\documentclass[reqno]{amsart}
\usepackage{hyperref}
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\AtBeginDocument{{\noindent\small
\emph{Electronic Journal of Differential Equations},
Vol. 2015 (2015), No. 130, pp. 1--9.\newline
ISSN: 1072-6691. URL: http://ejde.math.txstate.edu or http://ejde.math.unt.edu
\newline ftp ejde.math.txstate.edu}
\thanks{\copyright 2015 Texas State University - San Marcos.}
\vspace{9mm}}

\begin{document}
\title[\hfilneg EJDE-2015/130\hfil Non-existence of positive radial solution]
{Non-existence of positive radial solution for  semipositone
weighted p-Laplacian problems}

\author[S. Herr\'on,  E. Lopera \hfil EJDE-2015/130\hfilneg]
{Sigifredo Herr\'on,  Emer Lopera}

\address{Sigifredo Herr\'on \newline
Universidad Nacional de Colombia
Sede Medell\'in,  Apartado A\'ereo 3840,
Medell\'in, Colombia}
\email{sherron@unal.edu.co}

\address{Emer Lopera \newline
Universidad Nacional de Colombia
Sede Medell\'in,  Apartado A\'ereo 3840,
Medell\'in, Colombia}
\email{edlopera@unal.edu.co}

\thanks{Submitted January 12, 2015. Published May 7, 2015.}
\subjclass[2010]{35J92, 35J60, 35J62}
\keywords{Semipositone;  quasilinear weighted elliptic equation;
\hfill\break\indent  positive radial solution; non-existence result; 
 Pohozaev identity}

\begin{abstract}
 We prove the non-existence of positive radial solution to
 a semipositone weighted $p$-Laplacian problem whenever the weight is
 sufficiently large. Our main tools are a Pohozaev  type identity 
 and a comparison principle. 
\end{abstract}

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

\section{Introduction}

We consider the non-existence of positive radial  solutions to the problem
\begin{equation}\label{eq1}
\begin{gathered}
-\Delta_p u= W (\|x\|) f(u)  \quad \text{in }  B_1(0),\\
u=0 \quad \text{on }  \partial B_1(0),
\end{gathered}
\end{equation}
where $\Delta_pu=\nabla\cdot(|\nabla u|^{p-2}\nabla u)$ is the $p$-Laplace 
operator, $B_1(0)$ is the unit ball in $\mathbb{R} ^N$ and $2< p< N$.

Note that solving this problem is equivalent to solving the  problem
\begin{equation} \label{eq2}
[r^n\varphi_p (u')]'=-r^n W (r) f(u),  \quad 0<r<1, \quad u'(0)=0, \quad u(1)=0,
\end{equation}
where $r=\|x\|$, $n:=N-1$ and ${}'=\frac{d}{dr}$.
The differential equation in the last problem is equivalent to
\begin{equation}\label{eq3}
(p-1)|u'|^{p-2}u''+\frac{n}{r}|u'|^{p-2}u'+ W (r) f(u)=0, \quad 0<r<1.
\end{equation}

We assume that the nonlinearity satisfies the following hypotheses:
\begin{itemize}
\item [(F1)] $f:[0,\infty)\to \mathbb{R}$ is a continuous function 
with exactly three zeros $0<\beta_1<\beta_2<\beta_3$.

\item [(F2)] $f(0)<0$ and $f$ is increasing from $\beta_3$ on.

\item[(F3)] Set $F(t):=\int_0^t f(s)ds$. Then, $\beta_3 < \theta_1$ where 
$\theta_1$ is the unique positive zero of $F$.

\end{itemize}
Let us fix $\beta_3<\gamma<\theta_1$. We will say that a function $ W$
is an admissible weight if it satisfies the following conditions:
\begin{itemize}
\item[(W1)] $ W :[0,1]\to (0, \infty)$ is continuous  and differentiable in $(0,1)$.
\item[(W2)] $\widetilde{ W}(r):=N+r\frac{ W ' (r)}{ W (r)}$ is defined a.e. in 
 $[0,1]$.
\item[(W3)] If we define the following real numbers associated with 
$W$:
\begin{equation*}
\overline{\eta}:=\sup{ \widetilde{ W}}, \quad
\underline{\eta}:=\inf{\widetilde{ W}}, \quad
\overline{\lambda}:=\max{ W}, \quad   \underline{\lambda}:=\min{ W}, \quad
\widetilde{C}_\lambda:=\frac{\overline{\lambda}}{\underline{\lambda}},
\end{equation*}
then
\begin{gather}\label{weight2}
\frac{\underline{\eta}}{N}F(s)-\frac{1}{p*}f(s)s\leq\widetilde{C}_\lambda F(\gamma), 
\quad \text{for } \gamma\leq s <\theta_1, \\
\label{weight1}
\frac{\overline{\eta}}{N}F(s)-\frac{1}{p*}f(s)s \leq \widetilde{C}_\lambda F(\gamma), \quad
 \text{for }  s\geq \theta_1,
\end{gather}
where $p*=\frac{Np}{N-p}$ is the critical exponent.
\end{itemize}
For every fixed $T>0$ we consider the class of weights
\begin{equation*}
\mathcal{C}(T):= \{  W :   W \text{ is  an  admissible  weight, $\underline{\eta} >0$ 
 and $\widetilde{C}_\lambda \leq T$} \}.
\end{equation*}
We note this class contains admissible weights so that $\underline{\eta}>0$ and 
$\overline{\lambda}/\underline{\lambda}$ is bounded.

