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

\AtBeginDocument{{\noindent\small
\emph{Electronic Journal of Differential Equations},
Vol. 2013 (2013), No. 155, pp. 1--6.\newline
ISSN: 1072-6691. URL: http://ejde.math.txstate.edu or http://ejde.math.unt.edu
\newline ftp ejde.math.txstate.edu}
\thanks{\copyright 2013 Texas State University - San Marcos.}
\vspace{9mm}}

\begin{document}
\title[\hfilneg EJDE-2013/155\hfil Infinite semipositone problems]
{Infinite semipositone problems with indefinite
weight and asymptotically linear growth forcing-terms}

\author[G. A. Afrouzi, S. Shakeri \hfil EJDE-2013/155\hfilneg]
{Ghasem A. Afrouzi, Saleh Shakeri}  % in alphabetical order

\address{Ghasem Alizadeh Afrouzi \newline
Department of Mathematics, Faculty of Mathematical Sciences,
University of Mazandaran, Babolsar, Iran}
\email{afrouzi@umz.ac.ir}

\address{Saleh Shakeri \newline
Department of Mathematics, Faculty of Mathematical Sciences,
University of Mazandaran, Babolsar, Iran}
\email{s.shakeri@umz.ac.ir}

\thanks{Submitted September 17, 2012. Published July 8, 2013.}
\subjclass[2000]{35J55, 35J25}
\keywords{Infinite semipositone problem; indefinite weight; forcing term;
\hfill\break\indent asymptotically linear growth; 
sub-supersolution method}

\begin{abstract}
 In this work, we study the existence of positive solutions to the
 singular problem
 \begin{gather*}
 -\Delta_{p}u =  \lambda m(x)f(u)-u^{-\alpha} \quad \text{in }\Omega,\\
 u =  0  \quad \text{on }\partial \Omega,
 \end{gather*}
  where $\lambda$ is positive parameter,
 $\Omega $ is a bounded domain with smooth boundary,
 $ 0 <\alpha<1 $, and $ f:[0,\infty] \to\mathbb{R}$  is a continuous
 function which is asymptotically p-linear at $\infty $.
 The weight function is continuous satisfies $m(x)>m_0>0$,
 $\|m\|_{\infty}<\infty$. We prove the existence of a positive solution
 for a certain range of $\lambda $ using the method of sub-supersolutions.
\end{abstract}

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


\section{Introduction}

 In this article, we consider the positive solution to the
boundary-value problem
\begin{equation}\label{e1.1}
\begin{gathered}
-\Delta_{p}u =  \lambda m(x)f(u)-\frac{1}{u^{\alpha}} \quad \text{in }\Omega,\\
u =  0  \quad \text{on }\partial \Omega,
\end{gathered}
\end{equation}
where $\lambda $ is positive
parameter, $\Delta_{p}u=\operatorname{div}(|\nabla u|^{p-2}\nabla u)$,
$p>1 $, $\Omega $ is a bounded domain with smooth boundary
$\partial\Omega $, $ 0 <\alpha<1 $, and $ f:[0,\infty]
\to\mathbb{R} $  is a continuous function which is
asymptotically p-linear at $ \infty $. The weight function $ m(x)$
satisfies $ m(x) \in C(\Omega)$ and $ m(x)>m_0>0 $ for $x\in
\Omega$ and also $\|m\|_{\infty}= l<\infty$.
 We prove the existence of  a positive solution for a certain range
of $\lambda $.


