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

\AtBeginDocument{{\noindent\small
\emph{Electronic Journal of Differential Equations},
Vol. 2007(2007), No. 05, pp. 1--21.\newline
ISSN: 1072-6691. URL: http://ejde.math.txstate.edu or http://ejde.math.unt.edu
\newline ftp ejde.math.txstate.edu  (login: ftp)}
\thanks{\copyright 2007 Texas State University - San Marcos.}
\vspace{9mm}}

\begin{document}
\title[\hfilneg EJDE-2007/05\hfil Lane-Emden-Fowler equation]
{Multiple solutions to a singular Lane-Emden-Fowler equation with
convection term}

\author[C. C. Aranda, E. Lami D.\hfil EJDE-2007/05\hfilneg]
{Carlos C. Aranda, Enrique Lami Dozo}  % in alphabetical order

\address{Carlos C. Aranda \newline
Mathematics Department, Universidad Nacional de Formosa\\
Argentina}
\email{carloscesar.aranda@gmail.com}

\address{Enrique Lami Dozo \newline
CONICET-Universidad de Buenos Aires and Univ. Libre de Bruxelles} 
\email{lamidozo@ulb.ac.be}

\thanks{Submitted August 12, 2007. Published January 2, 2008.}
\subjclass[2000]{35J25, 35J60} 
\keywords{Bifurcation; weighted principal eigenvalues and eigenfunctions}

\begin{abstract}
 This article concerns  the existence of multiple
 solutions for the problem
 \begin{gather*}
 -\Delta u  =  K(x)u^{-\alpha}+s(\mathcal{A}u^\beta+\mathcal{B}
 |\nabla u|^\zeta)+f(x) \quad \text{in }\Omega\\
        u  >  0 \quad \text{in }\Omega\\
        u  =  0 \quad \text{on }\partial\Omega\,,
 \end{gather*}
 where $\Omega$ is a smooth, bounded domain in $\mathbb{R}^n$ with
 $n\geq 2$, $\alpha$, $\beta$, $\zeta$, $\mathcal{A}$,
 $\mathcal{B}$ and $s$ are real positive numbers, and $f(x)$ is a
 positive real valued and measurable function. We start with the
 case $s=0$ and $f=0$ by studying the structure of the range of
 $-u^\alpha\Delta u$. Our method to build $K$'s which give at least
 two solutions is based on positive and negative principal
 eigenvalues with weight. For $s$ small positive and for values
 of the parameters in finite intervals, we find
 multiplicity via estimates on the bifurcation set.
\end{abstract}

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

\section{Introduction}\label{intro}

 Singular bifurcation problems of the form
 \begin{equation}\label{maroon}
\begin{gathered}
-\Delta u  =  K(x)u^{-\alpha}+s\mathcal{G}(x,u,\nabla u)+f(x) \quad
  \text{in }\Omega \\
        u  >  0 \quad \text{in }\Omega\\
        u  =  0 \quad \text{on }\partial\Omega
\end{gathered}
\end{equation}
where $\alpha$ is a positive number, $K(x)$ is a bounded measurable function, $\mathcal{G}(x,\cdot,\cdot)$ a non-negative
Carath\'eodory function, $f(x)$ a non-negative bounded measurable function and $\Omega$ a bounded domain in $\mathbb{R}^n$, are
used in several applications. As examples, we mention: Modelling heat generation in electrical circuits \cite{fm}, fluid dynamics
\cite{cn1,cn2,lp}, magnetic fields \cite{l1}, diffusion in contained plasma \cite{l2}, quantum fluids \cite{gj}, chemical
catalysis \cite{ar,p}, boundary layer theory of viscous fluids \cite{jw}, super-diffusivity for long range Van der Waal
interactions in thin films spreading on solid surfaces \cite{deg}, laser beam propagation in gas vapors \cite{s,sz} and plasmas
\cite{ss}, exothermic reactions \cite{bgw,sw}, cellular automata and interacting particles systems with self-organized criticality
\cite{chor}, etc.

Our main concern in this paper is on the existence of multiple
solutions for the problem
\begin{equation}\label{amistades}
\begin{gathered}
-\Delta u  =
K(x)u^{-\alpha}+s(\mathcal{A}u^\beta+\mathcal{B}|\nabla
u|^\zeta)+f(x) \quad \text{in }\Omega\\
        u  >  0 \quad \text{in }\Omega\\
        u =  0 \quad \text{on }\partial\Omega\,,
\end{gathered}
\end{equation}
where $\Omega$ is a smooth, bounded domain in
$\mathbb{R}^n$ with $n\geq 2$, $\alpha$, $\beta$,
$\zeta$, $\mathcal{A}$, $\mathcal{B}$ and $s$ are
real positive numbers and $f(x)$ is a
non-negative measurable function.


