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

\AtBeginDocument{{\noindent\small
\emph{Electronic Journal of Differential Equations},
Vol. 2011 (2011), No. 26, 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-2011/26\hfil Nonexistence of radial positive solutions]
{Nonexistence of radial positive solutions for a nonpositone problem}

\author[S. Hakimi, A. Zertiti\hfil EJDE-2011/26\hfilneg]
{Said Hakimi, Abderrahim Zertiti}  % in alphabetical order

\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}

\address{Abderrahim Zertiti \newline
Universit\'e Abdelmalek Essaadi,
Facult\'e des sciences \\
D\'epartement de Math\'ematiques \\
BP 2121, T\'etouan, Morocco}
\email{zertitia@hotmail.com}

\thanks{Submitted June 14, 2010. Published February 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 problem when the
 nonliearity is superlinear and has more than one zero.
\end{abstract}

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

\section{Introduction}

We study the nonexistence of radial positive solutions
for the boundary-value problem
\begin{equation}
\begin{gathered}
-\Delta u(x)=\lambda f(u(x))\quad x\in \Omega, \\
u(x)=0\quad  x\in \partial \Omega,
\end{gathered}  \label{eq1}
\end{equation}
 where $\lambda >0$, $f: [ 0,+\infty ) \to \mathbb{R}$
is a continuous nonlinear function that has more than one zero,
and  $\Omega \subset \mathbb{R}^N$; is the annulus:
$\Omega =C(0,R,\widehat{R}) =\{x\in \mathbb{R}^N: R<|x| <\widehat{R}\}$
 ($N>2$, $0<R<\widehat{R}$).

When $f$ is a nondecreasing  satisfying
$f(0)<0$ (the nonpositone case) and has only one zero, problem
\eqref{eq1} has been studied by  Brown,  Castro and  Shivaji \cite{b1}
in the ball,
and by  Arcoya and  Zertiti \cite{a1} in the annulus.

We observe that the nonexistence of radial positive solutions of
\eqref{eq1} is equivalent to the nonexistence of positive solutions
of the  problem
\begin{equation}
\begin{gathered}
-u''(r)-\frac{N-1}ru'(r)=\lambda
f(u(r)) \quad R<r<\widehat{R} \\
u(R)=u(\widehat{R})=0,\label{eq2}
\end{gathered}
\end{equation}
where $\lambda >0$.

Our main objective in this article is to prove that the result of
nonexistence of radial positive solutions of  \eqref{eq1}
remains valid when $f$ has more than one zero and is not increasing
entirely on $[0,+\infty )$; see \cite[Theorem 3.1]{a1}.
 More precisely we assume that the map
 $f: [0,+\infty ) \to \mathbb{R}$ satisfies the following
hypotheses
\begin{itemize}
\item[(H1)] $f\in C^1([ 0,+\infty ),\mathbb{R})$ such that
$f$ has three zeros $\beta _1<\beta_2<\beta_3$ with
$f'(\beta _i) \neq 0$ for all $i\in \{1,2,3\}$.
 Moreover, $f'\geq 0$ on $[ \beta _3,+\infty )$.

\item[(H2)] $f(0)<0$.

\item[(H3)] $\lim_{u\to +\infty } f(u)/u=+\infty$,
\end{itemize}

\section{The main result}

In this section, we give the main result in this work. More
precisely we shall prove the following theorem.

\begin{theorem} \label{thm2.1}
Assume that the hypotheses {\rm (H1)--(H3)} are
satisfied. Then there exists a positive real number $\lambda _0$
such that if $\lambda >\lambda _0$, problem \eqref{eq1} has no radial
positive solution.
\end{theorem}

\noindent\textbf{Remark.} We do not know for what radius
$r\in (R,\widehat{R})$  the solution $u$ attains its
maximum. In addition, $f$ changes sign in $(\beta _1,+\infty )$.
These two facts make our study more difficult, and
change the proof of nonexistence in \cite{a1}.