Problems related to non-existence of positive radial solutions have been studied, 
most of them in the case $f(0)\geq 0$. 
For instance, in the semilinear case, in \cite{Bianchi} considering $f(u)=u^p$ 
and suitable conditions on the derivative of $ W$, and $p$; a result was 
showed in $\mathbb{R}^N$. See also \cite{Naito} and references therein. 
The case $f(0)<0$ is more complicated and in this direction, in \cite{HZ}, 
authors considered the non-existence of positive radial  solutions for a 
semipositone problem (i.e. $f(0) < 0$) if $f$ is  not increasing entirely. 
The nonlinearity was superlinear and had more than one zero. There, the domain 
was an annulus and they exhibited a positive constant weight. 
Authors in \cite{ShivajiYe}, studied a semipositone elliptic system which involves 
 positive parameters bounded away from zero, and the nonlinearities are smooth 
functions that  satisfy certain linear growth conditions at infinity. 
They established non-existence of positive solutions when two of the parameters  
are large. Other works can be found in \cite{AZ,BCShivaji, Hakimi} but,  
we refer the reader to the survey paper \cite{CMShivaji} and references therein 
for a review about semipositone problems. In the quasilinear case, some  
papers are known in this direction.  Chhetri  et al \cite{Chh} showed 
non-existence considering a $C^1$-nondecreasing non-linearity with a unique 
 positive zero. They extend the result in \cite{BCShivaji} for 
$1 <p< \infty$. Also,  Hai \cite{Hai} obtained a non-existence result if 
$f$ is locally Lipschitz continuous and $\liminf_{s\to\infty} f(s)/s^{p-1} > 0$. 
A very simple non-existence result for a semipositone quasilinear problem was 
showed in \cite{Rudd}, where $ W=1$ and a non-linearity given by step function
 was considered.

The study of such quasilinear equations with semipositone structure is open 
in the case of general bounded regions. Moreover,  questions on  uniqueness  
remain open even for radial solutions when the domain is a ball or an annulus.
We would like to mention that  Castro {\it et al.} \cite{CMShivaji} wrote: 
``In general, studying positive solutions for semipositone problems is more
 difficult compared to that of positone problems. The difficulty is due to 
the fact that in the semipositone case, solutions have to live in regions 
where the reaction term is negative as well as positive''. 
This makes remarkable our research and we emphasize that in this work we deal,
 mainly, with the case $2< p < N$ and  non-constant weight. 
Therefore, it is a generalization of \cite{HZ, Rudd, Chh, Hai}.

The main result reads as follows.

\begin{theorem}\label{theorem1}
 Assume hypotheses {\rm (F1)--(F3)}. Then for every $T>0$ there is a 
positive real number $\lambda_0$ such that if
 $ W \in \mathcal{C}(T)$ and $\overline{\lambda}\geq\lambda_0$, then  \eqref{eq1} 
has no positive radial solution.
\end{theorem}

A second theorem is obtained when $p= 2$ and $ W$ is constant.

\begin{theorem}\label{theorem2}
Set $\Omega=B_1(0)$. Assume that $p=2<N$,  $W\equiv\lambda$ and $f$ satisfies {\rm (F2), (F3)},
\begin{equation}\label{condition for p=2}
 F(s)-\frac{N-2}{2N}f(s)s \leq F(\gamma)\quad \forall s\geq \gamma,
\text{ and some } \gamma\in(\beta_3, \theta_1),
\end{equation}
and
\begin{itemize}
\item[(F1')] $f:[0,\infty)\to \mathbb{R}$  belongs to $C^1$ 
 with exactly three zeros $0<\beta_1<\beta_2<\beta_3$.
\end{itemize}
Then there exists $\lambda_0>0$ such that  \eqref{eq1} has no positive radial  
solution in $C^2(\bar{\Omega})$ provided $\lambda>\lambda_0$.
\end{theorem}

This article is organized as follows: 
In Section \ref{section2} we show some technical lemmas which will be useful 
in Section \ref{section3} for proving Theorems \ref{theorem1} and \ref{theorem2}.


