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

\AtBeginDocument{{\noindent\small
\emph{Electronic Journal of Differential Equations},
Vol. 2018 (2018), No. 178, pp. 1--14.\newline
ISSN: 1072-6691. URL: http://ejde.math.txstate.edu or http://ejde.math.unt.edu}
\thanks{\copyright 2018 Texas State University.}
\vspace{8mm}}

\begin{document}
\title[\hfilneg EJDE-2018/178\hfil Positive solution curves]
{Positive solution curves of an infinite semipositone problem}

\author[R. Dhanya \hfil EJDE-2018/178\hfilneg]
{Rajendran Dhanya}

\address{Rajendran Dhanya \newline
School of Mathematics and Computer Science,
Indian Institute of Technology, Goa 403401, India}
\email{dhanya.tr@gmail.com}

\dedicatory{Communicated by Ratnasingham Shivaji}

\thanks{Submitted May 3, 2018. Published November 1, 2018.}
\subjclass[2010]{35J25, 35J61, 35J75}
\keywords{Semipositone problems; topological methods; bifurcation theory}

\begin{abstract}
 In this article we consider the infinite semipositone problem
 $-\Delta u =\lambda f(u)$ in $\Omega$, a smooth bounded domain
 in $\mathbb{R}^N$,  and $u=0$ on $\partial\Omega$, where
 $f(t) = t^q-t^{-\beta}$ and $0 < q$, $\beta <1$.
 Using stability analysis we prove the existence of a connected branch
 of maximal solutions emanating from infinity. Under certain additional
 hypothesis on the extremal solution at $\lambda=\Lambda$ we prove a version
 of Crandall-Rabinowitz bifurcation theorem which provides a multiplicity
 result for $\lambda\in (\Lambda,\Lambda+\epsilon)$.
\end{abstract}

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

\section{Introduction}

Consider the  infinite semi-positone problem
\begin{equation} \label{ePl}
\begin{gathered}
-\Delta u = \lambda f(u)  \quad \text{in }\Omega\\
u>0 \quad\text{in }\Omega\\
u=0 \quad\text{on }\partial\Omega.
\end{gathered}
\end{equation}
where $f(t)=t^q-t^{-\beta}$, $0<q<1$ and $\beta\in (0,1)$ and $\lambda$  a
 positive parameter. Here $\Omega$ is assumed to be a bounded domain with
smooth boundary in $\mathbb{R}^N$. Note that $f(0)=-\infty$
(hence the name infinite semipositone problem) and $f$ is an increasing
 concave function in $\mathbb{R}^+$. Finding a positive solution for
semipositone problems are always challenging  and in fact proving the existence
of multiple positive solutions are even more difficult.
The existence of a positive solution for \eqref{ePl} when $\lambda$ large is
studied using sub-super solutions technique in \cite{SJE}.
Later in \cite{DES1}, it was additionally shown that when $\lambda$
is large there exists a maximal positive solution for \eqref{ePl} which
is in fact bounded below by the distance function
$d(x,\partial \Omega) = \inf \{|x-y|: y\in \partial\Omega \}$.
The aim of this work is to further understand this maximal branch of solution
of \eqref{ePl} which emanates  from $\infty$.

\begin{definition}\label{def1} \rm
We say $u$ is a solution of \eqref{ePl}, if
 $u\in C^2(\Omega)\cap C^{1}_0(\overline{\Omega})$ and 
$u(x)\geq c\, d(x,\partial\Omega)$
for some positive constant $c=c(\lambda)$.
\end{definition}

Suppose that $\partial\Omega$ is smooth and $u$ is a  solution of $(P_\lambda)$,
then the outward normal derivative $ \frac{\partial u}{\partial \nu}(x_0)<0$
for all $x_0\in \partial\Omega$. Conversely if we assume that
$ \frac{\partial u}{\partial \nu}\big|_{\partial\Omega}<0$ then  by the tubular
neighbourhood lemma  $u(x)\geq c$, $d(x,\partial\Omega)$ for some $c>0$.


\begin{definition} \rm
Let $\mathcal{S} = \{ (\lambda,u_\lambda) : u_\lambda \text{ is a solution 
to \eqref{ePl}, as in Definition \ref{def1}} \}$
and  let
$\Lambda = \inf \{\lambda>0: \text{\eqref{ePl} admits at least one solution}\}$.
\end{definition}

\begin{definition} \rm
We say $\lambda_{\infty}=\infty$ is a bifurcation point at infinity for \eqref{ePl}
if there exists a sequence $(\lambda_n,u_{\lambda_n})\in \mathcal{S}$  such that
$\lambda_n\to \lambda_{\infty}$ and $\|u_{\lambda_n}\|\to \infty$.
\end{definition}

The principal eigenvalue of the linearized operator associated to \eqref{ePl}
is denoted by $\Lambda_1(\lambda)$ and defined as
\begin{equation}
\Lambda_1(\lambda) = \inf_{\varphi \in H^1_0(\Omega), \|\varphi\|_2=1}
\Big(  \int_{\Omega} |\nabla \varphi|^2- \lambda \int_{\Omega} f'(u)\varphi^2 \Big).
\end{equation}
where $u$ solves \eqref{ePl} as in definition \ref{def1}.
Since the solution $u(x)$ behaves like $d(x)$ near $\partial\Omega$, by Hardy's inequality
the term $\int_{\Omega}f'(u) \varphi^2$ make sense.
The functional $  \int_{\Omega} |\nabla \varphi|^2- \lambda \int_{\Omega} f'(u)\varphi^2$
is bounded below and coercive on the set
$\{\varphi\in H^1_0(\Omega): \|\varphi\|_2=1\}$ and hence a minimizer exists.
Also one can show that $\Lambda_1(\lambda)$ satisfies the differential equation
$-\Delta \psi -\lambda f'(u)\psi = \Lambda_1(\lambda) \psi$ for some non-negative
$\psi\in H^1_0(\Omega)$. We say that a solution $u$ of \eqref{ePl} is stable
if $\Lambda_1(\lambda)$ is strictly positive.
Our main result is the following theorem.

\begin{theorem}  \label{thm1.4}
Assume that $\Omega$ is a bounded open set in $\mathbb{R}^N$ with smooth
 boundary and consider the infinite semipositone problem \eqref{ePl}
$-\Delta u= \lambda(u^q-u^{-\beta})$ in $\Omega$ for $0<q,\beta<1$ and $u=0$ on $\partial\Omega$.
\begin{itemize}
\item [(a)] There exists a $\Lambda\in (0,\infty)$ and for all
$\lambda>\Lambda$, there exists a maximal positive solution
$u_\lambda$  solving \eqref{ePl}.
And  $\|u_{\lambda}\|_{\infty} \to \infty$ as $\lambda \to \infty$,
 i.e.\ $\lambda_\infty$ is a bifurcation point at infinity.
Also if  $ \lambda\in (0,\Lambda)$, the problem \eqref{ePl} does not admit any
positive solution.

\item [(b)] The  maximal solution $u_\lambda$ is stable for all $\lambda>\Lambda$.

\item [(c)] There exists an unbounded connected branch $\mathcal{C}$ of
 solutions of \eqref{ePl} emanating from $(\infty,\infty)$ consisting of
the maximal solution $u_\lambda$. The map $(\Lambda,\infty) \ni \lambda \to u_{\lambda}$
is of class $C^2$ in $\mathbb{R}\times C_e(\overline{\Omega})$.
\end{itemize}
\end{theorem}

We prove  results (a) and (b) in Section 2 (see Theorems \ref{thm2.1}
 and \ref{thmstable}).
We introduce the  operator $\mathcal{A}$ and the space $C_e(\overline{\Omega})$
in  section 3 and prove the differentiability of the map $\mathcal{A}$
(in fact we prove $\mathcal{A}$ is  a $C^2$ map) in the Appendix.
Using the stability analysis and smoothness of the map $\mathcal{A}$
we prove $(c)$ in Theorem \ref{thm3.3}.  Existence of a positive solution
for large $\lambda$ for similar problems are well studied in literature.
For example  Shi-Yao\cite{ShiYao} and Hern\'andez et al.\ \cite{HERR}
consider the semipositone problem of the type $-\Delta u= \lambda u^q -u^{-\beta}$
 with Dirichlet boundary condition in an arbitrary smooth domain $\Omega$
and establish the existence of positive solution bounded below by the
distance function using sub-super solution techniques.
We also use similar techniques to prove the existence of solution for
large $\lambda$, but here in this work we additionally show that the maximal
solution curve $\lambda\to u_\lambda$ is in fact smooth.
Also see \cite{sauvy,davila,habib} for related problems where they prove
stability results for infinite semipositone problems. In \cite{AAB}
the authors discuss a bifurcation phenomenon for semipositone problems
($f(0)\in (-\infty,0)$) depending on the behaviour of $f(t)$ at infinity,
 i.e.\ depending on if $f$ is sublinear, superlinear or asymptotically
linear at infinity. Positive solutions curves of concave semipositone problems
are also studied in \cite{Ref2} and \cite{Ref3}.

