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\markboth{ Existence of non-negative solutions }
{ Cecilia S. Yarur }

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\title{\vspace{-1in}\parbox{\linewidth}{\footnotesize\noindent
USA-Chile Workshop on Nonlinear Analysis, \newline
Electron. J. Diff. Eqns., Conf. 06, 2001, pp. 359--367.\newline
http://ejde.math.swt.edu or http://ejde.math.unt.edu
\newline ftp  ejde.math.swt.edu or ejde.math.unt.edu (login: ftp)}
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%
 Existence of non-negative solutions \\ for a Dirichlet problem
% 
\thanks{ {\em Mathematics Subject Classifications:} 35C20, 35D10.
 \hfil\break\indent 
{\em Key words:} Dirichlet Problem, Non-negative solutions.
 \hfil\break\indent
\copyright 2001 Southwest Texas State University. 
\hfil\break\indent Published January 8, 2001.  \hfil\break\indent 
Partially supported by Fondecyt grant 1990877 and DICYT } } 

\date{}
\author{ Cecilia S. Yarur }
\maketitle
\begin{abstract} 
The aim of this paper is the study of existence of non-negative 
solutions of fundamental type for some systems without
sign restrictions on the non linearity.
\end{abstract}

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\section{Introduction}

We study the existence of non-negative non-trivial solutions to 
the boundary-value problem
\begin{eqnarray}\label{example0}
&\Delta u =  a_2 v^{p_2}-a_1 v^{p_1} \quad\mbox{in } B'&\nonumber\\
&\Delta v = b_2 u^{q_2}-b_1 u^{q_1} \quad\mbox{in }B' &\\
& u=  v= 0 \quad \mbox{on} \partial B\,, &\nonumber
\end{eqnarray}  
where $a_i,b_i$ are non-negative constants, $p_i>0$, $q_i>0$ for
$i=1,2$, $B$ is the unit ball centered at zero in
 $\mathbb{R}^N$, $N\ge 3$, and $B'= B\setminus \{0\}$.

The above problem involves many problems of a quite different nature
 depending on the  values of $a_i$,
$b_i$. For instance, if $a_2=b_2=0$ the solutions $u,v$ are
sub-harmonic functions, while if $a_1=b_1=0$ the solutions are
super-harmonic.

For a better understanding of this system, we recall that P.L.
Lions \cite{li},  Ni and Sacks \cite{nsa}, and Ni and Serrin
\cite{ns}, studied conditions for existence or non existence of
non-negative solutions $u$ to
\begin{equation}\label{lions}
-\Delta u  = u^q  \quad \mbox{in $B'$,} \quad u  = 0
\quad\mbox{on $\partial B$}. 
 \end{equation}
The range  of existence of solutions to (\ref{lions}) is
 $ q < (N+2)/(N-2)$. On the other hand, the problem
$$
 \Delta u  = u^q  \quad \mbox{in $B'$,} \quad
 u  = 0 \quad\mbox{on $
\partial B$},
 $$
 has a non-negative non-trivial solution if and only if $q < N/(N-2)$,
 see \cite{bv} for the non-existence and \cite{v}
 for existence and related problems.

We state next some known results concerning particular
cases of problem (\ref{example0}). Assume first that  $ a_1= b_1
=0$. Thus, we are concerned with
\begin{eqnarray}\label{example01}
&-\Delta u =  a_2 v^{p_2} \quad \mbox{in $B'$}&\nonumber\\
&-\Delta v  = b_2 u^{q_2} \quad \mbox{in $B'$}&\\ 
&u=v=0 \quad \mbox{on $\partial B$.}&\nonumber
\end{eqnarray}
The following result is well known. 

\begin{theorem} Assume that $a_1=b_1=0$, $p_2q_2>1$ and $a_2>0$, $b_2>0$.
 Then, there exists a  classical solution to
(\ref{example01}) if and only if $$ \frac{N}{p_2+1}+ \frac{N}{q_2+
1 } > N-2.$$
\end{theorem}