\section{Qualitative Analysis}\label{section2}

In this section we assume that there exists a positive radial solution to 
 \eqref{eq2}, which is denoted by $u(\cdot, W)$. 
We say that $u$ is a solution if $r\mapsto r^n\varphi_p(u^\prime)\in C^1$.

\begin{lemma} \label{lem2.1}
If $u(0, W) >\beta_3$, then $u(\cdot, W)$ is decreasing in $[0,1]$.
\end{lemma}

\begin{proof}
We note that by  \eqref{eq3},  if $u'(r, W)=0$ for some $0<r<1$ then 
$u(r, W)\in\{\beta_1,\beta_2,\beta_3\}$.
Since $u(0, W)>\beta_3$ then we set 
$t_3( W)=t_3:=\min\{r\in[0,1]:u(r, W)=\beta_3\}$. 
By the mean value Theorem, $u'(\xi, W)<0$ for some $\xi\in(0,t_3)$. 
Since $u'(\cdot, W)$ cannot change sign in this interval then $u(\cdot, W)$ 
is decreasing there.

Now, for all $r\geq t_3$, we claim $u(r, W)\leq \beta_3$.
Indeed, if  $u(r_1, W)>\beta_3$ for some $r_1\in(t_3,1)$ then there would 
be an interval $[r_2,r_3]\subseteq [t_3,1)$ such that $u(r, W)>\beta_3$ 
for all $r\in(r_2,r_3)$ and $u(r_2, W)=u(r_3, W)=\beta_3$.
 Again, the mean value Theorem implies $u'(\xi_0, W)=0$ for some
 $\xi_0\in(r_2,r_3)$. Thus $u(\xi_0,  W)\in\{\beta_1,\beta_2,\beta_3\}$,
 which is a contradiction and the claim is proved.

Let $t_2:=t_2( W):=\min\{r\in(t_3,1]: u(r, W)=\beta_2\}$. Then the same argument 
from above shows that $u(\cdot, W)$ is decreasing in the interval $[t_3,t_2]$. 
Afterwards we see that for all $r\in [t_2,1)$, $u(r, W)\leq \beta_2$. 
Assume on the contrary that there exists $r_1\in (t_2,1)$ such that 
$u(r_1, W)> \beta_2$. Then we can find an interval $[r_2,r_3]\subseteq [t_2,1)$ 
such that $u(r, W)>\beta_2$ for all $r\in (r_2,r_3)$ and 
$u(r_2, W)=u(r_3, W)=\beta_2$. The function $v(r, W):=u(r, W)-\beta_2$ satisfies
$$
-\Delta_p(v)= W(r)f(u), \text{ in } (r_2,r_3); \quad  v(r_2)=v(r_3)=0.
$$
 Since $u(\cdot, W)\in[\beta_2,\beta_3]$ in $(r_2,r_3)$, it follows
that $ W(r)f(u(r))\leq 0$. Therefore, a comparison principle 
(see \cite[Proposition 6.5.2]{GP}) lead us to $v(r, W)\leq 0$ for all 
$r\in(r_2,r_3)$. But this means $u(r, W)\leq \beta_2$ for $r$ in this interval. 
This is an absurd.

Now let $t_1:=t_1( W):=\min\{r\in(t_2,1]: u(r, W)=\beta_1\}$. 
Then, as  at the beginning of this proof,  $u(\cdot, W)$ is decreasing 
in $[t_2,t_1]$. We claim that for all $r\geq t_1$,  $u(r, W)\leq \beta_1$. 
Arguing by contradiction, if there exists $r_1\in (t_1,1)$ such that 
$u(r_1, W)>\beta_1$ then this forces $u(r, W)\geq\beta_1$ for all 
$r\in(t_1,r_1)$, otherwise there exists $r_0\in(t_1,r_1)$ with 
$u(r_0, W)<\beta_1$ and $u'(r_0, W)=0$. 
Thus $u(r_0, W)\in\{\beta_1,\beta_2,\beta_3\}$,  which cannot be. 
Hence there would exists $\gamma_0\in(\beta_1,\beta_2)$ such that 
$ \beta_1 \leq u(r) < \gamma_0$ in some neighborhood $(r_2,r_3)$ of $t_1$ 
and $u(r_2)=u(r_3)=\gamma_0$. Then $w(r, W):=u(r, W)-\gamma_0$ satisfies
$$
-\Delta_p(w)= W(r)f(u(r)), \text{ in } (r_2,r_3); \quad  w(r_2)=w(r_3)=0.
$$
 Because of behaviour of $u(\cdot, W)$ in $(r_2,r_3)$, $f(u(r))\geq 0$. 
Using again the same comparison principle  we deduce that $w(r, W)\geq 0$, 
that is $u(r, W)\geq \gamma_0$ for all $r\in (r_2,r_3)$. 
This contradiction shows the claim. 