In Section 4, existence of a non-negative weak solution at $\lambda=\Lambda$
is proved using a limiting argument (see Proposition \ref{prop4.1}).
We conclude our paper by proving the following result.

\begin{theorem} \label{thm1.5}
Either of the following two alternatives hold:
\begin{itemize}
\item[(a)] The extremal solution $u_{\Lambda}(x)$ does not belong to the
interior of $C_e(\overline{\Omega})$, or

\item[(b)] The point $\lambda=\Lambda$ is a bifurcation point, i.e.\ there exists a
$C^2$ curve $(\lambda(s),u(s))\in\mathcal{S}$ where $s\in (-\epsilon,\epsilon)$
with $\lambda(0)={\Lambda}$,  $\lambda'(0) = 0$, $\lambda''(0)<0$ and 
$ u(0)=u_{\Lambda}$.
\end{itemize}
\end{theorem}

To the best of our knowledge a complete bifurcation diagram for  semipositone
problem is understood in either of the following two situations:
 (a) in case of $f(0)=-\infty$ and dimension $N=1$ (see \cite{Diazmain})
or (b) in case of strictly semipositone problems, i.e.\
  $-\infty<f(0)<0$ in a ball (see \cite{GaCa}).
In the latter work the results were obtained by  using shooting methods
for ODE as any positive solution for a semipositone problem in a ball is
known to be radially symmetric.
In Theorem \ref{thm1.5} we make an attempt to understand  the bifurcation curve
in arbitrary domain $\Omega$ under certain additional hypothesis on extremal
function $u_{\Lambda}$. The second alternative gives a precise description of
the bifurcation branch at $\lambda=\Lambda$. At least in dimension $N=1$ and
$\beta \in (0,\frac{1}{2})$, it is clear from
\cite[Theorem 2]{Diazmain} that the first case does not arise.
The second alternative also suggests the existence of multiple positive
solutions for \eqref{ePl} when $\lambda\in (\Lambda, \Lambda+\epsilon)$ for some $\epsilon>0$.
In fact the solution in the lower branch (the non-maximal solution)
is also  bounded below by $\tilde{c}(\lambda) d(x,\partial\Omega)$. It is expected that
the solutions exhibit a "free boundary" condition(i.e.\ a non negative solution
becomes zero in a set of positive measure) beyond $\Lambda+\epsilon$.

\section{Stability analysis}

\begin{theorem} \label{thm2.1}
There exists a $\Lambda\in (0,\infty)$ and for all $\lambda>\Lambda$,
there exists a positive function $u_\lambda$  solving \eqref{ePl}
as defined in \ref{def1}.  In fact, the function $u_\lambda$ is the maximal
solution for \eqref{ePl}.
\end{theorem}

\begin{proof}
 For $\lambda$ large enough the existence of a positive solution bounded
 below by $d(x,\partial\Omega)$  is obtained in Section 5 of \cite{DES1} for more
general nonlinear function $f$. Here we briefly explain the sub and
supersolution to be chosen for our particular nonlinearity
$f(t)= t^q-t^{-\beta}$. Following the lines of proof of \cite[Example 5.6]{DES1}
we define $\psi = \lambda^r(\phi_1+\phi_1^{\frac{2}{1+\beta}})$,
where $\phi_1$  is the first eigenfunction of $-\Delta$,
 and $1<r<\frac{1}{1-q+\epsilon}$ is chosen so that
$-\Delta \psi \leq \lambda (\psi^q-\psi^{-\beta})$. We define a
 super-solution $\phi= v_\lambda$ where
\begin{equation}\label{glob}
-\Delta v_\lambda= \lambda v_\lambda^q  \text{ in } \Omega, \quad
  v_\lambda=0 \text{ on } \partial\Omega.
\end{equation}
Then we know that $v_\lambda= \lambda^{\frac{1}{1-q}} v_1$ and hence for
large $\lambda$ we have $\psi\leq \phi$. Now by \cite[Theorem 5.5]{DES1}
 there exists a maximal solution $u_\lambda$ in the ordered interval
$[\psi,\phi]$. Thus the solution is bounded below by $\psi$ and hence
 \begin{equation}\label{refeq1}
 u_{\lambda}(x) \geq \psi = \lambda^{r}(\phi_1+\phi_1^{\frac{1}{1+\beta}}) ,\quad
 \text{i.e. } \|u_{\lambda}\|_{\infty}\to \infty\text{ as } \lambda \to \infty.
 \end{equation}

Suppose $u$ is a solution of \eqref{ePl}. Then, $-\Delta u\leq \lambda u^q$ and
by comparison \cite[Lemma 2.2]{SS} $u\leq v_\lambda$.
Thus the $u_\lambda$ that we constructed via sub-super solution is
in fact the maximal positive solution of \eqref{ePl}.
Now define $\Lambda = \inf \{\lambda>0: (P_\lambda)$ admits at least one solution$\}$.
Next we claim that
\begin{equation}\label{refeq2}
0<\Lambda<\infty.
\end{equation}
Clearly from our previous discussion $\Lambda<\infty$.
We shall now prove that $\Lambda>0$.
Suppose on the contrary that $\Lambda=0$, then there exists a sequence
$(\lambda_m, u_{\lambda_m})\in \mathcal{S}$ and $\lambda_m\to 0$.
By comparison Lemma we have
$0<u_{\lambda_m}\leq v_{\lambda_m}$. Therefore  for large $m$,
since $v_{\lambda_m} = \lambda_m^{\frac{1}{1-q}} v_1$ we have
$0<u_{\lambda_m}<1$ and 
$-\Delta u_{\lambda_m}=\lambda_m(u_{\lambda_m}^q-u_{\lambda_m}^{-\beta}) <0$.
This leads to a contradiction, since by maximum principle any such solution
$u_{\lambda_m}$ has to be necessarily negative and hence $\Lambda>0$.

Next we claim that for any $\lambda>\Lambda$ there exists at least one solution
for \eqref{ePl}. Fix $\lambda>\Lambda$, then by definition there exists a
$\lambda'\in (\Lambda,\lambda)$ such that \eqref{ePl} with $\lambda=\lambda'$ admits at least one solution which
we call $\psi$. Note that we do not claim $\psi$ is a sub-solution for \eqref{ePl},
but still we prove that there exists a $u_\lambda>\psi$ solving \eqref{ePl}.
Clearly, $\psi< v_{\lambda'}<v_{\lambda}=:u_0$. Let
\begin{gather*}
-\Delta u_1 =  \lambda (u_0^q-u_0^{-\beta}) \quad\text{in }  \Omega\\
u_1 = 0 \quad \text{on } \partial \Omega.
\end{gather*}
By the standard weak comparison principle for the functions in  $W^{2,p}(\Omega)$
we obtain $u_1<u_0$. We claim that $\psi<u_1<u_0$. In fact,
\begin{align*}
-\Delta (u_1-\psi)
&= \lambda f(u_0)-\lambda' f(\psi)
 \geq  \lambda f(\psi)-\lambda' f(\psi)\\
&=\Big(\frac{\lambda-\lambda'}{\lambda'}\Big) \lambda' f(\psi)
 = -\Delta(\delta \psi)
\end{align*}
where $\delta= (\lambda-\lambda')/\lambda'>0$.
Thus once again by comparison method we prove the claim.
Iteratively if we define the sequence
\begin{gather*}
-\Delta u_{n+1}= \lambda (u_n^q-u_n^{-\beta}) \quad\text{in }  \Omega\\
u_{n+1}= 0 \quad \text{on } \partial \Omega.
\end{gather*}
by mathematical induction we can easily prove that
$$
\psi < \cdots \leq u_{n+1}\leq u_n \leq \cdots u_1<u_0.
$$
Thanks to the lower and upper bound of the sequence $\{u_n\}$, we have have
$u_n\in C^{1,\gamma}_0(\overline{\Omega})\cap C^2(\Omega)$
(see \cite[Theorem 5.2]{DES1} and \cite{GL}).
Hence the sequence $\{u_n\}$ is bounded say in $H^1_0(\Omega)$ and if we
define $u_\lambda= \lim_{n\to \infty} u_n$, then $u_\lambda$ is the maximal
 solution of \eqref{ePl}.
\end{proof}