 Troy \cite{tr} proved radial symmetry of positive classical
solutions to problem (\ref{example01}). The existence of positive
classical solutions of (\ref{example01}) was studied by Hulshof
and van der Vorst \cite{hv}  and de Figueiredo and Felmer
\cite{ff}. The behavior of solutions was studied by Bidaut-V\'eron
in \cite{biv1}.
 The existence of some singular solutions, that is solutions with
either \\ $\limsup_{x\to 0}u(x)=+\infty$ or $\limsup_{x\to
0}v(x)=+\infty$, is given by Garc\'{\i}a-Huidobro, Man\'asevich,
Mitidieri and Yarur, see \cite{gmmy}.
Using Pohozaev-Pucci-Serrin type identity,  Mitidieri \cite{m1,m2}
and van der Vorst
 \cite{vv}  proved non existence of classical solutions of
(\ref{example01}).
 Non-existence of radially symmetric singular positive
solutions was given by Garcia-Huidobro, Man\'asevich, Mitidieri
and Yarur in \cite{gmmy}.

We note that since $u$ and $v$ are super-harmonic functions, and
due to a result of Brezis and P.L.Lions \cite{bl}, $u^{q_2}\in
L^1(B)$ , $v^{p_2}\in L^1(B)$ and there exist $c\ge 0$ and $d\ge
0$ such that
\begin{eqnarray*}
&-\Delta u =  a_2 v^{p_2}+ c\delta_0 \quad \mbox{in $\mathcal{D'}(B)$}& \\ 
&-\Delta v =  b_2u^{q_2} + d\delta_0 \quad \mbox{in $\mathcal{D'}(B)$}&\\ 
& u= v= 0  \quad \mbox{on $\partial B$.}&
\end{eqnarray*}  
If   $(c, d) \not= (0, 0)$  we call this singularity of {\em
fundamental} type.

Let us consider now  $ a_1= b_2 = 0$, in (\ref{example0}). Hence,
we are looking for the solutions of:
\begin{eqnarray}\label{example02}
&-\Delta u =  a_2 v^{p_2} \quad \mbox{in $B'$} &\nonumber\\ 
&\Delta v = b_1 u^{q_1} \quad \mbox{in $B'$}&\\
& u= v=0  \quad \mbox{on $\partial B$.}&\nonumber 
\end{eqnarray} 
Since $v$ is sub-harmonic, there exists no non-negative classical
solutions to (\ref{example02}).

The following result is given in \cite{cidy1} for $p_2q_1>1$ and in
\cite{cidy2} for $p_2q_1 <1$.

\begin{theorem}  Assume $a_1=b_2=0$, $p_2>0$, $q_1>0$,
and $p_2q_1 \not=1$. Then there exists a non-trivial non-negative
solution to (\ref{example02}) if and only if 
$$ \frac{N}{p_2+1} +\frac{N-2}{q_1 + 1} > N-2, \quad \mbox{and}\quad
p_2 < N/(N-2).$$
\end{theorem}
The above result is based on the results given in \cite{bg}.

We say that $(u, v)$ has a {\em strong} singularity at $0$ if
either $$\limsup_{x\to 0} |x|^{N-2}u(x)= +\infty \quad \mbox{ or} 
\quad \limsup_{x\to 0} |x|^{N-2}v(x)= +\infty\,.$$

 It can be proved  that there exists a region in the plane $p_2-q_1$
  where there exist both {\em strong} and {\em fundamental}
non-negative singular solutions, see \cite{cidy2}.  This region is
given by
$$ \frac{N-2}{p_2+ 1}  +  \frac{N}{q_1 + 1} > N-2,
  \quad \mbox{and}\quad  p_2 < N/(N-2) < q_1 \,.$$
Assume now that $ a_2= b_2 =0$, and thus the problem
(\ref{example0}) is
\begin{eqnarray}\label{example03}
&\Delta u = a_1 v^{p_1}  \quad \mbox{in $B'$}&\nonumber\\
&\Delta v = b_1 u^{q_1} \quad \mbox{in $B'$}&\\
&u=v=0 \quad\mbox{on $\partial B$,}&\nonumber 
\end{eqnarray}
 Since $u$ and $v$ are sub-harmonic we have non
existence of non-negative solutions with either $u$ or $v$
bounded.

