exists $\delta\in 
C({\mathbb R}^+, {\mathbb R}^+)$ strictly increasing and such that for any $c>0$ there 
exists a radially symmetric  solution $(u, v)$ to $(\ref{general})$ such that
$$
\lim_{r\to \infty}\frac{u(r)}{h_0(r)}=\delta(c), \quad \mbox{and} \quad 
\lim_{r\to \infty}\frac{v(r)}{h_0(r)}=c\,.
$$
\end{theorem}

In particular for system (\ref{ralamenos2}) we have

\begin{corollary}\label{d1} Assume that there exists $(a_o, b_o)$ such that 
$r^{a_o}a(r)$ and $r^{b_o}b(r)$ are nondecreasing functions for some 
$(a_o, b_o)$ satisfying
$$
 \frac{N-a_o}{p+1}+\frac{N-b_o}{q+1}\ge N-2\,.
$$
Moreover, assume that $a$ and $b$ satisfy $(\ref{condicioneskelvinvno0})$ and 
$(\ref{condicioneskelvin1vno0})$.
Then, there exists $\delta \in C({\mathbb R}^+, {\mathbb R}^+)$ strictly increasing and 
such that for any $c>0$ there exists a radially symmetric  solution $(u, v)$ to 
$(\ref{ralamenos2})$ such that
$$
\lim_{r\to \infty}\frac{u(r)}{r^{\theta_0}}=\delta(c), \quad \mbox{and} \quad 
\lim_{r\to \infty}\frac{v(r)}{r^{\theta_0}}=c\,.
$$
\end{corollary}
\begin{thebibliography}{00}

\bibitem{as} M. Aizenman, B. Simon, Brownian motion and Harnack  inequality for 
Schroedinger operators,  {\em Comm. Pure Appl. Math.} {\bf 35} (1982), 209-273.

\bibitem{bly} R. Benguria, S. Lorca and C. Yarur, Nonexistence results for 
solutions of semilinear elliptic equations, {\em Duke Mathematical Journal}
 {\bf 74} (1994), 615-634.

\bibitem{bi} M.F. Bidaut-V\'eron, Local behaviour of the solutions of a class of
 nonlinear elliptic systems, {\em Advances in Differential Equations}, to 
 appear.

\bibitem{blions} H. Brezis and P.L. Lions, A note on isolated singularities for
 linear elliptic equations, {\em J. Math. Anal. and Appl.} {\bf 7A} (1981), 
 263-266.

\bibitem{gy} M. Garc\'{\i}a-Huidobro and C. Yarur, Classification of positive 
singular solutions for a  class of semilinear elliptic systems, {\em Advances 
in Differential Equations} {\bf 2} (1997), 383-402.

\bibitem{gt}D. Gilbarg and N.S. Trudinger, Elliptic Partial Differential 
Equations of Second Order, Springer-Verlag, 1977. 

\bibitem{ha} P. Hartman, Ordinary Differential Equations, John Wiley, New York,
 1964.

\bibitem{hvv} J. Hulshof and R. van der Vorst, Asymptotic Behaviour of ground 
states, preprint.

\bibitem{li} P.L. Lions, The concentration-compactness principle in the 
calculus of variations, part 1, {\em Rev. Iberoam. } {\bf 1} (1985), 145-201.

\bibitem{mi1} E. Mitidieri, Nonexistence of positive solutions of semilinear 
elliptic systems in $\mathbb{R}^N$, {\em Differential and Integral Equations},  
{\bf 9} (1996), 465-479.

\bibitem{ni} W.M. Ni,  On the elliptic equation 
$\Delta u + K(x)u^{(n+2)/(n-2)}=0$, its generalizations, and
 applications in geometry, {\em Indiana Univ. Math. J. } {\bf 31} (1982), 
 493-529.

\bibitem{sz1} J. Serrin and H. Zou,  Nonexistence of positive solutions of the 
Lane-Emden system, {\em Differential Integral Eqns.} {\bf 9} (1996), 635-653.

\bibitem{sz} J. Serrin and H. Zou, Existence of positive solutions of the 
Lane-Emden system, to appear.

\bibitem{sz13} J. Serrin and H. Zou, Existence of positive solutions of
elliptic Hamiltonian  systems, {\em Comm. Partial Diff. Eqns.,} to appear.

\bibitem{wi} D. Willett, Classification of second order linear differential 
equations with respect to oscillation, {\em Adv. Math. } {\bf 3} (1969), 
594-623.

\bibitem{y} C. Yarur, Nonexistence of positive singular solutions for a class 
of semilinear elliptic systems, {\em Electr. J. Diff. Eqns. 
Vol {\bf 1996}} (1996), No 8,  1-22. 

\end{thebibliography} 
\bigskip

\noindent {\sc Cecilia S. Yarur}\\
Departamento de Matem\'atica y C. C. \\
Universidad de Santiago de Chile\\ 
Casilla 307, Correo 2, Santiago, Chile \\
E-mail address: cyarur@fermat.usach.cl

\end{document}