 We consider problem \eqref{e1.1} under the following
assumptions.
\begin{itemize}
\item[(H1)] There exist $ \sigma_1> 0$, $k > 0 $ and $ s_0 >1 $ such that
$ f(s)\geq \sigma_1s^{p-1}-k $ for all $s\in[0,s_0]$;

\item[(H2)] $\lim_{s \to+\infty} \frac{f(s)}{s^{p-1}}= 0 $
for some $\sigma > 0 $.
\end{itemize}

Let $ F(u):=\lambda m(x)f(u)-\frac{1}{u^{\alpha}} $. The case when
$ F(0)< 0 $ (and finite) is referred to in the literature as a
semipositone problem. Finding a positive solution for a
semipositone problem is well known to be challenging
(see \cite{BeCaNi,Lion}).
Here we consider the more challenging case when
$\lim_{u\to 0^{+}} F(u)=-\infty $. which has received attention
very recently and is referred to as an infinite semipositone
problem. However, most of these studies have concentrated on the
case when the nonlinear function satisfies a sublinear condition
at $ \infty  $ (see \cite{LeShiYe,LeShiYe2,RamShiYe}).
The only paper to our knowledge dealing with an infinite semipositone
problem with an asymptotically linear nonlinearity is \cite{Hai},
where the author is restricted to the case $p = 2$.
Also here the existence of a
positive solution is focused near $\lambda_1/\sigma$ ,
where $\lambda_1$ is the first eigenvalue of$-\Delta $.
See also \cite{AmArBu,Zhang}, where asymptotically linear nonlinearities have
been discussed in the case of a nonsingular semipositone problem
and an infinite positone problem. Recently, in the case when
 $ m(x)=1 $ problem \eqref{e1.1} has been studied by  Shivaji et
al  \cite{HaSanShi}. The purpose of this paper is to improve
the result of \cite{HaSanShi} with weight $m$. We shall
establish our existence results via the method of sub and
super-solutions.


\begin{definition} \rm
We say that $(\psi)$ (resp. $ Z $) in $W^{1,p}(\Omega)
\cap C(\overline{\Omega})$ are called a
 sub-solution
(resp. super-solution) of \eqref{e1.1}, if $ \psi $
satisfies
\begin{equation}\label{2}
\begin{gathered}
\int_{\Omega}|\nabla \psi(x)|^{p-2}\nabla \psi(x).\nabla w(x) dx
\leq \int_{\Omega}(\lambda m(x)f(\psi )-\frac{1}{\psi^{\alpha}})
w(x)dx  \quad\forall w \in W,\\
\psi >0  \quad \text{in }\Omega,\\
 \psi=0  \quad \text{on }\partial \Omega,
\end{gathered}
\end{equation}
and
\begin{equation}\label{3}
\begin{gathered}
\int_{\Omega}|\nabla Z|^{p-2}\nabla Z .\nabla w(x) dx
\geq   \int_{\Omega} (\lambda m(x)f(Z )-\frac{1}{Z^{\alpha}}) w(x)dx  \quad
   \forall w \in W,\\
Z>0 \quad \text{in }\Omega,\\
Z=0  \quad \text{on }\partial \Omega,
\end{gathered}
\end{equation}
where  $W = \{\xi \in C_0^\infty(\Omega): \xi \geq 0 \text{in } \Omega  \}$
\end{definition}

 The following lemma was established by Cui \cite{SCui}.

\begin{lemma}[\cite{SCui}] \label{lem1.2}
If there exist sub-supersolutions $ \psi $ and $Z$, respectively,
such that $\psi\leq Z $ on $\Omega$, Then \eqref{e1.1} has a
positive solution $ u $ such that $\psi\leq u\leq Z $ in $\Omega$.
\end{lemma}

 In next Section, we will state and prove the existence of  a
positive solution for a certain range of $\lambda $.

\section{Main result}

Our main result for problem \eqref{e1.1} reads as follows.

\begin{theorem}\label{the2.1}
Assume {\rm (H1)--(H2)}. If there exist  constants
$s_0^*(\sigma,\Omega)$, $ J(\Omega)$, $\underline{\lambda}$, and
$\hat{\lambda}>\underline{\lambda}$ such that if $s_0\geq s_0^*$ and
$m_0\sigma_1/(l\sigma)\geq J $, then \eqref{e1.1} has a
positive solution for $\lambda \in
[\underline{\lambda},\hat{\lambda}]$.
Here $ \mu_1 $ is the principal eigenvalue of operator $-\Delta_{p} $
with zero Dirichlet boundary condition.
\end{theorem}