To prove Theorem \ref{thm2.1}, we need the next three technical
lemmas. We note that the proofs of the
first and the last lemma are analogous to those of
\cite[Lemma 3.2, Lemma 3.4]{a1}.
On the other hand, the proof of the second lemma is
different from that of \cite[Lemma 3.3]{a1}.
This is so because our  $f$ has no  constant sign in
$(\beta _1,+\infty )$.

 Denote by $u_\lambda (r)$ a positive solution of
\eqref{eq1} (if it exists) and let
$R_0=(R+\widehat{R})/2$.
Following the work \cite{a1}, we introduce the
following notation:
$\beta =\beta _1$,
$\theta =\min \{\beta _2\text{,}\min \theta _i\}$ where
$\theta _i$ are the zeros of $F$ ($F(x)=\int_0^xf(t)dt$).
\medskip

\noindent\textbf{Remark.} In \cite[Theorem B iii]{g1},
$F$ has at most one zero. On the opposite, in our case $F$ may
have more than one zero because $f$ has a finite number of zeros.
In this paper we assume, with out loss of generality,
that $f$ has three zeros.
In fact, the number of zeros of  $F$  depends on $f$,
but $F$ has at most three zeros.

\begin{lemma} \label{lem2.2}
Let $f\in C^1([ 0,+\infty)) $
satisfying {\rm (H3)} and consider $\lambda>2$.
If $u_\lambda$ is a positive solution of \eqref{eq2},
then for every $r\in(R_0,\widehat{R}]$ there exists a
positive number $M=M(r)>0$ (independent of $\lambda $) such that
$u_\lambda (r)\leq M$.
\end{lemma}

\begin{proof}
Let $\varphi _1$ be a positive eigenfunction associated to
the first eigenvalue $\mu _1>0$ of the eigenvalue problem
\begin{gather*}
 -(r^{N-1}v')'=\mu r^{N-1}v,\quad R<r<\widehat{R}\\
v(R)=0=v(\widehat{R}),
\end{gather*}
Multiplying the equation in \eqref{eq2} by $r^{N-1}\varphi _1(r)$
and integrating from $R$ to $\widehat{R}$, we obtain
\[
\int_R^{\widehat{R}}r^{N-1}u_\lambda '(r)\varphi'_1(r)dr
=-\int_R^{\widehat{R}}(r^{N-1}u_\lambda '(r))'\varphi
_1(r)dr,
\]
hence
\begin{equation}
\int_R^{\widehat{R}}r^{N-1}u_\lambda '(r)\varphi'_1(r)dr=\lambda
\int_R^{\widehat{R}}r^{N-1}f(u_\lambda
(r))\varphi_1(r)dr.\label{eq3}
\end{equation}
On the other hand, multiplying the equation $-(r^{N-1}\varphi
_1'(r))'=\mu _1r^{N-1}\varphi _1(r)$, $(R<r<\widehat{R})$
by $u_\lambda $ and integrating from $R$ to $\widehat{R}$, we obtain
\[
\int_R^{\widehat{R}}r^{N-1}u_\lambda'(r)\varphi'_1(r)dr
=-\int_R^{\widehat{R}}(r^{N-1}\varphi _1'(r))'u_\lambda
(r)dr,
\]
hence
\begin{equation}
\int_R^{\widehat{R}}r^{N-1}u_\lambda'(r)\varphi'_1(r)dr=\mu
_1\int_R^{\widehat{R}}r^{N-1}\varphi _1(r)u_\lambda (r)dr.
\label{eq4}
\end{equation}
Combining \eqref{eq3}, \eqref{eq4} and choosing
$\mu >\mu _1/2$, $c>0$ such that
\[
f(\zeta)\geq \mu \zeta -c,\quad \forall \zeta \geq 0
\]
(because $f$ is superlinear), we deduce
\begin{align*}
\mu _1\int_R^{\widehat{R}}r^{N-1}\varphi _1(r)u_\lambda(r)dr
&= \lambda \int_R^{\widehat{R}}r^{N-1}f(u_\lambda (r))\varphi_1(r)dr\\
&\geq \lambda \mu\int_R^{\widehat{R}}r^{N-1}\varphi _1(r)u_\lambda
(r)dr-\lambda  c\int_R^{\widehat{R}}r^{N-1}\varphi _1(r)dr,
\end{align*}
from which
\[
\int_R^{\widehat{R}}r^{N-1}\varphi _1(r)u_\lambda (r)dr
\leq \frac{\lambda k}{\lambda \mu -\mu _1}
\leq \frac k{\mu -\frac{\mu _1}2}:= A, \quad \forall \lambda >2,
\]
with $k=c\int_R^{\widehat{R}}r^{N-1}\varphi _1(r)dr>0$ and $A>0$ is
independent of $\lambda$. Now, let $r\in(R_0,\widehat{R}]$ and
choosing $\delta>0$ such that $R_0<r-\delta $ and using the fact
that $u_\lambda $ is non-increasing in $(R_0,\widehat{R})$ (see
\cite{g2}) implies
\begin{align*}
u_\lambda(r)
&\leq  \frac{\int_{r-\delta
}^rt^{N-1}u_\lambda(t)\varphi
_1(t)dt}{\int_{r-\delta }^rt^{N-1}\varphi _1(t)dt}\\
&\leq  \frac{\int_{R}^{\widehat{R}}t^{N-1}u_\lambda(t)\varphi
_1(t)dt}{\int_{r-\delta }^rt^{N-1}\varphi _1(t)dt}\\
&\leq  \frac A{\int_{r-\delta }^rt^{N-1}\varphi _1(t)dt}=M,\quad
\forall \lambda >2\,.
\end{align*}
The proof is complete.
\end{proof}