In \cite{bg2} and \cite{y}  it was proved non existence of
positive solutions if either 
$$ \frac{N}{p_1+1} + \frac{N-2}{q_1 +1} \le N-2, \quad
\mbox{or}\quad  
 \frac{N-2}{p_1+1}  +  \frac{N}{q_1 + 1}\le N-2.$$

If $ a_1= 0$ ( similarly for $b_1=0$) we have
\begin{eqnarray}\label{example04}
&-\Delta u = a_2 v^{p_2}  \quad \mbox{in $B'$}&\nonumber\\
& -\Delta v =b_2u^{q_2}- b_1 u^{q_1}\quad \mbox{in $B'$}&\\
&u=v=0 \quad\mbox{on $\partial B$}
\end{eqnarray}

The following result was proved in \cite{cidy3}.

\begin{theorem}  \label{corollaryintro} 
Let $p_2>0$, $q_1>0$ and $ q_2>0$. Let $a_1=0$, 
$a_2\ge 0$, $b_1\ge 0$ and $b_2 \ge 0$.
Assume that for $i=1,2$ we have
\begin{equation}\label{hipotesis0}
p_2 < \frac{N}{N-2}, \quad \frac{N}{p_2+1}+\frac{N-2}{q_i+1} >
N-2.
\end{equation} 
Assume that one of the following holds:
\begin{enumerate} \item[(i)]  $p_2q_i > 1$ for  all $i=1, 2$.
\item[(ii)] $p_2q_i < 1$, for all $i=1,2$.
\item[(iii)] If $p_2q_i=1$ for some $i=1,2$ then $a_2^{p_2}b_i$ is
sufficiently  small.
 \item [(iv)]  $p_2q_i < 1 < p_2q_j$, for some $i,j= 1, 2$, $i\not=j$,   and $a_2^{p_2}b_i$ is
sufficiently  small.
\end{enumerate}
Then, there exist $d_*\ge 0$,  $d^*> 0$ with $d_* < d^*$ such that
for any $d\in (d_*, d^*)$, there exists  $(u , v)$ a non-negative
solution to (\ref{example04})  satisfying $$\lim_{x \to
0}|x|^{N-2}(u(x), v(x))= (0, d).$$ Moreover, if $p_2q_i \ge 1,$
$i=1, 2$ then $d_*= 0$, and if $p_2q_i \le 1$,  $i=1, 2$ then
$d^*= \infty$.
\end{theorem}

For the general case we have the following previous result,  see
\cite{cidy3}.

\begin{theorem} \label{corollaryintro1} Let $p_1>0$, $p_2>0$, $q_1>0$ and $ q_2
>0$. Let $a_i$, $b_i$ $i=1, 2$ be non-negative constants.
Assume that \begin{equation}\label{hipotesis} p_i < \frac{N}{N-2},
\quad q_i < \frac{N}{N-2}, \quad i= 1, 2\end{equation} Assume that
one of the following holds:
\begin{enumerate} 
\item[(i)]  $p_iq_j > 1$,  for all $i,j=1, 2$.
\item[(ii)] $p_iq_j < 1$, for all $i,j =1,2$.
\item[(iii)] If $p_iq_j=1$ for some $i=1,2$ and some $j=1,2$ then  $a_i^{p_i}b_j$ is
sufficiently  small.
 \item [(iv)]   $p_iq_j <1 <p_kq_l$, for some $i,j,k,l=1,2$ and $a_i^{p_i}b_j$ is
sufficiently  small.
\end{enumerate}
Then, there exist $c > 0$, $d>0$ and $(u , v)$ a non-negative
solution to (\ref{example0})  such that 
$$\lim_{x \to 0}|x|^{N-2}(u(x), v(x))= (c, d).$$
\end{theorem}