\begin{proof}
By Anti-maximum principle \cite{Peter}, there exists
$\xi=\xi(\Omega)>0$ such that the solution $ z_{\mu}$ of
\begin{gather*}
-\Delta_{p} z-\mu |z|^{p-2}z=-1 \quad \text{in } \Omega,\\
z=0 \quad \text{on } \partial \Omega,
\end{gather*}\
for $\mu\in(\mu_1,\mu_1+\xi) $, is positive in $\Omega$ and is
such that $\frac{\partial z_{\mu}}{\partial \nu}<0$ on $
\partial\Omega $, where $\nu$ is outward normal vector at $\partial
\Omega$.

Since $z_{\mu}>0 $ in $ \Omega $ and $\frac{\partial
z_{\mu}}{\partial \nu}< 0 $ there exist $ m>0 $, $ A > 0 $, and
$\delta >0 $  such that
$ |\nabla z_{\mu}|\geq m$  in $\overline{\Omega}_{ \delta} $
and $ z_{\mu}\geq A$  in $\Omega \setminus \overline{
\Omega}_{\delta}$, where
$W = \{x \in \Omega: d(x,\partial \Omega)\leq \delta\}$.

We prove the existence of a solution by the comparison method
\cite{DraKerTak}. It is easy to see that any sub-solution of
\begin{equation}\label{e2.1}
\begin{gathered}
-\Delta_{p}u = \lambda m_0f(u)-\frac{1}{u^\alpha} \quad \text{in } \Omega,\\
u=0  \quad \text{on }\partial \Omega,
\end{gathered}
\end{equation}
is a sub-solution of \eqref{e1.1}. Also any super-solution of
\begin{equation}\label{e2.2}
\begin{gathered}
-\Delta_{p}u =\lambda lf(u)-\frac{1}{u^\alpha} \quad \text{in }\Omega,\\
u=0  \quad \text{on }\partial \Omega,
\end{gathered}
\end{equation}
is a super-solution of \eqref{e1.1}, where $l$ is as defined above.