\begin{lemma} \label{lem2.3}
Assume {\rm (H1)--(H3)} and let
$R_1\in (R_0,\widehat{R})$,
$c\in (\beta ,\theta )$. Then there exists $\lambda _1>0$
such that for all positive solutions $u_\lambda $ of
 \eqref{eq2} with $\lambda \geq \lambda _1$, there exists
$t_1=t_1(\lambda )\in (R_0,R_1)$ satisfying $u_\lambda (t_1)<c$.
\end{lemma}

\begin{proof}
We argue by contradiction. Suppose that there exists a sequence
$\{\lambda _n\}\subset (0,+\infty) $ converging to $+\infty $
such that
\[
u_{\lambda _n}(r)\geq c,\quad \forall r\in (R_0,R_1]
,\; \forall n\in \mathbb{N}.
\]
Consider $\overline{t}_n=\max \{r\in (R,\widehat{R}):
u_{\lambda _n}'(r)=0\}$. Then
$u_{\lambda _n}'(r)<0$\;for all $r\in (\overline{t}_n,\widehat{R}) $,
and we deduce
\[
{u_{\lambda _n}(r)\leq u_{\lambda _n}(\overline{t}_n),\quad
\forall r\in (\overline{t}_n,\widehat{R})}.
\]
It follows that $u_{\lambda _n}'(\overline{t}_n)=0$ and
$u_{\lambda _n}''(\overline{t}_n)\leq0$. So
$f(u_{\lambda _n}(\overline{t}_n))\geq0$ by \eqref{eq2}.
 Hence
\[
u_{\lambda _n}(\overline{t}_n)\leq \beta _2\quad \text{or}\quad
u_{\lambda _n}(\overline{t}_n)\geq \beta _3.
\]
Now, we study the following two cases:

\textbf{Case 1:}
 $u_{\lambda_n}(\overline{t}_n)\leq\beta _2$.