Here we prove the following general existence result of non
negative non-trivial solutions to (\ref{example0}).
Set 
\begin{equation}\label{defgamma} 
\Gamma(p,q):=\frac{N-2}{p+1}+\frac{N}{q+1}-(N-2).
\end{equation}

\begin{theorem} \label{teorema} Let $p_i, \ q_i$, $i=1,2$, positive
numbers. Then, there exists a nonnegative nontrivial solution $(u,
v)$ of (\ref{example0}) if one of the following holds:
\begin{itemize}
\item[(i)]  $a_1>0$, $b_1>0$, and $p_2 < N/(N-2), \ q_2< N/(N-2)$ $$\min\{\Gamma(p_1, q_1), \Gamma(q_1, p_1), \Gamma(p_2, q_1),\Gamma(q_2,
p_1)\}>0,$$ with small coefficient  $a_j$ ( respectively  $b_j$ )
for some $j=1,2$ if $p_j \le 1$ ( respectively $q_j\le 1$) and
  $1 \le \max\limits_{i=1,2}\{p_i, q_i\}$.
\item[(ii)] $a_1=0$, $b_1>0$, $p_2 < N/(N-2)$ and
$$\min\{\Gamma(q_1, p_2), \Gamma( q_2, p_2)\} >0,$$ with small
coefficient  $a_2$ ( respectively $b_j$) if  $p_2\le 1$
(respectively $q_j\le1$) for some $j= 1, 2$, and $ 1 \le
\max\limits_{i=1,2}\{p_2, q_i\}$.
 \item[(iii)]
$a_1=0 = b_1$, and $$\max\{\Gamma(p_2, q_2), \Gamma(q_2,
p_2)\}>0,$$ with small coefficient $a_2$ ( respectively $b_2$) if
 $p_2\le 1$ (respectively $q_2\le1$) and $1 \le\max\{p_2, q_2\}$.
\end{itemize}
\end{theorem}

\section{Proof of Theorem \ref{teorema}}

We note that for $p$ and $q$ non-negative numbers the condition
$\Gamma(p,q)>0$ is equivalent to 
$$ p(2-(N-2)q) + N >0.$$ 
Moreover, if $pq>1$,
 $$\displaylines{
\Gamma(p,q) = (\zeta -(N-2))(pq-1), \quad \zeta=\frac{2(p+1)}{pq-1}\cr
\Gamma(q,p) = (\xi -(N-2))(pq-1),\quad \xi= \frac{2(q+1)}{pq-1}.
}$$ 
Recall that $u(x)=C_1|x|^{-\zeta}$, $v(x)= C_2 |x|^{ -\xi}$
 for some positive
constants $C_1$ and $C_2$ is a non-negative solution of $$ -\Delta
u = v^p, \quad -\Delta v = u^q $$ 
if $\Gamma(p,q) < 0$ and
$\Gamma(q,p)<0$. This particular solution also plays a fundamental
role for example for the system 
$$ -\Delta u = v^p, \quad \Delta v= u^q,$$ 
where this solution exists if  $\Gamma(p, q) < 0$ and
$\Gamma(q, p)>0$.

\paragraph{Proof of Theorem \ref{teorema}.} Set  
$$ f_i(t)=a_i t^{p_i}, \ g_i(t)= b_i t^{q_i}, \ i=1,2.$$
 We will construct radially symmetric non-negative solutions to
(\ref{example0}), by monotone iteration as follows.  Let $d>0$,
$(u_1,v_1)= (0, d m), $ where $m(r):= |x|^{2-N}-1$ and let $(u_n,
v_n)$ be given by $(u_{n+1}, v_{n+1}) = T(u_n, v_n)$ where
$T=(T_1, T_2)$ is the operator given by
\begin{eqnarray}\label{operador2}
T_1(u, v)(r)&=&\int_r^1s^{1-N}\ds\int_s^1 t^{N-1}f_1(v(t))dt ds \nonumber\\ 
&& + \int_r^1s^{1-N}\int_0^s t^{N-1}f_2(v(t))dt ds ,\\
T_2(u,v)(r)&=& d m(r)+ \int_r^1s^{1-N}\ds\int_s^1 t^{N-1}g_1(u(t))dt 
ds \nonumber\\
&&+ \int_r^1s^{1-N}\int_0^s t^{N-1}g_2(u(t))dt ds,\nonumber
\end{eqnarray}
 We are looking for $\alpha$, $\delta$ and $C$ such that
$$T_1(C r^{\alpha},Cr^{\delta}) \le C r^{\alpha}, \ T_2(C
r^{\alpha},Cr^{\delta})\le Cr^{\delta}
$$ 
and $v_1= d m(r) \le C r^{\delta}$. Hence, the  sequence $(u_n, v_n)$ 
satisfies 
$$u_n \le C r^{\alpha}, \ v_n\le Cr^{\delta} \ \mbox{ for all $n\in
\mathbb{N}$},
$$ 
and the convergence of $(u_n, v_n)$ to a solution
of (\ref{example0}) follows.