We first construct a supersolution for\eqref{e1.1}.Let
$Z=M_{\lambda}e_{p}$, where $M_{\lambda} \gg 1 $ and $e_{p}$ is the
unique positive solution of
\begin{gather*}
-\Delta_{p}e_{p} =  1 \quad \text{in }\Omega,\\
e_{p} =  0  \quad \text{on }\partial \Omega,
\end{gather*}
Let $ \tilde{f}(s)=\max_{t\in [0,s]}f(t)$. Then
$ f(s)\leq \tilde{f}(s)$, $ \tilde{f}(s)$ is increasing, and
\[
 \lim_{u \to+\infty} \frac{\tilde{f}(u)}{u^{p - 1}} = 0\,.
 \]
 Hence, we can choose $ M_{\lambda} \gg 1 $ such that
$$
2\sigma \geq \frac{\tilde{f}(M_{\lambda}\|e_{p}\|_{\infty})}{(M_{\lambda}\|e_{p}\|_{\infty})^{p-1}}
$$
Now let $ \hat{\lambda}=1/\big(2l\sigma\|e_{p}\|_{\infty}^{p-1}\big)$.
For $\lambda \leq \hat{\lambda}$,
$$
-\Delta_{p}Z = M_{\lambda}^{p-1}
\geq \frac{\tilde{f}(M_{\lambda}\|e_{p}\|_{\infty})}{2\sigma
\|e_{p}\|_{\infty}^{p-1}}
\geq \lambda l \tilde{f}(M_{\lambda}e_{p})
\geq\lambda l f(M_{\lambda }e_{p})\geq \lambda l
f(Z)-\frac{1}{Z^{\alpha}}.
$$
Thus, $Z$ is a supersolution of \eqref{e2.2}; therefore $Z$ is a
supersolution of \eqref{e1.1}
Define
$$
\psi:=k_0z_{\mu}^\frac{p}{p-1+\alpha},
$$
where $ k_0> 0 $ is such that
\begin{gather*}
\frac{1}{k_0^{p-1+\alpha}}\Big(1+\frac{k
k_0^{\alpha}z_{\mu}^\frac{\alpha p}{p-1+\alpha}}
{2l\sigma\|e_{p}\|_{\infty}^{p-1} }\Big)
\leq
\min\Big\{\Big(\frac{m^{p}(1-\alpha)(p-1)p^{p-1}}{(p-1+\alpha)^{p}}\Big),
\Big(\frac{p}{p-1+\alpha}\Big)^{p-1}A \Big\}.
\end{gather*}
Then
\[
\nabla \psi=k_0\Big(\frac{p}{p-1+\alpha}\Big)z_{\mu}^\frac{
1-\alpha}{p-1+\alpha}\nabla z_{\mu},
\]
and
\begin{equation}\label{e2.3}
\begin{split}
-\Delta_{p} \psi
&= -\operatorname{div}(|\nabla \psi|^{p-2}\nabla \psi)\\
&=-k_0^{p-1}\Big(\frac{p}{p-1+\alpha}\Big)^{p-1}\operatorname{div}(
 z_{\mu}^\frac{(1-\alpha)(p-1)}{p-1+\alpha}|\nabla
z_{\mu}|^{p-2}\nabla
z_{\mu}) \\
&= -k_0^{p-1}\Big(\frac{p}{p-1+\alpha}\Big)^{p-1}\Big\{ (\nabla
z_{\mu}^\frac{(1-\alpha)(p-1)}{p-1+\alpha}).|\nabla
z_{\mu}|^{p-2}\nabla z_{\mu} \\
&\quad + z_{\mu}^\frac{(1-\alpha)(p-1)}{p-1+\alpha}\Delta_{p}z_{\mu} \Big\} \\
&=
 -k_0^{p-1}\Big(\frac{p}{p-1+\alpha}\Big)^{p-1}\Big\{\frac{(1-\alpha)(p-1)}{p-1+\alpha}
z_{\mu}^\frac{-\alpha p}{p-1+\alpha}|\nabla z_{\mu}|^{p}\\
&\quad + z_{\mu}^\frac{(1-\alpha)(p-1)}{p-1+\alpha}(1-\mu z_{\mu}^{p-1}) \Big\} \\
&=
 k_0^{p-1}\Big(\frac{p}{p-1+\alpha}\Big)^{p-1} \mu
 z_{\mu}^\frac{p(p-1)}{p-1+\alpha}-k_0^{p-1}\Big(\frac{p}{p-1+\alpha}\Big)^{p-1}
 z_{\mu}^\frac{(1-\alpha)(p-1)}{p-1+\alpha}\\
&\quad - \frac{k_0^{p-1}p^{p-1}(1-\alpha)(p-1)
|\nabla z_{\mu}|^{p}}{(p-1+\alpha)^{p}z_{\mu}^\frac{\alpha p}{p-1+\alpha}}.
\end{split}
\end{equation}

Now we let $ s_0^*(\sigma,\Omega)=  k_0\|z_{\mu}^\frac{p}{p-1+\alpha}\|_{\infty}$.
 If we can prove that
\begin{equation}\label{e2.4}
-\Delta_{p}\psi \leq \lambda m_0
\sigma_1k_0^{p-1}z_{\mu}^\frac{p(p-1)}{p-1+\alpha}- \lambda
k-\frac{1}{ k_0^{\alpha}z_{\mu}^\frac{\alpha p}{p-1+\alpha}},
\end{equation}
then (H1) implies that $-\Delta_{p}\psi \leq \lambda m_0
 f(\psi)-\frac{1}{\psi^{\alpha}}$,
and $ \psi  $ will be a subsolution of \eqref{e1.1}.