\textbf{(i)} If $\sup_{n} u_{\lambda _n}(\overline{t} _n)<\beta _2$,
then
\begin{align*}
-r^{N-1}u_{\lambda _n}'(r)
&= \lambda _n\int_{\overline{t}_n}^r
s^{N-1}f(u_{\lambda _n}(s))ds,\quad \forall r\in (R_0,R_1)
\\
&\geq \lambda _n\inf_{\xi \in (c,\sup_{n}
 u_{\lambda _n}(\overline{t}_n))} f(\xi) \int_{R_0}^rs^{N-1}ds.
\end{align*}
Since $\sup_{n} u_{\lambda _n}(\overline{t} _n)<\beta _2$,
it follows that
$\inf_{\xi \in (c,\sup_{n} u_{\lambda _n}(\overline{t}_n)) }f(\xi )>0$.
Therefore,
\begin{equation}
\lim_{n\to +\infty } u_{\lambda _n}'(r)=-\infty,\; \text{uniformly
on compact subsets of} \left( R_0,R_1\right) . \label{eq5}
\end{equation}
Now, let $r_1, r_2\in (R_0,R_1) $ such that
$R_0<r_1<r_2<R_1$.
By the mean value theorem, there exists $r_n\in (
r_1,r_2) $ such that
\[
u_{\lambda _n}(r_2)=u_{\lambda _n}(r_1)+(r_2-r_1)
u_{\lambda _n}'(r_n).
\]
Also, for all $r\in (R_0,R_1)$ we have $c\leq u_{\lambda
_n}(r)<\beta_2$  for all $n$, and by \eqref{eq5} the second summand
of the precedent equality tends to $-\infty $.
Hence
\[
\lim_{n\to +\infty }u_{\lambda _n}(r_2)=-\infty.
\]
This  contradicts $u_{\lambda _n}\geq 0$ for all
$n\in \mathbb{N}$.