Our next aim is to prove that the principal eigenvalue of the linearized
operator about the maximal solution $u_\lambda$  is positive.
As a first step towards it we prove the following proposition.

\begin{proposition}\label{semist}
The maximal solution  $u_\lambda$ is semi-stable or
the principal eigenvalue of the linearized operator
\[
\Lambda_1(\lambda) = \inf_{\varphi \in H^1_0(\Omega), \|\varphi\|_2=1}
\Big(  \int_{\Omega} |\nabla \varphi|^2
- \lambda \int_{\Omega} f'(u_\lambda)\varphi^2 \Big) \geq 0\,.
\]
\end{proposition}

\begin{proof}
For a fixed $\lambda>\Lambda$ we consider the $\epsilon$-approximate regular problem
\begin{equation} \label{ePle}
\begin{gathered}
-\Delta w = \lambda \left((w+\epsilon)^q-(w+\epsilon)^{-\beta})\right) \quad \text{in }\Omega,\\
w>0 \quad \text{in }\Omega, \\
w=0 \quad\text{on }\partial\Omega.
\end{gathered}
\end{equation}
Let
$$
-\Delta v_\lambda^\epsilon =\lambda (v_\lambda^\epsilon+\epsilon)^q \text{ in } \Omega\quad
v_\lambda^\epsilon>0 \text{ in } \Omega;\quad
v_\lambda^\epsilon=0 \text{ on } \partial\Omega.
$$
It is easy to check that $v_\lambda^\epsilon$ exists and $v_\lambda^\epsilon<v_\lambda$,
Note that $u_\lambda$ and $v_\lambda^\epsilon$ are respectively sub and super solutions
of \eqref{ePle} and by standard monotone iteration there exists a
$w_\epsilon\in [u_\lambda,v_\lambda]$ solving \eqref{ePle}. In fact $w_\epsilon$ is the maximal
solution of \eqref{ePle}. By Hopf's maximum principle for some
$\theta_1>0$ we have $w_\epsilon(x)+\theta_1 d(x,\partial\Omega)\leq v_\lambda^\epsilon$.
Next we observe that the sequence $\{w_\epsilon\}$ is bounded independent of $\epsilon$ since
$$
\int_{\Omega} |\nabla w_\epsilon|^2
\leq \lambda \int_{\Omega} (w_\epsilon+1)^{q+1}
\leq \lambda \int_{\Omega} (v_\lambda+1)^{q+1} <\infty.
$$
Clearly $w_\epsilon$ converges to some function $\tilde{w}$ which is a weak solution
of \eqref{ePl} and $u_\lambda \leq \tilde{w} \leq v_\lambda^\epsilon$.
 Since $u_\lambda$ is the maximal solution of \eqref{ePl} we must have
\begin{equation}\label{limeqn}
\lim_{\epsilon\to 0} w_\epsilon = u_\lambda.
\end{equation}
Let us write $f_\epsilon(t)= (t+\epsilon)^q-(t+\epsilon)^{-\beta}$.
\smallskip

\noindent\textbf{Claim:}
$\Lambda_1^\epsilon(\lambda) = \inf_{\varphi \in H^1_0(\Omega), \|\varphi\|_2=1}
\left(  \int_{\Omega} |\nabla \varphi|^2- \lambda \int_{\Omega} f_\epsilon'(w_\epsilon)\varphi^2 \right)
\geq 0$.
On the contrary suppose that $\Lambda_1^\epsilon(\lambda)<0$ and $\varphi_\epsilon\in H^1_0(\Omega)$
 be the associated non-negative eigenfunction of
$$
-\Delta \varphi_\epsilon-\lambda f_\epsilon'(w_\epsilon)\varphi_\epsilon = \Lambda^\epsilon_1(\lambda) \varphi_\epsilon.
$$
We will show that $(w_\epsilon+\theta \varphi_\epsilon)$ is a sub solution of \eqref{ePle}.
 For a non-negative $\varphi\in H^1_0(\Omega)$,
\begin{align*}
&\int_{\Omega} \nabla (w_\epsilon+\theta \varphi_\epsilon) \nabla \varphi-\lambda \int_{\Omega} f_\epsilon(w_\epsilon+\theta \varphi_\epsilon)\varphi \\
&= \lambda \int_{\Omega} f_\epsilon(w_\epsilon)\varphi -f_\epsilon(w_\epsilon+\theta \varphi_e)\varphi+\theta f_\epsilon'(w_\epsilon)\varphi_\epsilon\varphi
 +\theta \Lambda_1^\epsilon(\lambda)\int_{\Omega}\varphi\varphi_e\\
&= o(\theta)+\theta \Lambda^\epsilon_1(\Lambda) \int_{\Omega} \varphi\varphi_\epsilon
\end{align*}
Choosing $\theta>0$ small enough we have $(w_\epsilon+\theta \varphi_\epsilon)$
is a sub-solution of \eqref{ePle}. If required we may choose $\theta$
smaller so that $w_\epsilon(x)+\theta \varphi_\epsilon\leq v_\lambda^\epsilon$.
Thus $w_\epsilon+\theta \varphi_\epsilon$ and $v_\lambda^\epsilon$ forms an ordered pair of sub and
super solution of \eqref{ePle} and we   obtain a solution
$\tilde{w}_\epsilon\in [w_\epsilon+\theta \varphi_\epsilon,v_\lambda]$ of \eqref{ePle}.
This contradicts the fact that  $w_\epsilon$ is the maximal solution
of \eqref{ePle} and hence the claim is verified.
Thus for every $\varphi\in H^1_0(\Omega)$ such that $\|\varphi\|_2=1$,
$$
\int_{\Omega} |\nabla \varphi|^2-\lambda \int_{\Omega} f_\epsilon'(w_\epsilon) \varphi^2 \geq 0.
$$
Now passing through the limit using \eqref{limeqn} and Hardy's
inequality we obtain that $\Lambda_1(\lambda)\geq 0$.
\end{proof}

\begin{proposition} \label{prop2.3}
The semi-stable solution of \eqref{ePl} is unique.
\end{proposition}

\begin{proof}
 Let $u_\lambda$ be the maximal solution of  \eqref{ePl} and $v_\lambda$
be any other solution of \eqref{ePl}.
We know that $u_\lambda$ is semi-stable by Proposition \ref{semist} and
assume that $v_\lambda$ is also semi-stable. Then
$$
\int_{\Omega} | \nabla w|^2 \geq \lambda \int_{\Omega} f'(v_\lambda) w^2
$$
for all $w\in H^1_0(\Omega)$. In particular,
\begin{equation}
\int_{\Omega} |\nabla (u_\lambda-v_\lambda)|^2\geq \lambda \int_{\Omega} f'(v_\lambda) (u_\lambda-v_\lambda)^2.
\end{equation}
Since $v_\lambda$ and $u_\lambda$ are both the solutions of \eqref{ePl}
\begin{equation}
\int_{\Omega} |\nabla (u_\lambda-v_\lambda)|^2= \lambda \int_{\Omega} (f(u_\lambda)-f(v_\lambda))(u_\lambda-v_\lambda).
\end{equation}
Combining the above two equations  we have
$$
\int_{\Omega} \{ f(u_\lambda)-f(v_\lambda) -f'(v_\lambda)(u_\lambda-v_\lambda)\} (u_\lambda-v_\lambda)\geq 0.
$$
Since $u_\lambda$ is the maximal solution this implies
$$
\int_{\{u_\lambda>v_\lambda\}} \left\{ f(u_\lambda)-f(v_\lambda) -f'(v_\lambda)(u_\lambda-v_\lambda)\right\}
(u_\lambda-v_\lambda)\geq 0.
$$
Since $f$ is strictly concave the above integral is strictly negative if
the Lebesgue measure of  the set $\{x: u_\lambda(x)>v_\lambda(x)\}$ is non-zero.
Thus $u_\lambda\equiv v_\lambda$, or the semi-stable solution is unique.
\end{proof}