To find $C$, $d$, $\alpha$ and $\delta$  we use
the following: Let $\kappa $ be any number such that $\kappa +
N\not=0$, and define $$\phi(\kappa):=  \min\{ 2-N, \kappa+ 2 \}.
$$ Then
\begin{equation}\label{bound11}
m_{\kappa}(r):=\int_r^1s^{1-N}\ds\int_s^1 t^{N-1+\kappa}dt ds\le K
r^{\phi(\kappa)},
\end{equation}
where $K=K(N, \kappa)$.

Moreover, for any $\kappa$ satisfying
 $\kappa+N>0$,  and $\kappa +2 \not= 0, $ set $$\psi(\kappa):=  \min\{0, \kappa + 2 \}.
 $$We have
\begin{equation}\label{bound10}
h_{\kappa}:=\int_r^1s^{1-N}\ds\int_0^s t^{N-1+\kappa}dt ds\le K
r^{{\psi(\kappa)}},
\end{equation}
where $K=K(N, \kappa)$. Hence, $$ T_1(Cr^{\alpha},Cr^{\delta})=
a_1C^{p_1} m_{p_1\delta} + a_2C^{p_2} h_{p_2\delta}.$$ From
(\ref{bound11}) and (\ref{bound10}) and if we choose  $p_1\delta
+N\not=0$ and $p_2\delta +N >0$ we obtain
\begin{equation}\label{looking}
T_1(Cr^{\alpha},Cr^{\delta})\le K\left( a_1C^{p_1}
r^{\phi(p_1\delta)} + a_2C^{p_2} r^{\psi(p_2\delta)}\right).
\end{equation}
 We note that
$\phi(p_1\delta) \le 2-N < \psi(p_2\delta), $ and thus
\begin{equation} \label{found1}
T_1(Cr^{\alpha},Cr^{\delta})\le K\left( a_1C^{p_1} +
a_2C^{p_2}\right) r^{\sigma},
\end{equation}
where $$\sigma:= \left\{ \begin{array}{lll} \phi(p_1\delta) &&
\mbox{if $a_1\not=0$,}\\ \psi(p_2\delta) &&\mbox{if $a_1=
0$}.\end{array}\right. 
$$ 
Therefore, if $\alpha\le \sigma$ and $K( a_1C^{p_1} + a_2C^{p_2})\le 
C$, we obtain
$T_1(Cr^{\alpha},Cr^{\delta})\le C r^{\alpha}$.
 Arguing as above, we
have
\begin{equation}\label{looking2}
T_2(Cr^{\alpha},Cr^{\delta})\le dr^{2-N}+ K\left( b_1C^{q_1}
r^{\phi(q_1\alpha)} + b_2C^{q_2} r^{\psi(q_2\alpha)}\right),
\end{equation}
with $q_1\alpha + N \not=0$, $q_2\alpha + 2\not=0$, and
$q_2\alpha+N >0$. Therefore,
\begin{equation} \label{found2}
T_2(Cr^{\alpha},Cr^{\delta})\le \left( d+ K\left( b_1C^{q_1} +
b_2C^{q_2}\right)\right) r^{\rho},
\end{equation}
where 
$$\rho:= \left\{ \begin{array}{lll} \phi(q_1\alpha) &&
\mbox{if $b_1\not=0,$}\\ 2-N &&\mbox{if $b_1=
0$.}\end{array}\right.$$ Hence,
 $$T_2(Cr^{\alpha},Cr^{\delta})\le
C r^{\delta},$$ 
if $\delta \le \rho$ and $d+ K(b_1C^{q_1} + b_2C^{q_2})\le C$.