We will now prove \eqref{e2.4} by comparing terms in \eqref{e2.3} and
\eqref{e2.4}. Let $\underline{\lambda}=\frac{\mu
(\frac{p}{p-1+\alpha})^{p-1}  }{m_0\sigma_1}$.
For $\lambda \geq\underline{\lambda}$,
\begin{equation}\label{e2.5}
k_0^{p-1}\Big(\frac{p}{p-1+\alpha}\Big)^{p-1} \mu
 z_{\mu}^\frac{p(p-1)}{p-1+\alpha} \leq \lambda m_0
\sigma_1k_0^{p-1}z_{\mu}^\frac{p(p-1)}{p-1+\alpha}
\end{equation}
Also since $ \lambda \leq
\hat{\lambda}=\frac{1}{2l\sigma\|e_{p}\|_{\infty}^{p-1} }$,
\begin{equation}\label{e2.6}
\begin{split}
\lambda k+\frac{1}{ k_0^{\alpha}z_{\mu}^\frac{\alpha p}{p-1+\alpha}}
&\leq \frac{1}{ k_0^{\alpha}z_{\mu}^\frac{\alpha
p}{p-1+\alpha}}+\frac{k}{2l\sigma\|e_{p}\|_{\infty}^{p-1} }\\
&=\frac{k_0^{p-1}}{z_{\mu}^\frac{\alpha
p}{p-1+\alpha}}\Big[\frac{1}{k_0^{p-1+\alpha}}(1+\frac{k
k_0^{\alpha}z_{\mu}^\frac{\alpha
p}{p-1+\alpha}}{2l\sigma\|e_{p}\|_{\infty}^{p-1} })\Big]
\end{split}
\end{equation}
Now in ${\Omega}_{\delta}$, we have $ |\nabla z_{\mu}|\geq m $, and
by \eqref{e2.2},
$$
\frac{1}{k_0^{p-1+\alpha}}(1+\frac{k
k_0^{\alpha}z_{\mu}^\frac{\alpha
p}{p-1+\alpha}}{2l\sigma\|e_{p}\|_{\infty}^{p-1} })]\leq
\frac{m^{p}(1-\alpha)(p-1)p^{p-1}}{(p-1+\alpha)^{p}}.
$$
 Hence
\begin{equation}\label{e2.7}
\lambda k+\frac{1}{ k_0^{\alpha}z_{\mu}^\frac{\alpha
p}{p-1+\alpha}} \leq \frac{k_0^{p-1}p^{p-1}(1-\alpha)(p-1)|\nabla
z_{\mu}|^{p}}{(p-1+\alpha)^{p}z_{\mu}^\frac{\alpha
p}{p-1+\alpha}}\quad\text{ in }{\Omega}_{\delta}
\end{equation}
From \eqref{e2.5}, \eqref{e2.7} it can be seen that \eqref{e2.4} holds
in $ {\Omega}_{\delta}$.

We will now prove \eqref{e2.4} holds also in
 $\Omega \setminus { \Omega}_{\delta}.$ Since $z_{\mu}\geq A $ in
$\Omega \setminus {\Omega}_{\delta}$ and by \eqref{e2.2} and \eqref{e2.6}
 we obtain
\begin{equation}\label{e2.8}
\lambda k+\frac{1}{ k_0^{\alpha}z_{\mu}^\frac{\alpha p}{p-1+\alpha}}
\leq \frac{k_0^{p-1}}{ z_{\mu}^\frac{\alpha
p}{p-1+\alpha}}\Big(\frac{p}{p-1+\alpha}\Big)^{p-1}z_{\mu}
\leq k_0^{p-1}\Big(\frac{p}{p-1+\alpha}\Big)^{p-1}
 z_{\mu}^\frac{(1-\alpha)(p-1)}{p-1+\alpha}
\end{equation}
 in $\Omega \setminus {\Omega}_{\delta}$.

