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

\AtBeginDocument{{\noindent\small
\emph{Electronic Journal of Differential Equations},
Vol. 2011 (2011), No. 152, pp. 1--7.\newline
ISSN: 1072-6691. URL: http://ejde.math.txstate.edu or http://ejde.math.unt.edu
\newline ftp ejde.math.txstate.edu}
\thanks{\copyright 2011 Texas State University - San Marcos.}
\vspace{9mm}}

\begin{document}
\title[\hfilneg EJDE-2011/152\hfil
Nonexistence of radial positive solutions]
{Nonexistence of radial positive solutions for a nonpositone system
in an annulus}

\author[S. Hakimi \hfil EJDE-2011/152\hfilneg]
{Said Hakimi}

\address{Said Hakimi \newline
Universit\'e Abdelmalek Essaadi\\
Facult\'e des sciences \\
D\'epartement de Math\'ematiques \\
BP 2121, T\'etouan, Morocco}
\email{h\_saidhakimi@yahoo.fr}

\thanks{Submitted May 2, 2011. Published November 10, 2011.}
\subjclass[2000]{35J25, 34B18}
\keywords{Nonpositone problem; radial positive solutions}

\begin{abstract}
 In this article we study the nonexistence of radial
 positive solutions for a nonpositone system in an annulus by using
 energy analysis and comparison methods.
\end{abstract}

\maketitle
\numberwithin{equation}{section}
\newtheorem{theorem}{Theorem}[section]
\newtheorem{lemma}[theorem]{Lemma}
\allowdisplaybreaks

\section{Introduction}

We study the nonexistence of radial positive solutions for the
system
\begin{equation}
\begin{gathered}
 -\Delta u(x)=\lambda f(v(x)),\quad x\in \Omega \\
-\Delta v(x)=\mu g(u(x)),\quad x\in \Omega \\
u(x)=v(x)=0,\quad x\in \partial \Omega,
\end{gathered}  \label{eq1}
\end{equation}
 where $\lambda $, $\mu \geq \varepsilon _0>0$, $\Omega $
is an annulus in $\mathbb{R}^N$:
$\Omega =C(0,R,\widehat{R}) =\{x\in
\mathbb{R}^N: R<|x| <\widehat{R}\}$, ($0<R<\widehat{R}$, $N\geq2$),
$f$ and $g$ are smooth functions that grow at least
linearly at infinity.
 When $\Omega$ is a ball, problem \eqref{eq1} has been
studied by Hai, Oruganti and Shivaji \cite{h3}.

The nonexistence of radial positive solutions of \eqref{eq1} is
equivalent of the nonexistence of positive solutions of the system
\begin{equation}
\begin{gathered}
-(r^{N-1}u')'=\lambda r^{N-1}f(v),\quad R<r<\widehat{R}\\
-(r^{N-1}v')'=\mu r^{N-1}g(u),\quad R<r<\widehat{R}\\
u(R)=u(\widehat{R})=0=v(R)=v(\widehat{R}).
\end{gathered} \label{eq2}
\end{equation}
 The purpose of this paper is to prove that
the nonexistence of radial positive solutions of \eqref{eq1} remains
valid when $\Omega$ is an annulus and $f$ and $g$ satisfy the
following hypotheses
\begin{itemize}
\item[(C1)] $f$, $g : [0,+\infty )\to \mathbb{R}$ are continuous,
increasing and $f(0)<0$ and $g(0)<0$.

\item[(C2)] There exist two positive real numbers $a_i$ and $b_i$,
$i=1,2$ such that
$$
f(z)\geq a_1z-b_1,\quad
g(z)\geq a_2z-b_2,
$$
for all $z\geq0$.
\end{itemize}

\section{Main result}

Our main result is the following theorem.

\begin{theorem} \label{thm2.1}
Assume that  {\rm (C1)--(C2)} are satisfied. Then
there exists a positive real number $\sigma$ such that
\eqref{eq1} has no radial positive solution for $\lambda
\mu>\sigma$.
\end{theorem}

\noindent\textbf{Remark.} Existence result for positive
solutions with superlinearities satisfying (C1),
$\lambda=\mu$ and $\lambda$ small can be found in
\cite{h1, h2}.
Existence results, for the single equation case
can be found in \cite{a1,c1,h4},  and non-existence results
in \cite{a1,b1,h5}.