Next we prove the existence of $\alpha$, $\delta$, $C$ and $d$
under the hypothesis of the theorem. The existence of $C$ and $d$,
is classical. We can choose $d= C/2$ and for $i=1, 2$ $$
Ka_iC^{p_i}\le C/2, \quad Kb_iC^{q_i}\le C/4. $$ Therefore, if
either for all  $i$ , $p_i< 1$ and $q_i < 1$, or $p_i>1$ and
$q_i>1$, the existence of $C$ follows. By the contrary if
$\max\{p_i, q_i, i=1,2 \} \ge 1$ and $\min\{p_i, q_i, i=1,2 \}\le
1$, we obtain existence with a restriction on the coefficients.

We summarize the conditions that  $\alpha$ and $\delta$ must satisfy as follows:
$$ \alpha \le \left\{ \begin{array}{ll} \min\{2-N, p_1\delta +2
\} & \mbox{if $a_1\not= 0$ }\\
 \min\{0, p_2\delta+2\} &\mbox{if $a_1=0$ }
\end{array}\right. $$
$$ \delta \le \left\{ \begin{array}{ll} \min\{2-N, q_1\alpha +2
\} & \mbox{if $b_1\not= 0$ }\\
 2-N & \mbox{if $b_1=0$}.
\end{array}\right.
$$ 
Moreover, we need that
\begin{equation}\label{hiperbolas2}
 \ p_2\delta + N >0, \quad  q_2\alpha + N >0,
\end{equation}
We also used that $ p_1\delta + N \not=0, \ q_1\alpha + N \not=0,
\ p_2\delta + 2\not=0, \ q_2\alpha +2 \not=0$. These last
conditions are not relevant  since we can take $\alpha$ and
$\delta$ smaller and hence  these new $\alpha$ and $ \delta$
satisfy the conditions.

\paragraph{Case (i).} Assume first that $a_1>0$ and $b_1>0$.
If $p_1 < N/(N-2)$, and since $ \Gamma(p_1, q_1) >0$, and
$\Gamma(p_2, q_1)>0$,  we can take $$\alpha = 2-N, \ \delta=
\min\{2-N, 2-(q_1-\varepsilon)(N-2) \},$$ where $\varepsilon>0$ is
such that $$\Gamma(p_1, q_1-\varepsilon)>0, \ \mbox{ and $
\Gamma(p_2, q_1-\varepsilon) >0.$}$$ Now, since $q_2<N/(N-2)$, we
have that $q_2\alpha +N >0$. From $p_2< N/(N-2)$ and $\Gamma(p_2,
q_1-\varepsilon)>0$, we also have $p_2\delta+N>0$. It remains to
prove that $\alpha=2-N \le p_1\delta +2$, which follows easily
from $\Gamma(p_1, q_1-\varepsilon)>0$.

 If $p_1 \ge N/(N-2)$, from $\Gamma(p_1, q_1)>0$ we deduce that
 $q_1< N/(N-2)$. Thus, we may proceed as before but now with
$$ \delta= 2-N, \ \alpha= p_1(2-N)+2.$$

\paragraph{Case (ii).} Assume that $a_1= 0$ and $b_1 >0$. Let us
choose $$\delta = 2-N,\ \alpha= \min\{0,
2-(p_2-\varepsilon)(N-2)\},$$ where $\varepsilon>0$ is such that
$\Gamma(q_1, p_2-\varepsilon)>0$ and $\Gamma(q_2,
p_2-\varepsilon) >0$. Then the conclusion follows as in the
above case.

\paragraph{Case (iii).} Assume that $b_1=0$ and $a_1=0$. Assume that
$p_2\le q_2$. Since $\Gamma(q_2, p_2)>0$, then $p_2<N/(N-2)$. Let
us choose $$\delta= 2-N, \mbox{ and $\alpha =\min\{0,
2-(p_2-\varepsilon)(N-2) \} ,$}$$  and thus the conclusion follows
by taking $\Gamma(q_2, p_2-\varepsilon)>0$.

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\noindent{\sc  Cecilia S. Yarur}\\
Departamento de Matematicas \\
Universidad de Santiago de Chile \\
Casilla 307, Correo 2, Santiago, Chile  \\
email: cyarur@fermat.usach.cl 


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