Next we shall prove our main result of this section, the maximal
$u_\lambda$ is stable. We consider here a different  approximate problem
$\eqref{eQlt}$ for a parameter $\theta<0$.
\begin{equation} \label{eQlt}
\begin{gathered}
-\Delta z = \lambda \big(z^q-z^{-\beta}+\theta)\big) \quad \text{in }\Omega,\\
z>0 \quad \text{in }\Omega, \\
z=0 \quad \text{on }\partial\Omega.
\end{gathered}
\end{equation}

\begin{lemma}\label{dectheta}
For each $\theta\in (\theta_0,0)$ there exists a function
$z_\theta$ which is a maximal solution of \eqref{eQlt}.
If $\theta<\theta'$ then $z_\theta\leq z_{\theta'}$ and $z_\theta\not = z_{\theta'}$.
\end{lemma}

\begin{proof}
Fix a $\lambda\in (\Lambda,\infty)$ and choose $\lambda'\in (\Lambda,\lambda)$. Let
$$
-\Delta V_\lambda=\lambda \text{ in }\Omega; \quad
 V_\lambda =0 \text{ in } \partial\Omega
$$
and $u_\lambda$ be the maximal solution of \eqref{ePl}. Define
$\underline{z}_\epsilon= \frac{\lambda}{\lambda'}u_{\lambda'}-\epsilon V_\lambda$.
Then for some positive constants $C_1,C_2$
$$
\underline{z}_\epsilon-u_{\lambda'}= \Big(\frac{\lambda-\lambda'}{\lambda'}\Big)u_{\lambda'} -\epsilon V_\lambda
\geq \left(C_1 -\epsilon C_2\right)d(x,\partial\Omega).
$$
If we choose $0<\epsilon<|\theta_0|$ for some small $\theta_0<0$, we have
$\underline{z}_\epsilon>u_{\lambda'}$. For all $\theta\in (\theta_0,0)$ define
\begin{equation}
\underline{z}_\theta = \frac{\lambda}{\lambda'} u_{\lambda'} +\theta V_\lambda.
\end{equation}
Then $-\Delta \underline{z}_\theta = \lambda(u_{\lambda'}^q-u_{\lambda'}^{-\beta}+\theta)
 \leq \lambda ( \underline{z}_{\theta}^q-\underline{z}_{\theta}^{-\beta}+\theta )$
and hence a sub solution of \eqref{eQlt}. It is easy to check that
$\overline{z}_\theta= v_\lambda$ is a super solution of \eqref{eQlt} for all $\theta<0$.
Since $u_{\lambda'}<v_{\lambda'}$
$$
\underline{z}_{\theta}-\overline{z}_\theta = \frac{\lambda}{\lambda'} u_{\lambda'} +\theta V_\lambda - v_\lambda
< \frac{\lambda}{\lambda'} v_{\lambda'}-v_\lambda
= \Big(\frac{\lambda}{\lambda'}  (\lambda')^{\frac{1}{1-q}} -\lambda^{\frac{1}{1-q}}\Big)v_1 <0 .
$$
Thus there exists a solution $z_\theta$ of $\eqref{eQlt}$ in between the ordered
pair $[\underline{z}_{\theta},\overline{z}_\theta]$. As before using comparison lemma one can easily observe that
$z_\theta$ is the maximal solution of $\eqref{eQlt}$.
Now let $\theta<\theta'$ and $z_\theta, z_{\theta'}$ be the maximal solutions
of \eqref{eQlt} and \eqref{eQlt} with $\theta=\theta'$ respectively. Then
$$
-\Delta z_\theta  \leq \lambda (z_\theta^q-z_\theta^{-\beta}+\theta')\quad\text{and}\quad
 z_\theta\leq \overline{z}_{\theta'}.
$$
Since $z_{\theta'}$ is the maximal solution of \eqref{eQlt} with $\theta=\theta'$ we
conclude that $z_\theta\leq z_{\theta'}$.
\end{proof}

\begin{theorem}\label{thmstable}
The maximal solution $u_\lambda$ of \eqref{ePl} is stable.
\end{theorem}

\begin{proof}
Let ${\Lambda_1^\theta}(\lambda)$ denote the principal eigenvalue of $\eqref{eQlt}$.
 Repeating the calculations of Proposition \ref{semist} we can show that
$\Lambda_1^{\theta}(\lambda) \geq 0$. If $\theta_1<\theta_2$ using the strict concavity of
$f$ and Lemma \ref{dectheta} we have for all $\varphi\in H^1_0(\Omega)$, $\|\varphi\|_2=1$,
$$
\int_{\Omega} |\nabla \varphi|^2-\lambda \int_{\Omega} f'(z_{\theta_1})\varphi^2
< \int_{\Omega} |\nabla \varphi|^2-\lambda \int_{\Omega} f'(z_{\theta_2})\varphi^2 .
$$
Since $ \inf_{\varphi\in H^1_0(\Omega)} \int_{\Omega} |\nabla \varphi|^2-\lambda \int_{\Omega} f'(z_{\theta})\varphi^2 $
 is attained, we have $\Lambda_1^{\theta_1}(\lambda) < \Lambda_1^{\theta_2}(\lambda)$.
Observe that $z_\theta \to u_\lambda$ as $\theta\to 0^-$ and $
\lim_{\theta\to 0^-} \Lambda_1^\theta(\lambda) = \Lambda_1(\lambda)$. Thus
$$
\Lambda_1(\lambda) > \Lambda_1^\theta(\lambda) \geq 0
$$
which is the main result.
\end{proof}

\section{Bifurcation analysis}

In the previous section we have shown that for each $\lambda>\Lambda$
there exists a maximal solution for \eqref{ePl}.
In this section we try to understand this maximal branch of solution
using bifurcation theory.
For  $\lambda>\Lambda$, consider the function $u_{\lambda'}$
 which is a solution of \eqref{ePl} with $\lambda=\lambda'$ for some $\lambda'\in [\Lambda,\lambda)$
 and $v_\lambda$ as in \eqref{glob}. To ease notation we omit the subscript
$\lambda$ and denote $\psi=\psi_\lambda= u_{\lambda'}$ and $\phi=\phi_\lambda=v_\lambda$,
then clearly  $\psi < \phi $. Let
 \begin{equation}
 \mathcal{C}_{\lambda}=\{u\in C_0(\overline{\Omega}) : \psi \leq u \leq \phi \}.
\end{equation}

For each $u\in \mathcal{C}_{\lambda}$ there exists $w\in C^1_0(\overline{\Omega})\cap C^2(\Omega)$
 which is a solution of
\begin{equation}\label{defag}
-\Delta w= \lambda  f(u) \text{ in } \Omega, \quad  w=0 \text{ on } \partial\Omega.
\end{equation}
The existence of $w\in W^{2,p}(\Omega)$ easily follows from the lower estimate
on $u$ and the regularity of $w$ by \cite{GL} (see Section 5 of \cite{DES1}
for the details). Since we would repeatedly use the regularity result
of Gui-Lin \cite{GL}, for the sake of completeness we quote the result below.

\begin{theorem}[{Gui-Lin \cite[Prop. 3.4]{GL}}]  \label{guilin}
 Let $\Omega$ be a bounded Lipschitz domain in $\mathbb{R}^n$, and suppose
 $u\in C^2(\Omega)\cap C(\overline{\Omega})$ satisfies
$$
|\Delta u(x)|\leq M d(x)^{-\beta} \quad \text{and} \quad
|u(x)|\leq M d(x)^{\alpha}
$$
for some positive constants $M, \alpha$. Then there exists some
$\gamma \in (0,1)$ depending upon $\beta$ and $\alpha$ such that
$\|u\|_{C^{1,\gamma}(\overline{\Omega})}\leq C(M,\alpha,\beta)$.
\end{theorem}

We can in fact prove that the solution $w$ of \eqref{defag} belongs to $ \mathcal{C}_{\lambda}$.
One can observe that $w\leq \phi$ since $\phi$ is a supersolution of \eqref{defag}.
It is not clear if $\psi$ is a sub solution of \eqref{defag} or not.
But still by the specific choice of $\psi$ we can show that
\begin{equation}\label{subs}
-\Delta(w-\psi)=\lambda f(u)-\lambda' g(\psi) \geq   \frac{\lambda-\lambda'}{\lambda'}(-\Delta \psi )
\end{equation}
Since $\lambda'<\lambda$ it follows that $w>\psi$ and hence $w\in \mathcal{C}_{\lambda}$.
For a fixed $\lambda\in (\Lambda,\infty)$ we define the map
\begin{equation}\label{defag1}
\mathcal{A}{:} \mathcal{C}_{\lambda} \to \mathcal{C}_{\lambda} \text{ is defined as } \mathcal{A}(u)
= w \text{ if } w \text{ is a solution of } \eqref{defag}.
\end{equation}

We aim to employ the well known abstract setting of bifurcation theory to
prove the existence of a connected branch of solutions.
If we consider the map $\mathcal{A}{:} \mathcal{C}_{\lambda} \to \mathcal{C}_{\lambda}$ it is not possible to use
the implicit function theorem since the set $\mathcal{C}_{\lambda}\subset C_0(\overline{\Omega})$
has empty interior. Hence we introduce the space $C_e(\overline{\Omega})$ as in \cite{amann}
and consider the set $\mathcal{C}_{\lambda}$ with the topology induced from $C_e(\overline{\Omega})$ in which
$\mathcal{C}_{\lambda}$ has nonempty interior.