 To prove Theorem \ref{thm2.1}, we need the next three
lemmas. Here we use ideas adapted from
Hai, Oruganti and Shivaji \cite{h3}.

\begin{lemma} \label{lem2.2}
There exists a positive constant $C$ such that
for $\lambda \mu $ large,
\begin{equation*}
u(R_0)+v(R_0)\leq C,
\end{equation*}
where $R_0=(R+\widehat{R})/2$.
\end{lemma}

\begin{proof} Multiplying the first equation in \eqref{eq2}
by a positive eigenfunction, say $\phi$ corresponding to
$\lambda _1$, and using (C1) we
obtain
\begin{equation*}
-\int_R^{\widehat{R}}(r^{N-1}u')'\phi dr
\geq\int_R^{\widehat{R}}\lambda (a_1v-b_1)\phi r^{N-1}dr;
\end{equation*}
that is,
\begin{equation}
\int_R^{\widehat{R}} \lambda_1 u r^{N-1}\phi dr
\geq\int_R^{\widehat{R}}\lambda (a_1v-b_1)\phi
r^{N-1}dr. \label{eq3}
\end{equation}
Similarly, using the second equation in \eqref{eq2} and (C2), we
obtain
\begin{equation}
\int_R^{\widehat{R}} \lambda_1 v r^{N-1}\phi dr
\geq\int_R^{\widehat{R}}\mu (a_2u-b_2)\phi r^{N-1}dr.
\label{eq4}
\end{equation}
Combining \eqref{eq3} and \eqref{eq4}, we obtain
$$
\int_R^{\widehat{R}}[ \lambda _1-\lambda \mu
\frac{a_1a_2}{\lambda _1}] v\Phi r^{N-1}dr\geq
\int_R^{\widehat{R}}\mu [ -\lambda \frac{a_2b_1}{\lambda
_1}-b_2] \Phi r^{N-1}dr.
$$
Now, if $\lambda \mu a_1a_2/2 \geq \lambda _1^2$, then
$$
\int_R^{\widehat{R}}\mu [ -\lambda a_2b_1-b_2\lambda _1]
\Phi r^{N-1}dr\leq \int_R^{\widehat{R}}-\frac{\lambda \mu
}2a_1a_2v\Phi r^{N-1}dr;
$$
that is,
\begin{equation}
\int_R^{\widehat{R}}\frac{a_1a_2}2v\Phi r^{N-1}dr\leq
\int_R^{\widehat{R}}[ a_2b_1+\frac{b_2\lambda _1}{\varepsilon
_0}] \Phi r^{N-1}dr, \label{eq5}
\end{equation}
(because $\lambda \geq \varepsilon _0$).
Similarly
\begin{equation}
\int_R^{\widehat{R}}\frac{a_1a_2}2u\Phi r^{N-1}dr\leq
\int_R^{\widehat{R}}[ a_1b_2+\frac{b_1\lambda _1}{\varepsilon
_0}] \Phi r^{N-1}dr. \label{eq6}
\end{equation}
Adding \eqref{eq5} and \eqref{eq6}, we obtain the
inequality
$$
\int_R^{\widehat{R}}(u+v)\Phi r^{N-1}dr\leq
\frac{2}{a_1a_2}\int_R^{\widehat{R}}[ a_1b_2+\frac{b_1\lambda
_1}{\varepsilon _0}+a_2b_1+\frac{b_2\lambda _1}{\varepsilon
_0}] \Phi r^{N-1}dr.\\
$$
Then
\begin{align*}
(u+v)(R_0)\int_{\overline{t}}^{R_0}\Phi r^{N-1}dr
&\leq  \int_{\overline{t}}^{R_0}(u+v)\Phi r^{N-1}dr\\
&\leq \int_{R}^{\widehat {R}}(u+v)\Phi r^{N-1}dr\\
&\leq  \frac{2}{a_1a_2}\int_R^{\widehat{R}}[
a_1b_2+\frac{b_1\lambda _1}{\varepsilon _0}+a_2b_1+\frac{b_2\lambda
_1}{\varepsilon _0}] \Phi r^{N-1}dr,
\end{align*}
where $\overline{t}=\max (\overline{t}_1,\overline{t}_2)$ with
$\overline{t}_1$ and $\overline{t}_2$ are such that
\[
\overline{t}_1=\max \{ r\in (R,\widehat{R}) :u'(r)=0\},\quad
\overline{t}_2=\max \{ r\in (R,\widehat{R}): v'(r)=0\}.
\]
The proof is complete.
\end{proof}