From \eqref{e2.5} and \eqref{e2.8}, \eqref{e2.4} holds also in
$\Omega \setminus \overline{ \Omega}_{\delta}$. Thus $ \psi $ is a
positive subsolution of\eqref{e1.1} if
$\lambda \in [\underline{\lambda},\hat{\lambda}]$.
 We can now choose $M_{\lambda} \gg 1 $ such that
$ \psi \leq Z $. Let
$J(\Omega)=2\|e_{p}\|_{\infty}^{p-1}\mu \Big(\frac{p}{p-1+\alpha}\Big)^{p-1}$.
if $ \frac{m_0\sigma_1}{l\sigma}\geq J $ it is easy to see that
$\underline{\lambda}\leq \hat{\lambda}$, and for
$\lambda \in [\underline{\lambda},\hat{\lambda}]$ we have a positive solution.
This completes the proof
\end{proof}

\begin{thebibliography}{99}

\bibitem{AmArBu} A. Ambrosetti, D. Arcoya,  B. Buffoni;
\emph{Positive solutions for some semi-positone problems via bifurcation
theory},  Diff. Int. Eqs, \textbf{7} (1994), 655-663.

\bibitem{BeCaNi} H. Berestycki, L. A. Caffarelli,  L. Nirenberg;
\emph{Inequalities for second order elliptic equations with
applications to unbounded domains}, Duke Math. J., A Celebration of
John F. Nash Jr, \textbf{81} (1996), 467-494.

\bibitem{SCui} S. Cui;
\emph{Existence and nonexistence of positive solutions for singular
semilinear elliptic boundary value problems}, {\it Nonlinear Anal.},
\textbf{41} (2000), 149-176.

\bibitem{DraKerTak} P. Drabek, P. Kerjci, P. Takac;
\emph{Nonlinear differential equations}, Chapman and Hall/CRC, 1999.

\bibitem{GheRad} M. Ghergu, V. Radulescu;
\emph{Multi-parameter bifurcation and asymptotics for the singular
Lane-Emden-Fowler equation with a convection term},  Proc. Roy.
Soc. Edin, \textbf{135}(A) (2005), 61–83.

\bibitem{GheRad.} M. Ghergu, V. Radulescu;
\emph{Singular elliptic problems: bifurcation and asymptotic analysis},
 Oxford Lecture Series in Mathematics and its Applications, 37.
The Clarendon Press, Oxford University Press, Oxford (2008).

\bibitem{Hai} D. D. Hai;
\emph{On an asymptotically linear singular boundary value problem},
Topo. Meth. in Nonlin. Anal, \textbf{39} (2012), 83-92.

\bibitem{HaSanShi}  D. D. Hai, L. Sankar, R. Shivaji;
\emph{Infinite semipositone problem with asymptotically linear growth
 forcing terms },  Differential Integral Equations, \textbf{25}(12)
(2012), 1175-1188.

\bibitem{Lae} T. Laetsch;
\emph{The number of solutions of a nonlinear two point boundary value
problem},  Indiana Univ. Math. J., \textbf{20} (1970), 1-13.

\bibitem{Lion} P. L. Lion;
\emph{On the existence of positive solutions of semilinear elliptic
equations},  SIAM Rev, \textbf{24} (9) (1982), 441-467.

\bibitem{LeShiYe} Eunkyoung Lee, R. Shivaji, J. Ye;
\emph{Positive solutions for elliptic equations involving nonlinearities
with falling zeros},  Applied Mathematics Letters, \textbf{22}(12)
(2009), 846-851.

\bibitem{LeShiYe2} Eunkyoung Lee, R. Shivaji, J. Ye;
\emph{Classes of infinite semipositone systems,},  Proc. Roy. Soc.
Edin, \textbf{139}(A) (2009), 815-853.


\bibitem{Peter} Takac Peter;
\emph{Degenerate elliptic equations in ordered Banach spaces and
applications},  Nonlinear differential equations, \textbf{404} II
Chapman, Hall/CRC, Research Notes in Mathematic, 119-196.


\bibitem{RamShiYe} M. Ramaswamy, R. Shivaji, J. Ye;
\emph{Positive solutions for a class of infinite semipositone problems},
 Differential Integral Equations, \textbf{20}(12)
(2007), 1423-1433.

\bibitem{Zhang}  Z. Zhang;
\emph{Critical points and positive solutions of singular elliptic
boundary value problems,},  J. Math. Anal. Appl, \textbf{302}
 (2005), 476-483.

\end{thebibliography}

\end{document}