\textbf{(ii)} If $\sup_{n}  u_{\lambda _n}(\overline{t}_n)=\beta
_2$. Consider the  following two sets:
\begin{gather*}
\Phi _n=\{r\in [R_1,\widehat{R}] : \beta \leq u_{\lambda
_n}(r)\leq \frac{3\beta +c}4\},\\
\Psi_n=\{r\in [R_1,\widehat{R}] : \frac{2(\beta
+c)}4\leq u_{\lambda _n}(r)\leq \frac{\beta +3c}4\}.
\end{gather*}
 Since $(\beta,\frac{3\beta+c}4),
(\frac{2(\beta +c)}4,\frac{\beta +3c}4) \subset
u_{\lambda _n}((R_1,\widehat{R}))$, by the
intermediate value theorem,
$\Phi_n$ and $\Psi_n$  are not empty.
Consider $\underline{a}(n),\;\overline{a}(n),\;
\underline{b}(n)$\;and
$\overline{b}(n)$ such that\\
$\underline{a}(n)=\inf_r \Psi_n,\; \overline{a}(n)=\sup_r \Psi_n,\;
\underline{b}(n)=\inf_r \Phi_n\;\text{and}\;
\overline{b}(n)=\sup_r \Phi_n$.\\
Let $r_0\in \left[\underline{a}(n),\overline{b}(n)\right]$. Then
\begin{align*}
-r_0^{N-1}u_{\lambda _n}'(r_0) &=\lambda
_n\int_{\overline{t}_n}^{r_0} s^{N-1}f(u_{\lambda _n}(s))ds \\
&\geq \lambda _nR^{N-1}\int_{R_0}^{r_0}f(u_{\lambda _n}(s))ds \\
&\geq \lambda _nR^{N-1}\int_{u_{\lambda _n}(R_0)}^{u_{\lambda
_n}(r_0)}\frac{f(t)} {u_{\lambda _n}'(u_{\lambda _n}^{-1}(t))}dt\\
&= \lambda _nR^{N-1}\int_{u_{\lambda _n}(r_0)}^{u_{\lambda
_n}(R_0)}\frac{f(t)} {-u_{\lambda _n}'(u_{\lambda _n}^{-1}(t))}dt,
\end{align*}
hence
\begin{align*}
-r_0^{N-1}u_{\lambda _n}'(r_0)(-u_{\lambda _n}'(s_0))
&\geq \lambda _nR^{N-1}\int_{u_{\lambda _n}(r_0)}^{u_{\lambda
_n}(R_0)}f(t)dt\\
&\geq \lambda_nR^{N-1}\int_{\frac{\beta +3c}4}^cf(t)dt,
\end{align*}
where $s_0$ satisfies $u_{\lambda
_n}'(s_0)=\underset{[R_0,r_0]}{\inf }u_{\lambda _n}'(s)$. Since the
function $r\longmapsto -r^{N-1}u_{\lambda _n}'(r)\;$is increasing on
$\left(\underline{a}(n),\overline{b}(n)\right)$,
$$
-r_0^{N-1}u_{\lambda _n}'(r_0)(-u_{\lambda _n}'(s_0))\leq
(-r_0^{N-1}u_{\lambda _n}'(r_0))^2\frac 1{s_0^{N-1}}.
$$
Then
$$
(-r_0^{N-1}u_{\lambda _n}'(r_0))^2\frac
1{s_0^{N-1}}\geq\lambda_nR^{N-1}\int_{\frac{\beta +3c}4}^cf(t)dt.
$$
Therefore,
\begin{equation}
\lim_{n\to +\infty }u_{\lambda _n}'(r_0)=-\infty. \label{eq6}
\end{equation}
Now, Let $r_1\in\left[\underline{a}(n),\overline{a}(n)\right]\;$and
$r_2\in \left[\underline{b}(n),\overline{b}(n)\right]$, then by the
mean value theorem, there exists $r*\in (r_1,r_2)$ such that
\begin{align*}
u_{\lambda _n}(r_2)&=u_{\lambda _n}(r_1)+\left(r_2-r_1\right)
u_{\lambda _n}'(r*)\\
&<u_{\lambda _n}(R_1)+\left(\underline{b}(n)-\overline{a}(n)\right)
u_{\lambda _n}'(r*)\\
&\leq u_{\lambda
_n}(R_1)+\inf_n \left(\underline{b}(n)-\overline{a}(n)\right)
u_{\lambda _n}'(r*).
\end{align*}
Since $u_{\lambda _n}(R_1)\leq M,$ for all $n$ and some $M=M(R_1)>0$
(see Lemma \ref{lem2.2}) and
$\inf_n \left(\underline{b}(n)-\overline{a}(n)\right)>0$
and $\lim_{n\to +\infty } u_{\lambda_n}'(r*)=-\infty $ (by
\eqref{eq6}), it follows that 
$\lim_{n\to +\infty }u_{\lambda _n}(r_2)=-\infty $, which contradicts 
$u_{\lambda _n}\geq0$ for all $n\in \mathbb{N}$.

\textbf{Case 2:} $u_{\lambda _n}(\overline{t}_n)\geq\beta _3$. Let
$r_0\in \left[\underline{a}(n),\overline{b}(n)\right]$, then
$$
-r_0^{N-1}u_{\lambda _n}'(r_0)=\lambda _n\int_{\overline{t}
_n}^{r_0}s^{N-1}f(u_{\lambda _n}(s))ds.
$$
Consider $t_{\beta _2}$ such that
$t_{\beta _2} =\max \{r_n\in
(R,\widehat{R}] : u_{\lambda_n} (r_n)=\beta_2\}$. Then
\begin{align*}
-r_0^{N-1}u_{\lambda _n}'(r_0)
&= \lambda _n\int_{\overline{t} _n}^{r_0}s^{N-1}f(u_{\lambda _n}(s))ds\\
&= \lambda_n\Big[\int_{\overline{t}
_n}^{t_{\beta _2}}s^{N-1}f(u_{\lambda _n}(s))ds+\int_{t_{\beta
_2}}^{r_0}s^{N-1}f(u_{\lambda _n}(s))ds\Big] \\
&\geq \lambda _n\int_{t_{\beta _2}}^{r_0}s^{N-1}f(u_{\lambda
_n}(s))ds,
\end{align*}
because $\int_{\overline{t}_n}^{t_{\beta _2}}s^{N-1}f(u_{\lambda
_n}(s))ds\geq0$.
Then as in Case 1, we obtain a
contradiction with the positivity of $u_{\lambda _n}$.
\end{proof}