Finally, using the same arguments as before it follows that $u(\cdot, W)$ 
is decreasing in $[t_1,1]$. Thus, the lemma is proved.
\end{proof}

\begin{remark}\label{remark1} \rm
If $p=2$ our previous demonstration does not work. However, it is well known 
that when $p=2$,  $W\equiv \lambda$ (a constant) and $f\in C^1$, a regular 
positive solution turns out to be radially symmetric and decreasing 
(see \cite{GNN}).
\end{remark}

Let us define the Energy associated to the problem
\eqref{eq2} by
\[
E(t, W)  :=\frac{| u'(t, W )  | ^p}{p' W(  t)  }+F(u(t, W )  )  .
\]
Also, we define
\[
H(t, W )  :=t W(  t)  E(t, W )
+\frac{N-p}{p}\varphi_{p}(  u'(t, W )  )
u(t, W )  .
\]
Suppose that $u(\cdot, W)$ is a solution of \eqref{eq2}.
Then  a Pohozaev type identity takes place,
\begin{equation}\label{pohozaev}
\begin{aligned}
&  t^n H( t, W )  -s^n H(  s, W ) \\
&  =\int_{s}^{t}r^n  W(  r)  \big[  \big(
N+r\frac{ W'(  r)  }{ W(  r)  }\big)  F(
u)  -\frac{N-p}{p}f(  u)  u\big]  dr,
\end{aligned}
\end{equation}
whenever $0\leq s\leq t\leq1$ (see \cite[formula (2.2)]{HL}).
Now, from the definition of the energy we have
\begin{equation*}
 E'(r, W)  =-\frac{|u'(r, W)|^p}{p' W (r)r}[\frac{np}{p-1}
+\frac{r W '(r)}{ W(r)}]\leq 0,
\end{equation*}
since $\underline{\eta} >0$ and $p< N$. Thus the energy is a decreasing function of $r$. 
Hence $E(r, W) \geq E(1, W) =\frac{|u'(1, W)|^p}{p'  W(1)} \geq 0$ for all 
$r\in[0,1]$. In particular $E(0, W) = F(u(0, W)) \geq 0$. 
Thus $u(0, W) \geq \theta_1$. So, in view of (F3) and the previous lemma, we
conclude the following result.

\begin{lemma}\label{newlemma}
 Under assumption {\rm (F3)}, every positive radially symmetric solution of 
 \eqref{eq1} is radially decreasing.
\end{lemma}

\begin{lemma}\label{lemmat_3}
Let $u(\cdot, W) $ be a positive solution of problem \eqref{eq2} and 
set $t_\gamma=t_\gamma( W)$ the unique number in $(0,1)$ such that 
$u(t_\gamma, W)=\gamma$. Then there exists a constant $C$ independent on 
$ W$ such that $ | u'(t_\gamma, W ) |t_\gamma  \leq C $.
\end{lemma}

\begin{proof}
Using the Pohozaev identity (see \eqref{pohozaev}) with $s=0$ and $t=t_\gamma$ 
we have
\begin{equation}\label{pohozaev2}
\begin{aligned}
& t_\gamma^n | u'(t_\gamma, W ) | ^{p-1} 
\big[ \frac{t_\gamma}{p'} | u'(t_\gamma, W ) |  - \frac{N-p}{p} \gamma \big] \\
& =\int_{0}^{t_\gamma}r^n  W (  r)  \big[  \widetilde{ W}  F(
u)  -\frac{N-p}{p}f(  u)  u\big]  dr - t_\gamma^N  W(t_\gamma)F(\gamma ) .
\end{aligned}
\end{equation}
Now, from \eqref{weight2} and \eqref{weight1} we have that for all $r\in[0,t_\gamma]$,
\begin{equation*}
\widetilde W (r) F(u)-\frac{N-p}{p}f(u)u \leq N\widetilde{C}_\lambda F(\gamma).
\end{equation*}
Taking into account $0< W (t_\gamma) \leq \widetilde{C}_\lambda  W (r)$ 
for all $r\in [0,t_\gamma]$ (indeed it is true for all $r\in [0,1]), F(\gamma)<0$ 
and the above inequality, we obtain
\begin{equation*}
r^n  W (r) \big(\widetilde W (r) F(u(r))-\frac{N-p}{p}f(u(r))u(r)\big) 
\leq r^n  W (t_\gamma) NF(\gamma).
\end{equation*}
Integrating over the interval $[0,t_\gamma]$,
\begin{equation*}
\int_{0}^{t_\gamma}r^n  W (r) \big(\widetilde W (r) F(u(r))-\frac{N-p}{p}f(u(r))u(r)\big)
 dr \leq F(\gamma) W (t_\gamma) t_\gamma^N.
\end{equation*}
Therefore, from \eqref{pohozaev2},
\begin{equation*}
t_\gamma^n | u'(t_\gamma, W ) | ^{p-1} 
\big[ \frac{t_\gamma}{p'} | u'(t_\gamma, W ) |  - \frac{N-p}{p} \gamma \big] \leq 0;
\end{equation*}
that is,
\begin{equation*}
\frac{t_\gamma}{p'} | u'(t_\gamma, W ) |  \leq \frac{N-p}{p} \gamma.
\end{equation*}
The lemma is proved
\end{proof}