Let $e\in C^2(\overline{\Omega})$ denote the unique positive solution of
\begin{gather*}
-\Delta e = 1\quad \text{in }\Omega\\
e= 0\quad \text{on }\partial \Omega.
 \end{gather*}
Then $e(x)>0$ in $\Omega$, $\frac{\partial e}{\partial \nu} <0$ on $\partial\Omega$ and
thus $e(x)\geq k d(x,\partial\Omega)$ for some constant $k>0$.
 $C_e(\overline{\Omega})$ is the set of functions in $u \in C_0(\overline{\Omega})$
such that   $-t e \leq u\leq t e$ for some $t\geq 0$.
$C_e(\overline{\Omega})$ equipped with
$ \|u\|_e = \inf\{ t > 0 : -t e \leq u \leq t e\} $
is a Banach space. Also the following continuous embedding holds:
$$
C_0^1(\overline{\Omega}) \hookrightarrow C_e(\overline{\Omega})
\hookrightarrow C_0(\overline{\Omega}).
$$
Further $C_e(\overline{\Omega})$ is an ordered Banach space(OBS) whose positive
 cone $P_e = \{u \in C_e(\overline{\Omega}) : u(x)\geq 0\}$
is normal and has non empty interior. In particular the interior of $P_e$
consists of all those functions $u\in C(\overline{\Omega})$
with $t_1 e\leq u\leq  t_2 e$ for some $t_1,t_2 >0$. Define
\begin{equation}\label{mlambda}
\mathcal{M}_{\lambda}=\{u\in C_e(\overline{\Omega}) : \psi \leq u \leq \phi \}
\end{equation}
 Using the lower and upper bounds for $\psi$ and $\phi$ in terms of
$d(x,\Omega)$ we find that set theoretically $\mathcal{C}_{\lambda}$ is same as $\mathcal{M}_\lambda$.
But topologically they are different and in fact $\mathcal{M}_\lambda$
has non empty interior which we denote by $\mathcal{U}_\lambda$ where
\begin{equation}\label{ulambda}
\mathcal{U}_{\lambda}= \{ u\in \mathcal{M_\lambda}
: \psi+ t_1 e  \leq  u  \leq \phi-t_2  e \text{ for some } t_1, t_2 >0 \}.
\end{equation}
By definition the set $\mathcal{U}_\lambda$ is open and we denote the restriction
of the map $\mathcal{A}$ to $\mathcal{U}_{\lambda}$ as $\mathcal{A}$ itself.
From \eqref{defag1} $\mathcal{A}$ maps $\mathcal{U}_\lambda$ to $\mathcal{C}_{\lambda}$.
In the next proposition we prove that $\mathcal{A}$ maps $\mathcal{U}_{\lambda}$ to itself and it is
a $C^2$ map.

\begin{proposition} \label{prop3.2}
The map $\mathcal{A} {:} \mathcal{U}_{\lambda} \to \mathcal{U}_{\lambda}$ is twice continuously differentiable.
The map $\mathcal{A}'(u) {:} C_e(\overline{\Omega})\to C_e(\overline{\Omega})$ is continuous linear and compact.
\end{proposition}

\begin{proof}
 Let $u\in \mathcal{U}_{\lambda}$, i.e there exists some $t_1, t_2 >0$ such that
$\psi+ t_1 e \leq  u  \leq \phi-t_2  e$ and let $\mathcal{A}(u)=w$.
Then $-\Delta(w-\phi)<0$ in $\Omega $ and $w-\phi =0$ on $\partial\Omega$, and by
 Hopf Maximum principle there exists a $\tilde{t}_2>0$ for which
$w\leq \phi-\tilde{t}_2 e$. From our previous discussion \eqref{subs}
if we take $\tilde{t}_1 =\frac{\lambda-\lambda'}{\lambda}$ we find $w\geq \psi+\tilde{t}_1 e$.
Thus $\mathcal{A}$ maps $\mathcal{U}_{\lambda}$ into itself.  Proof of the smoothness of the map $\mathcal{A}$
and the compactness of $\mathcal{A}'(u)$ is much technical and we shall give
the details in the Appendix.
\end{proof}

Next we shall treat $\lambda$ as a variable and define the map
$A{:} (\Lambda,\infty)\times \mathcal{U}_{\lambda} \to \mathcal{U}_{\lambda}$ as $A(\lambda, u) =w$ if $w$ is a solution of
\begin{equation}\label{defag2}
-\Delta w= \lambda f(u) \text{ in } \Omega, \quad  w=0 \text{ on } \partial\Omega.
\end{equation}

Fix $\lambda_1,\lambda_2 $ such that $\Lambda < \lambda_1<\lambda_2<\infty$.
Then for all $\lambda\in [\lambda_1,\lambda_2]$ we can in fact fix the indexed set $\mathcal{U}_{\lambda}$
independent of $\lambda$ in the following way.
By the definition of $\Lambda$ there exists a $\lambda'\in [\Lambda,\lambda_1)$
and \eqref{ePl} with $\lambda=\lambda'$ is solvable. Let $\psi = u_{\lambda'}$ and
$\phi =v_{\lambda_2}$ and  let $\mathcal{M}_{\lambda}$
and $\mathcal{U}_{\lambda}$ defined as before in \eqref{mlambda} and \eqref{ulambda}
for this choice of $\psi$ and $\phi$. Now $\mathcal{U}_{\lambda}$ is independent of
$\lambda$ for all $\lambda\in [\lambda_1,\lambda_2]$. For this particular choice of
$\mathcal{U} = \mathcal{U}_{\lambda}$ we can prove that the map $A$ is $C^2$ in
$\lambda$ and $u$ variable in $(\lambda_1,\lambda_2)\times \mathcal{U}$.

\begin{theorem} \label{thm3.3}
There exists a connected branch of positive maximal solutions of \eqref{ePl}
 bifurcating from $\lambda_\infty = \infty$.
\end{theorem}

\begin{proof}
Fix an open interval  $I \subset (\Lambda,\infty)$ and $\overline{\text{I}}$
compactly contained in  $(\Lambda,\infty)$. Let $I=(\lambda_1,\lambda_2)$ and $\psi=u_{\lambda'}$
and $\phi=v_{\lambda_2}$ as before. Thus for all $\lambda\in I$  we define
$\mathcal{M}=\{u\in C_e(\overline{\Omega}) : \psi\leq u \leq \phi\}$ and $\mathcal{U}$
to be the interior of $\mathcal{M}$.
Consider the map $F{:} I\times \mathcal{U} \to \mathcal{U}$ defined as
\begin{equation}\label{mapf}
F(\lambda,u) = u -A(\lambda,u).
\end{equation}
Clearly the zeroes of $F$ are the solutions of \eqref{ePl}
 and $F(\lambda,u_{\lambda})=0$ where $u_{\lambda}$ is the maximal solution of \eqref{ePl}.
 Note that $F{:} I\times \mathcal{U} \to \mathcal{U}$ is a $C^2$ map and
 $\partial_u F(\lambda,u)=I - \partial_u A(\lambda ,u)$ is a compact perturbation of identity.
 Fix $\lambda_0\in I$ and let $u_0=u_{\lambda_0}$ be the maximal solution of
\eqref{ePl} with $\lambda=\lambda_0$, then $F(\lambda_0,u_0)=0$.
From Theorem \ref{thmstable}  we know that $u_0$ is a stable solution
and hence   $\partial_u F(\lambda_0,u_0)$ is one-one. Now by Fredholm alternative
it is onto as well. Thus the linear map $\partial_u  F(\lambda_0,u_0)$ is
 bijective and continuous, hence by open mapping theorem
$\partial_u  F(\lambda_0,u_0)$ has a continuous inverse. Now we can apply
implicit function theorem around $(\lambda_0,u_0)$ and deduce that there
exists a $C^2$ curve $(\lambda,u(\lambda))\in (\lambda_0-\epsilon,\lambda_0+\epsilon) \times \mathcal{U}$
such that the set of all solutions of $F(\lambda,u)=0$ in a neighbourhood
of $(\lambda_0,u_0)$ is given by $(\lambda,u(\lambda))$.
Note that this $u(\lambda)$ may be different from the maximal solution $u_\lambda$.