We remark that $\overline{t}_i\leq R_0$, for $i=1,2$, was shown
in \cite{g1}.
Now, assume that there exists $z\geq 0$ ($z\not\equiv 0$) on $\overline{I}$ 
where
$I=(\alpha,\beta)$, and a constant $\gamma$ such that
\begin{equation}
-(r^{N-1}z')'\geq \gamma r^{N-1}z\,,\quad r\in I.
\label{eq7}
\end{equation}
Let $\lambda _1=\lambda _1(I)>0$ denote the principal eigenvalue of
\begin{equation}
 \begin{gathered}
-(r^{N-1}\Psi')'=\lambda r^{N-1}\Psi,\quad r\in (\alpha,\beta)\\
\Psi(\alpha)=0=\Psi (\beta),
\end{gathered} \label{eq8}
\end{equation}
where $0<\alpha<\beta\leq 1$.

\begin{lemma} \label{lem2.3}
Let \eqref{eq7} hold. Then $\gamma \leq \lambda _1(I)$.
\end{lemma}

\begin{proof}
Multiplying \eqref{eq7} by $\Psi$ ($ \Phi>0$), an eigenfunction
corresponding to the principal eigenvalue $\lambda _1(I)$, and
 integrating by parts (twice) we obtain
\begin{equation}
\int_\alpha^\beta[ \gamma -\lambda _1(I)] r^{N-1}z\Psi
dr\leq \beta^{N-1}\Psi
'(\beta)z(\beta)-\alpha^{N-1}\Psi'(\alpha)z(\alpha). \label{eq9}
\end{equation}
However, $\Psi'(\beta)<0$ and $\Psi'(\alpha)>0$; hence the
right-hand side of \eqref{eq9} is less than or equal to zero.
Then $\gamma\leq\lambda _1(I)$, and the proof is complete.
\end{proof}

Now, we define
\[
R_1=R_0+\frac{\widehat{R}-R_0}3, \quad
R_2=R_0+\frac{2(\widehat{R}-R_0)}3.
\]

\begin{lemma}\label{lem2.4}
 For $\lambda \mu $ sufficiently large,
$u(R_2)\leq \beta _2$ or $v(R_2)\leq \beta _1$, where $\beta _1$ and
$\beta _2$ are the unique positive zeros of $f$ and $g$
respectively.
\end{lemma}

\begin{proof}
We argue by contradiction. Suppose that $u(R_2)>\beta _2$
and $v(R_2)>\beta _1$.

\noindent\textbf{Case 1:} $u(R_1)>\rho _2$ or $v(R_1)>\rho _1$,
where $\rho _1=\frac{\beta_1 +\theta_1}2$ and
$\rho _2=\frac{\beta_2 +\theta_2}2$
($\theta_1$ and $\theta_2$ are the unique zeros of
 $F$ and $G$ respectively where $F(x)=\int_0^xf(t)dt$
and $G(x)=\int_0^xg(t)dt$).
If $u(R_1)>\rho _2$ then
\[
-(r^{N-1}v')' =\mu r^{N-1}g(u)
\geq \varepsilon _0r^{N-1}g(\rho _2)\quad\text{in }J=(R_0,R_1)
\]
and $v(r)\geq \beta _1$ on $\bar{J}$.