\begin{lemma}\label{lemmab}
Let $\alpha \in(0, \beta_1 )$ be fixed and set $b=b( W,\alpha)$ the unique 
number in $(0,1)$ such that $u(b, W)=\alpha$. Then 
$b( W,\alpha)\to 1$ as $\overline{\lambda} \to \infty$.
\end{lemma}

\begin{proof}
Integrating the equation in \eqref{eq2}  on $[b,1]$ we obtain
\begin{equation}\label{b2}
\begin{aligned}
 \varphi_p(u'(1))-b^n\varphi_p(u'(b))
&=-\int_{b}^{1} r^n W (r)f(u)dr \\
& \geq K\int_{b}^{1} r^n W (r)dr \geq \frac{K\underline{\lambda}}{N}(1-b^N),
\end{aligned}
\end{equation}
where $-K=-K(\alpha):=\max \{f(s):s\in[0,\alpha]\} <0$.
Multiplying the differential equation in \eqref{eq2} by $r^m u'$, with $n+m:=np'$, 
and integrating by parts the left-hand side of the resulting equation, we have
\begin{align*}
& b^{n+m}|u'|^p-1^{n+m}|u'|^p
+ \int_b^1 r^n \varphi_p(u')[mr^{m-1}u'(r)+r^mu''(r)]dr \\
& =\int_b^1 r^{n+m} W (r)[F(u)]'dr.
\end{align*}
Now, integrating by parts the right-hand side of the above equation we 
can estimate it as
\begin{align*}
& \int_b^1 r^{n+m} W (r)[F(u)]'dr = -b^{n+m}  W (b)F(u(b))-\int_b^1 F(u)(r^{n+m} W (r))' dr \\
&  \leq -b^{n+m}  W (b)F(u(b))-F(\alpha)\int_b^1 (r^{n+m} W (r))' dr.
\end{align*}
This estimate is due to the fact that $F(\alpha)\leq F(u(r))$ for all 
$r\in(b,1)$ and the assumption that $\widetilde{ W}$ is positive, 
which implies that $[r^{n+m} W (r)]'>0$. In consequence,
\begin{equation}\label{b1}
\begin{aligned}
& b^{n+m}|u'|^p-1^{n+m}|u'|^p \\
&  \leq -F(\alpha) W (1)- \int_b^1 r^n \varphi_p(u')[mr^{m-1}u'(r)+r^mu''(r)]dr \\
&  = -F(\alpha) W (1) + \int_b^1 r^{n+m-1} |u'|^{p-1}[mu'(r)+ru''(r)]dr,
\end{aligned}
\end{equation}
where we have noted that by Lemma \ref{newlemma}, $u'(r)\leq 0$ for all 
$r\in[b,1]$. On the other hand, from $\eqref{eq3}$ and 
$M=M(\alpha):=\max_{s\in[0,\alpha]}|f(s)| >0$, we have for all $r\in[b,1]$
\begin{equation*}
(p-1)|u'|^{p-2}u''+\frac{n}{r}|u'|^{p-2}u'=- W (r) f(u)= W (r) |f(u)|\leq \overline{\lambda} M.
\end{equation*}
Then
\begin{equation*}
|u'|^{p-1}[ru''+mu'] \leq \frac{\overline{\lambda} M}{p-1}r|u'|.
\end{equation*}
Therefore, from \eqref{b1},
\begin{equation}\label{b3}
\begin{aligned}
 b^{n+m}|u'|^p-|u'(1)|^p
&  \leq -F(\alpha)\overline{\lambda}+ \frac{\overline{\lambda} M}{p-1}\int_b^1 r^{n+m} |u'| dr \\
&  \leq -F(\alpha)\overline{\lambda}+ \frac{\overline{\lambda} M}{p-1}\int_b^1 |u'| dr \\
& = [-F(\alpha)+ \frac{\alpha M}{p-1} ]\overline{\lambda}=:C_0 \overline{\lambda},
\end{aligned}
\end{equation}
where $C_0=C_0(\alpha)>0$. Now, from \eqref{b2} and taking into account 
that $ u' \leq 0$,
\begin{equation}\label{positivity of b3}
0<b^{np'}|u'(b)|^p-|u'(1)|^p.
\end{equation}
Hence, combining  \eqref{b3} and \eqref{positivity of b3},
\begin{equation*}
0<b^{np'}|u'(b)|^p-|u'(1)|^p \leq C_0 \overline{\lambda}.
\end{equation*}
Thus, using \eqref{b2} again,
\begin{align*}
0&\leq\frac{\underline{\lambda} K}{N}(1-b^N)\\
&\leq \varphi_p(u'(1))-b^n\varphi_p(u'(b))\\
&\leq|b^n |u'(b)|^{p-1}-|u'(1)|^{p-1}|\\
&\leq \big(b^{np'} |u'(b)|^{(p-1)p'}-|u'(1)|^{(p-1)p'}\big)^{1/p'}\\
&= \big(b^{np'} |u'(b)|^p-|u'(1)|^p\Big)^{1/p'} \leq \overline{\lambda}^{1/p'}C_0^{1/p'},
\end{align*}
which implies
\begin{equation*}
0\leq \frac{K}{N}(1-b^N) \leq \frac{\overline{\lambda}}{\underline{\lambda}}^{1/p'}C_0^{1/p'} 
\leq \frac{1}{\overline{\lambda}^\frac{1}{p}}TC_0^{1/p'},
\end{equation*}
where $T>0$ satisfies $\overline{\lambda}/\underline{\lambda}\leq T$. The statement of the lemma follows.
\end{proof}