\begin{lemma}\label{lem2.4}
Assume {\rm (H2)}. Let $R_2\in (R_0,\widehat{R}) $ and
$\overline{c} >1$. Then there exists $\lambda _2>0$ such that
every positive solution $ u_\lambda$ of \eqref{eq2} satisfies
$\frac \beta {\overline{c}}\in u_\lambda([ R_2,\widehat{R}] )$,
for all $\lambda \geq \lambda _2$.
Where $b_\lambda =\max \{r\in (R,\widehat{R})
:u_\lambda (r)=\frac \beta {\overline{c}}\}$.
\end{lemma}

\begin{proof}
 This lemma will be proved if we show that
 \begin{equation}
\lim_{\lambda \rightarrow +\infty } b_\lambda =\widehat{R}
\label{eq7}
\end{equation}
To do this, we multiply the equation in \eqref{eq2} by $r^{N-1}$,
integrate it from  $b_\lambda $ to $\widehat{R}$ and use that
$u_\lambda(r)<\frac \beta {\overline{c}}$, for all
$r\in (b_\lambda , \widehat{R}]$,
 to deduce that
$$
\int_{b_\lambda }^{\widehat{R}}(r^{N-1}u_\lambda '(r)) 'dr
\geq \int_{b_\lambda }^{\widehat{R}}\lambda r^{N-1}Kdr
$$
where $K=-\max \{f(\zeta ):\zeta \in [ 0,\frac \beta {\overline{c}}]\}>0$.
Hence
\begin{equation}
\widehat{R}^{N-1}u_\lambda '(\widehat{R})-b_\lambda ^{N-1}u_\lambda
'(b_\lambda )\geq \frac \lambda NK(\widehat{R}^N-b_\lambda
^N) >0\,.  \label{eq8}
\end{equation}
 On the other hand, multiplying  the same equation by
$r^{2(N-1)}u_\lambda '(r)$ and integrating from
$b_\lambda $ to $\widehat{R}$, we have
$$
-\int_{b_\lambda}^{\widehat{R}}[ r^{N-1}u_\lambda '(r)] 'u_\lambda
'(r)r^{N-1}dr=\lambda \int_{b_\lambda }^{\widehat{R}}[
F(u_\lambda (r))] 'r^{2(N-1)}dr
$$
Computing the two integrals by parts, we obtain
\begin{align*}
&\frac 12[ b_\lambda ^{2(N-1)}u_\lambda '(b_\lambda )^2-
\widehat{R}^{2(N-1)}u_\lambda '(\widehat{R})^2]\\
& =-\lambda b_\lambda ^{2(N-1)}F(\frac \beta {\overline{c}})
-2(N-1)\lambda \int_{b_\lambda }^{\widehat{R}}F(u_\lambda (r))
r^{2N-3}dr
\end{align*}
Since  $u_\lambda (r)<\frac \beta {\overline{c}}$,
for all $r\in (b_\lambda ,\widehat{R}] $ and $F$ is
decreasing in $(0,\beta ) $ by (H2), we deduce that
\begin{align*}
&\frac 12[ b_\lambda ^{2(N-1)}u_\lambda '(b_\lambda )^2-
\widehat{R}^{2(N-1)}u_\lambda '(\widehat{R})^2]\\
&\leq -\lambda b_\lambda ^{2(N-1)}F(\frac \beta
{\overline{c}})-2(N-1)F(\frac
\beta {\overline{c}})\lambda \int_{b_\lambda }^{\widehat{R}}r^{2N-3}dr \\
&= -\lambda \widehat{R}^{2(N-1)}F(\frac \beta {\overline{c}})
\end{align*}
By \eqref{eq8}, the left hand of the precedent inequality
is positive (because $u_\lambda'(b_\lambda )\leq 0$ by definition
of $b_\lambda $ and $u_\lambda '(\widehat{R})\leq 0$ by \cite{g2}).
Consequently we can take square roots and using that
$A-B\leq \sqrt{A^2-B^2}$ for all $A\geq B\geq 0$, we obtain
(by \eqref{eq8} again)
\[
\frac 1{N\sqrt{2}}K\frac 1{
\sqrt{-F(\frac \beta {\overline{c}})}}\sqrt{\lambda }(\widehat{R}
^N-b_\lambda ^N) \leq \widehat{R}^{N-1}
\]
 and as a consequence \eqref{eq7} is satisfied. So the
proof is complete.
\end{proof}