Let $\omega $ be the unique solution of
\begin{gather*}
-(r^{N-1}\omega')' = \varepsilon _0r^{N-1}g(\rho
_2)\quad \text{in }J\\
\omega =\beta _1\quad \text{in }\partial J.
\end{gather*}
Then by comparison arguments,
$ v(r)\geq \omega (r)=\varepsilon _0g(\rho _2)\omega _0(r)+\beta _1$
on $\bar{J}$, where
$\omega _0$ is the unique (positive) solution of
\begin{gather*}
-(r^{N-1}\omega _0')' =r^{N-1}\quad\text{in }J \\
\omega _0 = 0\quad \text{on }\partial J.
\end{gather*}
In particular, there exists $\overline{\beta }_1>\beta _1$
(we choose $\overline{\beta }_1$ such that
$f(\overline{\beta }_1)\neq 0$) such that
\[
v(R_0+\frac{2(R_1-R_0)}3) \geq \omega (R_0+\frac{2(R_1-R_0)}3)
\geq \overline{\beta }_1
\]
in $J^{*}=(R_0+\frac{R_1-R_0}3,R_0+\frac{2(R_1-R_0)}3)$.
Then
\begin{align*}
-(r^{N-1}(u-\beta _2)')'
&=\lambda r^{N-1}f(v)\\
&\geq \lambda r^{N-1}f(\overline{\beta }_1)\\
&\geq (\frac{\lambda f(\overline{\beta }_1)}C)
r^{N-1}(u-\beta _2)\quad \text{on }J^{*},
\end{align*}
(where $C$ is as in Lemma \ref{lem2.2}).
 Since $u-\beta _2>0$ on $\bar{J}^*$, it follows that
\begin{equation}
\frac{\lambda f(\overline{\beta }_1)}C\leq \lambda _1(J^{*}),
\label{eq10}
\end{equation}
where $\lambda _1(J^{*})$ is the principal value of \eqref{eq8}
(with $(\alpha,\beta)=J^{*}$).

Next we consider
\begin{align*}
(r^{N-1}(v-\beta _1)')'
&= \mu r^{N-1}g(u)\\
&\geq \mu r^{N-1}g(\rho _2)\\
&\geq (\frac{\mu g(\rho _2)}C) r^{N-1}(v-\beta
_1)\quad \text{on }J.
\end{align*}
Since $v-\beta _1>0$ on $\bar{J}$, it follows that
\begin{equation}
\frac{\mu g(\rho _2)}C\leq \lambda _1(J), \label{eq11}
\end{equation}
where $\lambda _1(J)$ is the principal value of \eqref{eq8}
(with $(\alpha,\beta)=J$).
Combining \eqref{eq10} and \eqref{eq11}, we obtain
$$
\frac{\lambda \mu f(\overline{\beta }_1)g(\rho _2)}{C^2}\leq \lambda
_1(J^{*})\lambda _1(J),
$$
But $f(\overline{\beta }_1)$, $g(\rho _2)$ and $C$ are fixed positive
constants.
This is a contradiction for $\lambda \mu $ large.
A similar contradiction can be reached for the case
$v(R_1)>\rho _1$.

\noindent\textbf{Case 2:} $u(R_1)\leq \rho _2$ and $v(R_1)\leq
\rho _1$.
Then $\beta _2<u\leq \rho _2$ and $\beta _1<v\leq \rho _1$ on
$J_1=[R_1,R_2]$. Then by the mean value theorem, there exist
$c_1,c_2\in (R_1,R_2)$ such that
$$
|u'(c_2)|\leq \frac{\rho_2}{R_2-R_1}, \quad
|v'(c_1)|\leq \frac{\rho_1}{R_2-R_1}.
$$
Since $-(r^{N-1}u')'\geq 0$ on $[R_1,R_2)$, we have
$$
-r^{N-1}u'(r)\leq -c_2^{N-1}u'(c_2)\quad \text{on }J_2=[R_1,c_2);
$$
thus
\[
| u'(r)|  \leq \frac{c_2^{N-1}}{r^{N-1}}u'(c_2)
\leq (\frac{R_2}{R_1}) ^{N-1}\frac{\rho_2}{R_2-R_1}\quad
\text{in }J_2.
\]
Similarly, we obtain
$$
| v'(r)| \leq (\frac{R_2}{R_1}) ^{N-1}
\frac{\rho _1}{R_2-R_1}\quad \text{in }J_3=[R_1,c_1).
$$
Hence there exists $r_0\in (R_1,R_2)$ such that
$$
| u'(r_0)| \leq \widetilde{c}, \quad
v'(r_0)| \leq \widetilde{c},
$$
where
$$
\widetilde{c}=\frac 1{R_2-R_1}(\frac{R_2}{R_1})
^{N-1}\max (\rho _2,\rho _1).
$$
Now, we define the energy function
$$
E(r)=u'(r)v'(r)+\lambda F(v(r))+\mu G(u(r)).
$$
Then
\[
E'(r)=-\frac{2(N-1)}ru'(r)v'(r)\leq 0,
\]
and hence $E\geq 0$ on $[R,\widehat{R}]$, (because
$u'(\widehat{R})v'(\widehat{R})\geq 0$).
However,
\begin{equation}
E(r_0)\leq \widetilde{c}^2+\lambda F(\rho _1)+\mu G(\rho _2),
\label{eq14}
\end{equation}
and $F(\rho _1)<0$ and $G(\rho _2)<0$. Hence $E(r_0)<0$ for
$\lambda \mu$ large which is a contradiction. The proof is complete.
\end{proof}