\section{Proof of  results and examples}\label{section3}

\begin{proof}[Proof of theorem \ref{theorem1}]
Let us argue by contradiction. Suppose there exist $T>0$ and  a sequence 
of admissible weights $\{ W_m\}_{m=1}^\infty$ with
 $\overline{\lambda}_m:=\| W_m\|_\infty\to +\infty$ as $m\to\infty$ and such that 
for all $m$  problem $\eqref{eq1}$ has a positive radial solution
 $u(\cdot, W_m)$. If we suppose that there is a positive constant $L$ such that 
$t_\gamma(m):=t_\gamma( W_m)\geq L$ for all $m$ sufficiently large then,  
from Lemma \ref{lemmat_3}, $|u'(t_\gamma(m), W_m)|\leq \frac{C}{L}=:C_1$, 
where $C$ is independent on the weight. This would imply
\begin{align*}
E(t_\gamma(m), W_m) 
& =\frac{|u'(t_\gamma(m), W_m)|^p}{p' W(t_\gamma(m))}+F(u(t_\gamma(m), W_m)) \\
& \leq \frac{C_1^p}{p' W(t_\gamma(m))}+F(\gamma) \\
&\leq \frac{C_1^p}{p'\underline{\lambda}_m}+F(\gamma)\\
& \leq \frac{C_1^p T}{p'\overline{\lambda}_m}+F(\gamma).
\end{align*}
Hence $0\leq\limsup_{m\to\infty}E(t_\gamma(m), W_m)\leq F(\gamma)<0$, 
what is an absurd. In consequence there is a sub-sequence of 
$\{t_\gamma(m)\}_{m=1}^\infty$ (denoted in the same way) which converges to zero.  
Let us fix $\alpha >0$ and we define $b_m=b( W_m)$ as in Lemma \ref{lemmab}. 
Then by  that lemma we can choose $k>0$ such that for all $m\geq k$, 
$t_\gamma(m)<1/4$ and $3/4< b_m$. 
Set $\widetilde{L}:=\max_{s\in[\alpha ,\gamma]}F(s)<0$. 
Thus, there exists $\xi_m\in(t_\gamma(m),b_m)$ such that
\begin{equation*}
|u'(\xi_m, W_m)|=\frac{|u(t_\gamma(m), W_m)-u(b_m, W_m)|}{|t_\gamma(m)-b_m|}
\leq 2(\gamma+\alpha)=:\mu>0.
\end{equation*}
Then, taking into account that $u(\cdot, W_m)$ is decreasing we have 
$\alpha\leq u(r, W_m)\leq \gamma$ for all $r\in[t_\gamma(m),b_m]$. Therefore,
\begin{align*}
E(\xi_m, W_m) 
& =\frac{|u'(\xi_m, W_m)|^p}{p' W(\xi_m)}+F(u(\xi_m, W_m)) \\
& \leq \frac{\mu^p}{p' W(\xi_m)}+\widetilde{L} \\
& \leq \frac{\mu^p}{p'\underline{\lambda}_m}+\widetilde{L}\\
& \leq \frac{\mu^p T}{p'\overline{\lambda}_m}+\widetilde{L}.
\end{align*}
Hence $0\leq\limsup_{m\to\infty}E(\xi_m, W_m)\leq \widetilde{L}<0$. 
This is a contradiction and so the theorem is proved.
\end{proof}