If we can show that $\lambda \longmapsto u_\lambda$ (where $u_{\lambda}$ is the maximal solution)
is continuous then by the uniqueness of the solution near $(\lambda_0,u_0)$
we have a $u(\lambda)=u_\lambda$. On the contrary suppose $\lambda\longmapsto u_{\lambda}$
is not continuous at $\lambda_0$. i.e.\ there exists a sequence
$\lambda_n\to \lambda_0$ such that $u_{\lambda_n} \not \to u_0$. One can use Hardy's
inequality to prove that $\{u_{\lambda_n}\}$ is bounded in $H^1_0(\Omega)$
and hence up to a sub sequence $u_{\lambda_n} \rightharpoonup \tilde{u}$ in $H^1_0(\Omega)$.
It is also easy to check that $\tilde{u}$ is a solution of $(P_{\lambda_0})$.
Since $u_{0}$ is the maximal solution of $(P_{\lambda_0})$ we have
\begin{equation}\label{comp}
\tilde{u} \leq u_0 \quad\text{and}\quad \tilde{u} \not = u_0.
\end{equation}
On the other hand we have $u(\lambda_n) \to u_0$ and $u(\lambda_n)\leq u_{\lambda_n}$.
Taking limit as $n\to \infty$ we find $u_0 \leq \tilde{u}$ which contradicts
\eqref{comp}. We have now $u(\lambda)=u_\lambda$ and hence by implicit function
theorem $\lambda\to u_{\lambda}$ is a $C^2$ map which completes the proof of
theorem.
\end{proof}

\begin{remark} \label{rmk3.4}\rm
The smoothness of the map $\lambda \to u_{\lambda}$ for $\lambda\in (\Lambda,\infty)$
is completely determined by the smoothness of the operator $\mathcal{A}$.
We can in fact prove that the map is infinitely many times differentiable,
hence $\lambda\to u_\lambda$ is a $C^\infty$ map.
\end{remark}

The proof of our main result now follows from Theorem \ref{thm2.1},
equations \eqref{refeq1}, \eqref{refeq2}, Theorems \ref{thmstable},
\ref{thm3.3} and Remark \ref{rmk3.4}.

\section{Bifurcation analysis at $\lambda=\Lambda$}

\begin{proposition} \label{prop4.1}
There exists a non-negative solution $u_{\Lambda}$ solving \eqref{ePl}
with $\lambda=\Lambda$ in the weak sense. The Lebesgue measure of the set
$\{x: u_{\Lambda}(x)=0\}$ is zero.
\end{proposition}

\begin{proof}
Let $\{u_n\}$ denote the sequence of maximal solutions of $(P_{\lambda_n})$ where
$\lambda_n \downarrow \Lambda$ and $\lambda_n<\overline{\lambda}$. If $\overline{v}$ denote
the solution of \eqref{glob} for $\lambda=\overline{\lambda}$, we have
$0<u_{n+1} \leq u_n\leq {\overline{v}}$ and
\begin{equation}
  \int_{\Omega} |\nabla u_n|^2 = \lambda_n\int_{\Omega}(u_n^{q+1}- u_n^{1-\beta})
\leq {\lambda_n}\int_{\Omega}u_n^{q+1}\leq\overline{\lambda} \int_{\Omega} {\overline{v}}^{q+1}.
\end{equation}
Thus the sequence $\{u_n\}$ is bounded in $H^1_0(\Omega)$ and denote the weak limit
of $u_n$ as
\begin{equation}
u_{\Lambda} : = \lim_{n\to \infty} u_n.
\end{equation}
We will show that $u_{\Lambda}$ is in fact a solution of \eqref{ePl} with
$\lambda=\Lambda$ in the weak sense. As a first step we shall prove that
$\{x\in \Omega: u_{\Lambda}(x)=0\}$ has Lebesgue measure zero.
Let $\phi_1$ be the first eigenfunction of $-\Delta$ and $\gamma\in (0,1), \epsilon>0$.
Consider the function $\psi =(\phi_1+\epsilon)^\gamma-\epsilon^{\gamma} \in H^1_0(\Omega)$.
Then from a direct computation we find $-\Delta \psi \geq 0$ and hence
$ <-\Delta u_n$, $\psi >_{{H_0^1(\Omega)}\times H^{-1}(\Omega)} \geq 0$
 which implies
\begin{equation}
\lambda_n \int_{\Omega} (u_n^{q}-u_n^{-\beta}) \psi \geq 0.
\end{equation}
Thus
\[
 \int_{\Omega} u_n^{-\beta} ((\phi_1+\epsilon)^\gamma-\epsilon^{\gamma} )
\leq \int_{\Omega} u_n^{q} ((\phi_1+\epsilon)^\gamma-\epsilon^{\gamma}).
\]
 Now letting  $\epsilon\to 0$ and $\gamma\to 0$ we have
$\int_{\Omega} u_n^{-\beta} \leq \int_{\Omega} u_n^q
\leq \int_{\Omega} \overline{v}^q<\infty$. Once again using Fatou's lemma,
\begin{equation}
\int_{\Omega} u_{\Lambda}^{-\beta} <\infty
\end{equation}
which in turn implies $\{x\in \Omega : u_{\Lambda}(x) =0\}$ is of Lebesgue measure zero.
Now we will prove that $u_{\Lambda}$ is a weak solution of \eqref{ePl} with
$\lambda=\Lambda$. We have
$$
\int_{\Omega} \nabla u_n \nabla \varphi = \lambda_n\int_{\Omega} (u_n^{q}-u_n^{-\beta}) \varphi
\quad \text{for all } \varphi \in C_c^\infty (\Omega).
$$
The only difficulty arises while passing through the limit in the term
involving $u_n^{-\beta}$. But note that
$u_n^{-\beta} |\varphi| \leq u_{\Lambda}^{-\beta} \|\varphi\|_{\infty}\in L^1(\Omega)$
and by dominated convergence theorem $u_{\Lambda}$ is a weak solution of
\eqref{ePl} with $\lambda=\Lambda$.
\end{proof}

Next we shall discuss a sufficient condition that ensures the existence of
multiple solutions for \eqref{ePl}. We make a crucial assumption
that the non-negative solution $u_{\Lambda}$ belongs to $C_e(\overline{\Omega})$
and is bounded below by $ c  d(x,\partial\Omega)$ for some $c>0$. By the above
assumption $u_\Lambda$ is positive and it can be shown that the \eqref{ePl}
with $\lambda=\Lambda$ admits a unique positive solution.
Indeed, if $\tilde{u}_{\Lambda}$ is another positive solution of \eqref{ePl}
with $\lambda=\Lambda$ then we can show that a convex combination of
$u_{\Lambda}$ and $\tilde{u}_{\Lambda}$ is a positive solution of \eqref{ePl}
with $\lambda=\lambda'$
 for some $\lambda'<\Lambda$ which is impossible (see \cite[Proposition 5]{sauvy}
for details).  Now the uniqueness in the class of positive solutions imply
that $u_{\Lambda}$ is maximal and by Proposition \ref{semist},
$\Lambda_1(\Lambda)=0$. Indeed, since $u_{\Lambda}$ is maximal it is clear that
$\Lambda_1(\Lambda)\geq 0$. Suppose $\Lambda_1(\Lambda)>0$, then  implicit function
theorem would guarantee the existence of a positive solution for some
$\lambda<\Lambda$ which would contradict the definition of $\Lambda$.
Next we shall prove a local bifurcation result of Crandall-Rabinowitz \cite{CR}
for an infinite semipositone problem. Similar ideas of the proof were used
in \cite{Ref1,DR} when the authors studied a positone convex
 non-linearity.

\begin{lemma}\label{bending}
The solutions of $F(\lambda,u)= 0$ near $(\Lambda,u_\Lambda)$ are described
by  a curve $(\lambda(s),u(s))=(\Lambda+\tau(s),u_\Lambda +s\phi_\Lambda + x(s))$
where $s\rightarrow (\tau(s),x(s)) \in\mathbb{R} \times C_e(\overline{\Omega})$ is a
continuously differentiable function near $s=0$ with
$\tau(0)=\tau'(0) = 0$, $\tau''(0)>0$ and $ x(0)=x'(0)=0$. Moreover
$\tau$ is of class $C^2$ near $0$.
\end{lemma}

\begin{proof}
Consider the map $F(\lambda,u)$ and the Gateaux derivative of $F$ at
$(\Lambda,u_{\Lambda})$. Clearly
$\partial_{\lambda}F(\Lambda,u_{\Lambda})=-\partial_{\lambda} A(\Lambda,u_{\Lambda})=-\frac{u_{\Lambda}}{\Lambda}$.
Now consider the null space of the linear operator $\partial_u F(\Lambda, u_{\Lambda})$.
Since $\Lambda_1(\Lambda)=0$, there exists a $\phi_{\Lambda}\in H^1_0(\Omega)$ 
such that
 \begin{gather*}
-\Delta \phi_{\Lambda} = \Lambda f'(u_{\Lambda})\phi_{\Lambda} \quad \text{in } \Omega,\\
\phi_{\Lambda} = 0 \quad \text{on }\partial\Omega.
\end{gather*}
By the interior regularity results  the eigenfunction
$\phi_{\Lambda}\in C^2(\Omega)\cap H^1_0(\Omega)$ itself.
Now by \cite[Theorem 8.16]{ghergu}, the principal eigenvalue $\Lambda_1(\Lambda)$
is simple and the corresponding  eigenfunction $\phi_\Lambda$ is positive.
 Hence $\ker (\partial_u F(\Lambda,u_{\Lambda}))$ is one dimensional and is spanned by $\phi_{\Lambda}$.
We claim that $\partial_\lambda F(\Lambda,u_{\Lambda})\not \in \ker \partial_u F(\Lambda,u_{\Lambda})$.
If so, then for some constant $k$ we have $u_{\Lambda}= k \phi_{\Lambda}$.
This implies $f(u_{\Lambda})= k f'(u_{\Lambda})\phi_{\Lambda}$ which is impossible since RHS
is has a constant sign and LHS changes its sign inside $\Omega$ and hence that
$\partial_\lambda F(\Lambda,u_{\Lambda})\not \in \ker \partial_u F(\Lambda,u_{\Lambda})$.