\begin{proof}[Proof of Theorem \ref{thm2.1}]
Assume $\lambda \mu $ is large enough so that both
lemmas \ref{lem2.2}, \ref{lem2.4} hold.
We take the case when  $u(R_2)\leq \beta _2$. Then
\begin{gather*}
-(r^{N-1}v')' =\mu r^{N-1}g(u)\leq 0\quad \text{on }
 J_3=(R_2,\widehat{R})\\
v(R_2) \leq C,\quad v(\widehat{R})=0,
\end{gather*}
hence, by a comparison argument, $v(r)\leq \widetilde{\omega }(r)$,
where $\widetilde{\omega }$ is the solution of
\begin{gather*}
-(r^{N-1}\widetilde{\omega }')' =0\quad \text{on }J_3\\
\widetilde{\omega }(R_2)=C,\quad
\widetilde{\omega}(\widehat{R})=0.
\end{gather*}
However, $\widetilde{\omega }(r)=
C\int_r^{\widehat{R}}s^{1-N}ds/\int_{R_2}^{\widehat{R}}s^{1-N}ds$
decreases from $C$ to $0$ on
$[R_2,\widehat{R}]$, hence there exists $r_1\in
(R_2,\widehat{R})$ (independent of $\lambda \mu $) such that
$\widetilde{\omega }(r_1)=\beta _1/2$.

{\bf Remark.} Here, we assume that $\beta _1/2<C$,
unless we can choose $N_0$ such that $\beta _1/N_0<C$.