\begin{proof}[Proof of Theorem \ref{thm2.1}]
Let $c\in (\beta ,\theta ) $, $\overline{c}>1$ and $R_1$,
$R_2\in (R_0,\widehat{R})$ such that $R_1<R_2$.
Consider $\lambda _1,\lambda _2$  given
respectively by lemmas \ref{lem2.3} and \ref{lem2.4}, and choose
$\lambda ^{*}\geq \max \{\lambda _1,\lambda _2\}$ such
that
$$
\lambda ^{*}L+\frac{\mu ^2}2<0 ,
$$
where
$$
L=\max \{F(\zeta ):\frac \beta {\overline{c}}\leq \zeta \leq c\}.
$$
Hence \eqref{eq2} has no positive solutions for
$\lambda \geq\lambda ^{*}$. Otherwise, there exists
$\lambda \geq\lambda ^{*}$ such
that \eqref{eq2} has at least one positive solution $u_\lambda$.

Since $\lambda\geq\lambda_i$, $i=1, 2$ we deduce from lemmas
\ref{lem2.3}, \ref{lem2.4} the existence of $t_1\in(R_0,R_1]$ and
$t_2\in[R_2,\widehat{R}]$ satisfying $u_\lambda(t_1)<c$ and
$u_\lambda(t_2)=\frac \beta {\overline{c}}$.
Then by the mean value theorem there exists
$t_3\in [ t_1,t_2]$ such that
\[
|u_\lambda '(t_3)|
= \frac{|u_\lambda (t_2)-u_\lambda (t_1)| }{t_2-t_1}
\leq \mu,
\]
where $\mu =(\frac \beta {\overline{c}}+c)/(R_2-R_1)$.

Consider the energy function
$E(r)=\lambda F(u_\lambda (r))+\frac{u_\lambda '(r)^2}2$.
Then for all $\lambda \geq \lambda ^{*}$,
\[
E(t_3)\leq \lambda L+\frac{\mu ^2}2\leq \lambda ^{*}L+\frac{\mu
^2}2<0
\]
(because $L<0$ and $u_\lambda (t_3)\in [ \frac \beta {\overline{c}},c] $).
This is a contradiction, since $E$ is a non-increasing
function (recall that
$E'(r)=-\frac{N-1}ru'(r)^2\leq 0$) and $E(\widehat{R}
)=\frac{u'(\widehat{R})^2}2\geq 0$. Hence the result follows.
\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{g1} X. Garaizar;
\emph{Existence of Positive Radial Solutions for
Semilinear Elliptic Equations in the Annulus}, Journal of
Differential Equations, 70 (1987), 69-92.

\bibitem{g2} B. Gidas, W.M. Ni, L. Nirenberg;
\emph{Symmetry and related properties via the maximum principle},
Commun. Maths Phys., 68 (1979), 209-243.
\end{thebibliography}

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