Let $X$ be any complement of the span of $\{\phi_\Lambda\}$ in
$C_e(\overline{\Omega})$ and the map
$\theta{:} \mathbb{R}\times \mathbb{R}\times X \to C_e(\overline{\Omega})$ be defined as
$$
\theta(s,\tau, x)= F(\Lambda+\tau, u_{\Lambda}+s \phi_{\Lambda} + x)
$$
Then, we claim that $\partial_{\tau,x}\theta(0,0,0)=(\partial_\lambda
F(\Lambda,u_\Lambda),\partial_u F(\Lambda,u_\Lambda))$ is an isomorphism from
$\mathbb{R} \times X$ on to $X$.
Since $\partial_\lambda F(\Lambda,u_{\Lambda})\not \in \operatorname{Range}\partial_uF(\Lambda,u_{\Lambda})$
the map $\partial_{\tau,x}\theta(0,0,0)$ is one-one in $\mathbb{R} \times X$.
Now by Fredholm alternative $\partial_{\tau,x}\theta(0,0,0)$ is also onto.
Now by implicit function theorem there exists an $\epsilon>0$ and a
$C^2$ function $p{:}(-\epsilon,\epsilon) \to  \mathbb{R}\times X$ such that
$p(s)= (\tau(s), x(s))$ and $\theta(s, p(s))=0$, $\tau(0)=0$ and $x(0)=0$. i.e.,
$F(\Lambda+ \tau(s), u_\Lambda+ s \phi_\Lambda+ x(s))=0$. Now differentiating
with respect to $s$ variable and evaluating at $s=0$, we obtain
$$
\partial_\lambda F(\Lambda,u_{\Lambda}) \tau'(0)+ \partial_uF(\Lambda,u_{\Lambda})x'(0)=0.
$$
Since $\partial_\lambda F(\Lambda,u_{\Lambda})\not \in Range( \partial_u F(\Lambda,u_{\Lambda}))$
we have $\tau'(0)=x'(0)=0$. Once again differentiating
$F(\Lambda+ \tau(s), u_\Lambda+ s \phi_\Lambda+ x(s))$ we obtain
\begin{equation}\label{secondd}
\partial_\lambda F(\Lambda,u_{\Lambda}) \tau''(0)+ \partial_{uu} F(\Lambda,u_{\Lambda})\phi_\Lambda^2
+\partial_u F(\Lambda,u_{\Lambda})x''(0)=0.
\end{equation}
Let us write the middle term in the above expression as
$W= \partial_{uu} F(\Lambda,u_{\Lambda})\phi_\Lambda^2$.
 Then one can easily check that
 \begin{gather*}
\Delta W = \Lambda f''(u_{\Lambda}) \phi_{\Lambda}^2\quad \text{in } \Omega,\\
W = 0 \quad \text{on }\partial\Omega.
\end{gather*}
Since $f$ is concave, by maximum principle $W\geq 0$. Now
call $w=\partial_u F(\Lambda,u_{\Lambda})x''(0)$ which by definition is equal to
$ x''(0)- \partial_u A(\Lambda,u_{\Lambda})x''(0)$. If $w_1= \partial_u A(\Lambda,u_{\Lambda})x''(0)$
then $w_1$ solves
 \begin{gather*}
-\Delta w_1 = \Lambda f'(u_{\Lambda}) x''(0)\quad \text{in } \Omega,\\
w_1 = 0 \quad \text{on }\partial\Omega.
\end{gather*}
Thus $ \int_{\Omega} \nabla w_1 \nabla \phi_{\Lambda}= \int_{\Omega}  \Lambda f'(u_{\Lambda})  x''(0) \phi_{\Lambda}$.
 From the definition of $\phi_{\Lambda}$, we also have
$ \int_{\Omega} \nabla w_1 \nabla \phi_{\Lambda}= \int_{\Omega}  \Lambda f'(u_{\Lambda})  w_1 \phi_{\Lambda}$. Thus
$$
\int_{\Omega}\Lambda f'(u_\Lambda)\phi_{\Lambda} w=0.
$$
 Now multiplying \eqref{secondd} by $\Lambda f'(u_\Lambda) \phi_{\Lambda}$ and integrating over $\Omega$,
$$
-\tau''(0) \int_{\Omega} u_\Lambda f'(u_{\Lambda}) \phi_{\Lambda}
+\int_{\Omega} W \Lambda f'(u_{\Lambda}) \phi_{\Lambda} =0.
$$
We know $f$ is monotonically increasing and $\phi_{\Lambda}$ is a non-negative function
and $W\geq 0$. Thus $\tau''\geq 0$ which completes the proof.
\end{proof}

\begin{proof}[Proof of Theorem \ref{thm1.5}]
 Suppose that  alternative (a) does not hold. Then from the properties of
$C_e(\overline{\Omega})$ (see Section 3) there exists a constant
 $c_{\Lambda}>0$ such that $u_{\Lambda}(x) \geq c_{\Lambda} d(x,\partial\Omega)$.
Thus $\Lambda_1(\Lambda)$ is well defined and is non-negative.
Now by the definition of $\Lambda$ the  principal eigenvalue $\Lambda_1(\Lambda)$
cannot be positive and hence the proof of  Lemma \ref{bending} is
applicable and which completes the Theorem \ref{thm1.5}.
\end{proof}

\section{Appendix}

 \begin{proposition} \label{prop5.1}
The map $\mathcal{A} {:} \mathcal{U}_{\lambda} \to \mathcal{U}_{\lambda}$ is a $C^2$ map.
\end{proposition}

\begin{proof}
Let $u\in \mathcal{U}_{\lambda}$, i.e.\ there exists some $t_1, t_2 >0$ such that
$\psi+ t_1 e \leq  u  \leq \phi-t_2  e$ and let $\mathcal{A}(u)=w$.
Then $-\Delta(w-\phi)<0$ in $\Omega $ and $w-\phi =0$ on $\partial\Omega$,
and by Hopf Maximum principle there exists a $\tilde{t}_2>0$ for
which $w\leq \phi-\tilde{t}_2 e$. From our previous discussion \eqref{subs}
if we take $\tilde{t}_1 =\frac{\lambda-\lambda'}{\lambda}$ we find
$w\geq \psi+\tilde{t}_1 e$. Thus $\mathcal{A}$ maps $\mathcal{U}_{\lambda}$ 
into itself.
\smallskip

\noindent\textbf{Step I.}   
$\mathcal{A} {:} \mathcal{U}_{\lambda} \to \mathcal{U}_{\lambda}$ is continuous.
Let $h\in C_e(\overline{\Omega})$ with $\|h\|_{C_e(\overline{\Omega})}$ small so that $u+h\in \mathcal{U}_{\lambda}$ and
$\mathcal{A}(u+h) = w_h$. Then $(w_h-w)$ satisfies
$-\Delta(w_h-w)=  \lambda\left(f(u+h)-f(u)\right) $ in $\Omega$ and
$w_h-w= 0$ on $\partial\Omega$. For $p\in (1, \frac{1}{\beta})$ using $L^p$
estimate and dominated convergence theorem  we find
\begin{equation}\label{w2p}
\|w_h-w\|_{W^{2,p}(\Omega)} \leq C  \|f(u+h)-f(u)\|_{L^p(\Omega) } \to 0 \quad
\text{ as } \|h\|_{C_e(\overline{\Omega})}\to 0.
\end{equation}
Now since $w_h$ and $w$ belongs to $\mathcal{U}_{\lambda}$ we have $|w_h-w|\leq C d(x,\partial\Omega)$.
Now we can apply Theorem \ref{guilin} and obtain
$\|w_h-w\|_{C^{1,\gamma}(\Omega)}$ is bounded. Thanks to Ascoli-Arzela
theorem and \eqref{w2p} we have $w_h \to w$ in $C^1_0(\overline{\Omega})$.
Finally using the continuity of the embedding
$C^1_0(\overline{\Omega})\hookrightarrow C_e(\overline{\Omega})$ we conclude that
$\mathcal{A} {:} \mathcal{U}_{\lambda} \to \mathcal{U}_{\lambda}$ is continuous.
\smallskip