\begin{proof}[Proof of theorem \ref{theorem2}]
Assume  on the contrary that there exist a sequence $\lambda_m\to\infty$ 
and positive solutions $u_m\in C^2(\bar{\Omega})$ of the problem
\begin{equation*}
\Delta u+\lambda f(u)=0,  \text{ in }\ \Omega; \quad 
u=0\ \text{ on }\ \partial\Omega.
\end{equation*}
A celebrated result in \cite{GNN} (see Remark \ref{remark1}) implies that 
$u_m$ is radially symmetric and decreasing. Now, after a detailed 
reading of proofs Lemmas \ref{lemmat_3} and \ref{lemmab}, one can see 
that their conclusions hold for $p=2$. In this case we must use 
hypothesis \eqref{condition for p=2}. 
The proof  follows the same argument as in proof of theorem \eqref{theorem1}.
\end{proof}

\subsection*{Examples} 
Next, we  exhibit a nonlinearity $f$ and two weights $ W$, holding all
 conditions. First, we consider  a constant weight.

Let $g:[0,3]\to \mathbb{R}$ defined by $g(t)=(t-1)(t-2)(t-3)$. Let
\[
f(t)  :=\begin{cases}
g(t) &\text{if }\ 0 \leq t\leq 3 \\
(t-3)^q  &\text{if } t > 3,
\end{cases}
\]
with $p^\ast <q+1$. Thus $f$ has exactly three zeros, $f(0)<0$ and $f(3)=0$.
 Also, $F(t):=\int_{0}^t f(s) ds= \frac{t^4}{4}-2t^3+\frac{11t^2}{2}-6t<0$ 
for all $t\in [0,3]$. Then we have $\beta_3=3$ and $4=\theta_1 >\beta_3$. 
We fix $\gamma\in(\beta_3, \theta_1)$ and note that a simple computation shows
\begin{equation}\label{Fs-Fgamma}
F(t)=F(\gamma)+\frac1{q+1}[(t-3)^{q+1}-(\gamma-3)^{q+1}],\quad 
\forall t\geq \gamma.
\end{equation}
We define the family of constant weights $ W _\lambda \equiv \lambda>0$, 
which belongs to the class of weights $\mathcal{C}(1)$.
 Because of $p^\ast <q+1$ then $\frac{(t-3)^{q+1}}{q+1}\leq \frac{t(t-3)^q}{p*}$. 
Therefore
\begin{equation*}
F(t)-F(\gamma)= \frac{(t-3)^{q+1}-(\gamma-3)^{q+1}}{q+1}
\leq \frac{t(t-3)^q}{p*} = \frac{t f(t)}{p*}, \quad \text{for all }  t \geq \gamma.
\end{equation*}
This gives us \eqref{weight2} and \eqref{weight1}.
 In consequence, by Theorem \ref{theorem1}, there exists $\lambda_0 >0$ 
such that for all $\lambda > \lambda_0$, problem \eqref{eq2} has no 
radial positive solution.