Hence $v(r_1)\leq \beta _1/2$, and
\begin{align*}
-(r^{N-1}(\beta _2-u)')'
&= -\lambda r^{N-1}f(v)\\
&\geq -\lambda r^{N-1}f(\frac{\beta _1}2)\\
&\geq \lambda \big(-f(\frac{\beta _1}2)\big) r^{N-1}\frac{\beta
_2-u}{\beta _2}\quad \text{on }J_4=(r_1,\widehat{R}).
\end{align*}
Since $\beta _2-u>0$ on $\bar{J}_4$, we have
\begin{equation}
\frac{\lambda \widetilde{K}_1}{\beta _2}\leq \lambda _1(J_4),
\label{eq15}
\end{equation}
where $\widetilde{K}_1=-f(\beta _1/2)$ and
$\lambda_1(J_4)$ is the principal eigenvalue of \eqref{eq8}
(with $(\alpha,\beta)=J_4$).
Similarly, there exists $r_2\in (r_1,\widehat{R})$
(independent of $\lambda \mu $) such that
$$
v(r_2)<\frac{\beta _1}2.
$$
Hence
\begin{gather*}
-(r^{N-1}u')' = \mu r^{N-1}f(v)\leq 0\quad \text{on }
 J_5=(r_2,\widehat{R})\\
u(r_2) \leq C,\quad u(\widehat{R})=0,
\end{gather*}
then, by a comparison argument we obtain
$$
u(r)\leq \omega _1(r)=\frac
C{\int_{r_2}^{\widehat{R}}s^{1-N}ds}\int_r^{\widehat{R}}s^{1-N}ds;
$$
thus
\begin{gather*}
-(r^{N-1}\omega _1')'=0,\quad \text{on }J_5,\\
\omega_1(r_2)=C,\quad \omega _1(\widehat{R})=0.
\end{gather*}
Arguing as before there exists $r_3\in (r_2,\widehat{R})$
(independent of $\lambda \mu $) such that
$$
u(r_3)\leq \omega _1(r_3)\leq \frac{\beta _2}2<C.
$$
Hence
\begin{align*}
-(r^{N-1}(\beta _1-v)')'
&= -\mu r^{N-1}g(v)\\
&\geq -\mu r^{N-1}g(\frac{\beta _2}2)\\
&\geq \mu \big(-g(\frac{\beta _2}2)\big) r^{N-1}\frac{\beta
_1-v}{\beta _1}\quad \text{on }J_6=(r_3,\widehat{R}).
\end{align*}
Since $\beta _1-v>0$ on $\bar{J}_6$, it follows that
\begin{equation}
\frac{\mu \widetilde{K}_2}{\beta _1}\leq \lambda_1(J_6),\label{eq16}
\end{equation}
where $\widetilde{K}_2=-g(\beta _1/2)$ and
$\lambda_1(J_6)$ is the principal eigenvalue of \eqref{eq8}
(with $(\alpha,\beta)=J_6$).
Combining \eqref{eq15} and \eqref{eq16}, we obtain
$$
\frac{\lambda \mu \widetilde{K}_1\widetilde{K}_2}{\beta _1\beta
_2}\leq \lambda _1(J_4)\lambda _1(J_6),
$$
which is a contradiction to $\lambda \mu $ being large.

A similar contradiction can be reached for the case
$v(R_2)\leq \beta_1$. Hence Theorem \ref{thm2.1} is proven.
\end{proof}


\begin{thebibliography}{00}

\bibitem{a1} D. Arcoya and A. Zertiti;
\emph{Existence and non-existence
of radially symmetric non-negative solutions for a class of
semi-positone problems in annulus}, Rendiconti di Mathematica, serie
VII, Volume 14, Roma (1994), 625-646.

\bibitem{b1} K .J. Brown, A. Castro and R. Shivaji;
\emph{Non-existence of radially symmetric non-negative solutions
for a class of semi-positone problems},
Diff. and Int. Equations,2. (1989), 541-545.

\bibitem{c1} A. Castro and R. Shivaji;
\emph{Nonnegative solutions for a class of radially symmetric
nonpositone problems}, Proc. AMS, 106(3) (1989), pp. 735-740.

\bibitem{g1} B. Gidas, W. M. Ni and  L. Nirenberg;
\emph{Symmetry and related properties via the maximum principle},
Commun. Maths Phys., 68 (1979), 209-243.

\bibitem{h1} D. D. Hai;
\emph{On a class of semilinear elliptic systems},
Journal of Mathematical Analysis and
Applications. Volume 285, issue 2, (2003), pp. 477-486.

\bibitem{h2} D. D. Hai and R. Shivaji;
\emph{Positive solutions for semipositone systems in the annulus},
Rocky Mountain J. Math., 29(4) (1999), pp. 1285-1299.

\bibitem{h3} D. D. Hai, R. Shivaji and S. Oruganti;
\emph{Nonexistence of Positive Solutions for a Class of Semilinear
Elliptic Systems}, Rocky Mountain Journal of Mathematics.
 Volume 36, Number 6 (2006), 1845-1855.

\bibitem{h4} S. Hakimi and A. Zertiti;
\emph{Radial positive solutions for a
nonpositone problem in a ball}, Eletronic Journal of Differential
Equations, Vol. 2009 (2009), No. 44, pp. 1-6.

\bibitem{h5} S. Hakimi and A.  Zertiti;
\emph{Nonexistence of radial positive solutions for a
nonpositone problem}; Eletronic Journal of Differential Equations,
Vol. 2011 (2011), No. 26, pp. 1-7.


\end{thebibliography}

\end{document}