\noindent\textbf{Step II.}
The map $\mathcal{A}{:} \mathcal{U}_{\lambda} \to \mathcal{U}_{\lambda}$ is $C^1$.
For a given $u\in \mathcal{U}_{\lambda}$ and $h\in C_e(\overline{\Omega})$  
consider the solution operator $z$ defined as
\begin{equation}
-\Delta z = \lambda f'(u) h \text{ in } \Omega  \quad \text{and} \quad u=0
\text{ on } \partial \Omega.
\end{equation}
Let us denote $\xi_\lambda \in C^1_0(\overline{\Omega})\cap C^2(\Omega)$ be the
unique solution of
\[
-\Delta \xi_{\lambda} = \lambda \xi_{\lambda}^{-\beta} \text{ in } \Omega  \quad
 \text{and}\quad \xi_{\lambda}=0  \text{ on } \partial \Omega.
\]
The existence and  behaviour of the solution $\xi_\lambda$ near $\partial\Omega$
is studied in \cite{ctr}. It is well known that
$\xi_\lambda \sim d(x,\partial\Omega)$ and $d(x,\partial\Omega) \sim e(x)$
and thus $\xi_\lambda \sim e(x)$. We can estimate
$f'(u) h$ in terms of $ \xi_\lambda$ as
\[
|f'(u) h| \leq C_0 .e(x)^{-(\beta+1)} |h(x)|
\leq\frac{ C_1 \|h\|_{C_e(\overline{\Omega})}}{  \xi_{\lambda}^\beta}
\]
 for some positive constant $C_1$. Thus,
$$
C_1 \|h\| \Delta \xi_\lambda \leq -\Delta z =\lambda f'(u)h
\leq \lambda C_1 \|h\| \xi_{\lambda}^{-\beta} = -C_1 \|h\| \Delta \xi_\lambda,
$$
By the comparison principle and since $\xi_\lambda(x)\sim e(x)$ we have
for some $C>0$,
\begin{equation}\label{estz}
|z(x)| \leq C \|h\|_{C_e(\overline{\Omega})}  e(x)
\end{equation}
Now as in Step I, let $w_h = A(u+h)$ and $w=A(u)$, then using Taylor's theorem
$$
-\Delta(w_h-w-z) = \lambda f''(u+\theta h) \frac{h^2}{2} \quad \text{for some }
\theta(x)\in(0,1).
$$
Since $|f''(u+\theta h) h^2| \leq C \|h\|_{C_e(\overline{\Omega})}^2 e(x)^{-\beta}$ 
we have
\[
\| \frac{w_h-w-z}{\|h\|_{C_e(\overline{\Omega})}} \|_{W^{2,p}(\Omega)}
 \leq C \|h\|_{C_e(\overline{\Omega})}.
\]
Up to a sub sequence $(w_h-w-z)/ \|h\|_{C_e(\overline{\Omega})}$ converges to $0$ as
$\|h\|_{C_e(\overline{\Omega})}\to 0$. It can be shown that
$|w_h-w-z|/\|h\|_{C_e(\overline{\Omega})} \leq  C d(x,\partial\Omega)$ and thus
$ (w_h-w-z)/\|h\|_{C_e(\overline{\Omega})}$ satisfies the assumptions of theorem \ref{guilin}.
 Hence,
\begin{equation}\label{bddc1}
 \frac{w_h-w-z}{\|h\|_{C_e(\overline{\Omega})}} \text{ is bounded in } C^{1,\gamma}(\overline{\Omega})
\end{equation}
Now by using  Ascoli-Arzela theorem and continuity of the embedding
$C^1_0(\Omega)\hookrightarrow C_e(\overline{\Omega})$ we deduce that
$ \frac{w_h-w-z}{\|h\|_{C_e(\overline{\Omega})}} \to 0$ in $C_e(\overline{\Omega})$. If we call $\mathcal{A}'(u) h =z$ then
$$
\| \mathcal{A}(u+h)-\mathcal{A}(u)-\mathcal{A}'(u) h \|_{C_e(\overline{\Omega})} =o(\|h\|).
$$
Now  from \eqref{estz} we note that $\mathcal{A}'(u) {:} C_e(\overline{\Omega}) \to C_e(\overline{\Omega})$ is a
 bounded linear map and hence the map $\mathcal{A}{:} \mathcal{U}_{\lambda} \to \mathcal{U}_{\lambda}$
is differentiable. It remains to show that $\mathcal{A}$ is continuously
differentiable, i.e.\ $u \to \mathcal{A}'(u)$ is continuous.
Let $\tilde{u}\in C_e(\overline{\Omega})$ such that $\|\tilde{u}-u\|<\delta$ and
$\mathcal{A}'(\tilde{u})h =\tilde{z}$ for some $h\in C_e(\overline{\Omega})$.
Using Taylor's theorem there exists some $\theta(x) \in [u,\tilde{u}]$ and
$$
| f'(\tilde{u})-f'(u)|=\lambda |f''(\theta)| \,|(\tilde{u} -u) h|
\leq \frac{C_0 e(x)^2}{d(x)^{\beta+2}} \delta \|h\|_{C_e(\overline{\Omega})}
\leq\frac{ C_1 \delta }{\xi_1^\beta}  \|h\|_{C_e(\overline{\Omega})}
$$
where the constant $C_1$ is independent of $u$ and $\tilde{u}$.
 As before estimating $-\Delta (\tilde{z}-z)$ from above and below and
using maximum principle we have
$|\tilde{z}(x)-z(x)| \leq C \delta \|h\|  e(x)$.
Now taking supremum over $\|h\|_{C_e(\overline{\Omega})} \leq 1$ we have
$$
\|\mathcal{A}'(\tilde{u}) -\mathcal{A}'(u)\| \leq C \|\tilde{u}-u\|_{C_e(\overline{\Omega})}
 $$
and thus $\mathcal{A}$ is continuously differentiable.
\smallskip

\noindent\textbf{Step III.}
The map $\mathcal{A}$ is $C^2$.
Now that we have proved $\mathcal{A}{:}\mathcal{U}_{\lambda} \to \mathcal{U}_{\lambda}$ is $C^1$,
using the same idea we can prove that $\mathcal{A}$ is twice continuously
differentiable.In order to avoid the repetition of the same arguments
we skip the details of the proof of step III.
\end{proof}

From  \eqref{bddc1} of above proposition we know that
$\| \frac{w_h-w-z}{\|h\|}\|_{C^{1,\gamma}(\overline{\Omega})}$ is bounded
and similarly $\|\frac{w_h-w}{\|h\|}\|_{C^{1,\gamma}}$ is also bounded. So
\begin{align*}
\|\mathcal{A}'(u) h\| _{C^{1,\gamma}(\overline{\Omega})}
& = \|z\|_{C^{1,\gamma}(\overline{\Omega})}
 \leq  \|w_h-w-z\|_{C^{1,\gamma}(\overline{\Omega})}
  + \|w_h-w\|_{C^{1,\gamma}(\overline{\Omega})}\\
&= \|h\|  \| \frac{w_h-w-z}{\|h\|}\|_{C^{1,\gamma}(\overline{\Omega})}
+ \|h\|   \| \frac{w_h-w}{\|h\|}\|_{C^{1,\gamma}(\overline{\Omega})}\\
&\leq M \|h\|_{C_e(\overline{\Omega})}
\end{align*}
which implies $\mathcal{A}'(u) \in BL(C_e(\overline{\Omega}),
C^{1,\gamma}(\overline{\Omega}))$
and hence $\mathcal{A}'(u) {:} C_e(\overline{\Omega})\to C_e(\overline{\Omega})$
 is compact.

\begin{corollary}
$\mathcal{A}'(u) {:}  C_e(\overline{\Omega})  \to C_e(\overline{\Omega})$
is  continuous linear and compact.
\end{corollary}

\subsection*{Acknowledgments}
This research was supported by INSPIRE faculty fellowship
(DST/INSPIRE/04/2015/003221) at the Indian Statistical Institute,
Bangalore Centre.

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\end{document}