For the same non-linearity $f$  we introduce another family of weights that 
satisfy the hypotheses of Theorem \ref{theorem1}. 
For  $\lambda \geq 1$, define $ W _\lambda (r):=\lambda +r$. 
Then $\widetilde { W}_\lambda (r)=N+\frac{r}{\lambda +r}$, 
$\widetilde {C}_\lambda = \frac{\lambda +1}{\lambda}$, $\underline{\eta} =N$ 
and $\overline{\eta} = N+\frac{1}{\lambda +1}$. First of all, since 
$\lim_{\gamma \to \theta_1} f(\gamma)= f(\theta_1)>0$ and 
$\lim_{\gamma \to \theta_1} F(\gamma)=0$, we can take $\gamma$ 
between $\beta_3$ and $\theta_1$ such that
\begin{equation*}
-\frac{f(\gamma)\gamma}{p*} < \frac{\lambda +1}{\lambda} F(\gamma),
\end{equation*}
where $\gamma $ is independent on $\lambda$ due to the estimate 
$1< \frac{\lambda+1}{\lambda}\leq 2$.
Since $F(s)\leq 0$ and $-\frac{f(s)s}{p*}\leq -\frac{f(\gamma)\gamma}{p*}$ 
for all $\gamma \leq s \leq \theta_1$ 
($t\mapsto tf(t)$ is non-decreasing for $t>3$), then
\begin{equation*}
F(s)-\frac{f(s)s}{p*} < \frac{\lambda +1}{\lambda} F(\gamma),
\end{equation*}
which gives us \eqref{weight2}.
On the other hand, for $\lambda$ sufficiently large,
\begin{gather*}
\big(\frac{1}{N(\lambda +1)}-\frac{1}{\lambda}\big)F(\gamma)
 \leq  \frac{3(\theta_1-3)^q}{p*}, 
\\
\frac{1}{p*}-\frac{1}{q+1}-\frac{1}{N(\lambda +1)(q+1)} >0.
\end{gather*}
Then, for all $s\geq \theta_1$,
\begin{align*}
& \big(\frac{1}{N(\lambda +1)}-\frac{1}{\lambda}\big)F(\gamma) \\
& \leq  \frac{3(s-3)^q}{p*}
+\big(\frac{1}{p*}-\frac{1}{q+1}-\frac{1}{N(\lambda +1)(q+1)}\big)(s-3)^{q+1} \\
&\quad +\big(\frac1{q+1}+\frac{1}{N(\lambda +1)}\frac{1}{q+1}\big)(\gamma-3)^{q+1}.
\end{align*}
So, keeping in mind \eqref{Fs-Fgamma} we have \eqref{weight1}.

\subsection*{Acknowledgments}
This research was partially supported by  Fondo Nacional de Financiamiento
para la Ciencia, la Tecnolog\'\i a y la  Innovaci\'on Francisco
Jos\'e de  Caldas, contrato Colciencias FP44842-087-2015.


\begin{thebibliography}{00}

\bibitem{AZ} D. Arcoya, 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{Bianchi} G. Bianchi;
 \emph{Non-existence of positive solutions  to semilinear elliptic equations 
on $R^n$ or $R^n_+$ through the method of moving planes},
 Commun. in Partial Differential Equations, 22(9\&10), 1671-1690 (1997).

\bibitem{BCShivaji} K. J. Brown, A. Castro, 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{CMShivaji} A. Castro,  M. Chhetri, R. Shivaji; 
\emph{Nonlinear eigenvalue problems with semipositone structure}, 
Electron. J. Diff. Eqns., Conf. 05, 2000, pp. 33-49.

\bibitem{Chh} M. Chhetri, P. Girg;
 \emph{Nonexistence of nonnegative solutions for a class of $(p-1)$-superhomogeneous 
semipositone problems}, Journal of Mathematical Analysis and Applications
 Volume 322, Issue 2, 15 October 2006, pp. 957-963.

\bibitem{GP} L. Gasi\'nski, N. S. Papageorgiou; 
\emph{Nonlinear analysis}, Volume 9, Series in mathematical analysis 
and applications, Chapman \& Hall/CRC, 2005.

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

\bibitem{Hai} D. D. Hai;
\emph{Nonexistence of positive solutions for a class of p-Laplacian boundary 
value problems}, Applied Mathematics Letters, 31 (2014), pp.  12-15.

\bibitem{Hakimi} S. Hakimi;
\emph{Nonexistence of radial positive solutions for a nonpositone system 
in an annulus}, Electron. J. Diff. Equ., Vol. 2011 (2011), No. 152, pp. 1-7.

\bibitem{HZ} S. Hakimi, A. Zertiti; 
\emph{Nonexistence of radial positive solutions for a nonpositone problem},
 Electron. J. Diff. Equ., Vol. 2011 (2011), No. 26, pp. 1-7.

\bibitem{HL} S. Herr\'on, E. Lopera; 
\emph{Signed Radial Solutions for a Weighted $p$-Superlinear Problem}, 
Electron. J. Differential Equations 2014, No. 24, pp. 1-13.

\bibitem{Naito} Y. Naito;
\emph{Nonexistence results of positive solutions for semilinear elliptic 
equations in $\mathbb{R}^N$}, J. Math. Soc. Japan
Vol. 52, No. 3, 2000, pp. 637-644.

\bibitem{Rudd} M. Rudd; 
\emph{Existence and nonexistence results for quasilinear semipositone 
Dirichlet problems}, Proceedings of the Seventh Mississippi State-UAB 
Conference on Differential Equations and Computational Simulations,  
Electron. J. Differ. Equ. Conf., 17, 207-212,  2009.

\bibitem{ShivajiYe} R. Shivaji, J. Ye; 
\emph{Nonexistence results for classes of $3\times3$ elliptic systems},
 Nonlinear Analysis: Theory, Methods \& Applications, Volume 74, 
Issue 4, 15 February 2011, Pages 1485-1494.

\end{thebibliography}

\end{document}