We start with the case $s=0$ and $f\equiv 0$. The situation with
positive $K$ has been widely studied by several authors. For
example in \cite{ag1,crt,fm,g,lm,delp},
 under different hypothesis on $K$, they prove the
existence and unicity of solutions for equation \eqref{amistades}.
In Theorem \ref{basf}, we build a family of $K$'s, such that
problem \eqref{amistades}, with $s=0$, $f\equiv 0$ and $\alpha$
positive small enough has at least two solutions. We apply the
classical Lyapunov-Schmidt method to the map
$F:\mathcal{C}^+\to\mathcal{D}$,
\begin{equation}
F(u)=-u^\alpha\Delta u
\end{equation}
where $\mathcal{C}^+$ is defined in (\ref{banach1}, \ref{banach2})
and $\mathcal{D}$ is defined in (\ref{banach3}) to search a
bifurcation point for $F(u)$. This point will be an eigenfunction
corresponding to a negative principal eigenvalue of a linear
weighted eigenvalue problem. To prove it, we give a  Lemma
concerning the localization of the maximum value of such an
eigenfunction (see Lemma \ref{yo}). We also use a Harnack
inequality to establish a necessary estimate (see Lemma
\ref{hanson}). A final technical matter is differentiability of
$F(u)$ (Lemma \ref{francia}). To our knowledge there are no
previous similar results for
\eqref{amistades} with $s=0$ and $f\equiv 0$.

Concerning the existence of at least one solution
to (\ref{maroon}) or \eqref{amistades} we may
recall:

For $K(x)\equiv 1$, $\mathcal{A}=1$, $\mathcal{B}=0$, $f\equiv 0$,
$\alpha>0$ and $\beta>0$ in \eqref{amistades},  Coclite-G.
Palmieri   \cite{cp}
have shown that there exists $0<s^*\leq\infty$ such that this
problem \eqref{amistades} has at least one solution for all $s\in
(0,s^*)$.

Similar results for problem \eqref{amistades} can
be found in Zhang and Yu \cite{zy} under the
conditions $K(x)\equiv 1$, $\alpha>0$,
$\mathcal{A}\equiv0$, $\mathcal{B}\equiv1$,
$0<\zeta\leq 2$ and $f(x)$ equivalent to a
non-negative constant.

In a  recent work about (\ref{maroon}), Ghergu and R\u adulescu
\cite{gr} prove existence and nonexistence results for a more
general singular equation. They study
\begin{equation}\label{amistades1}
\begin{gathered}
-\Delta u  =  g(u)+\lambda|\nabla u|^\zeta+\mu f(x,u) \quad
 \text{in }\Omega\\
        u  >  0 \quad \text{in }\Omega\\
        u  =  0 \quad \text{on }\partial\Omega\,,
\end{gathered}
\end{equation}
where  $g:(0, \infty)\to(0, \infty)$ is a H\"older
continuous function which is non-increasing and
$\lim_{s\searrow 0}g(s)=\infty$. They prove in \cite[Theorem 1.4]{gr})
that for $\zeta=2$, $f\equiv1$ and fixed $\mu$, (\ref{amistades1}) has
a unique solution. Under the assumption
$\mathop{\rm lim\,sup}_{s\searrow 0}s^\alpha g(s)<+\infty$,
they also prove  existence of a
bifurcation at infinity for some $\lambda^*<\infty$.
In this article we also obtain bifurcations from infinity at $s=0$
(see  Theorems \ref{bono}
and \ref{williams}).

Concerning existence of multiple solutions for
problem \eqref{amistades},  Haitao \cite{h},
using a variational method, proves existence of
two classical solutions  under the assumptions
$K(x)\equiv1$, $0<\alpha<1<\beta\leq
\frac{N+2}{N-2}$, $\mathcal{A}=1$ $s\in (0,s^*)$
for some $s^*>0$, $\mathcal{B}\equiv0$ and
$f\equiv0$. We remark that our problem
\eqref{amistades} has not a variational structure
because of the convection  term
$\mathcal{B}|\nabla u|^\zeta$.


Aranda and Godoy \cite{ag2} proved the existence of two
weak solutions for the  problem,
involving the $p$-laplacian,
\begin{equation}\label{amistades3}
\begin{gathered}
-\Delta_p u  =  g(u)+s\mathcal{G}(u) \quad \text{in }\Omega\\
        u  >  0 \quad \text{in }\Omega\\
        u  =  0 \quad \text{on }\partial\Omega\,,
\end{gathered}
\end{equation}
where $s>0$ is small enough. This is done under the assumptions
\begin{itemize}
\item[(i)] $g:(0,\infty)\to(0,\infty)$ is a locally Lipschitz and
non-increasing function such that $\lim_{s\searrow 0}g(s)=\infty$.
\item[(ii)] $1<p\leq2$, $\mathcal{G}$ is a locally Lipschitz on $[0,\infty)$,
$\inf_{s>0}\mathcal{G}(s)/s^{p-1} >0$  and
$\lim_{s\to\infty}\mathcal{G}(s) /s^q <\infty$
for some $q\in \big(p-1,n(p-1)/(n-p)\big]$.
\item[(iii)] $\Omega$ is a bounded convex domain.
\end{itemize}

We remark that for $p=2$ and using  the change of variable
$v=e^u-1$ (see \cite{gr}), we can immediately obtain  existence of
two classical solutions of the singular problem with a particular
convection term
\begin{gather*}
-\Delta u  = \frac{g(e^u-1)}{e^u}+s\frac{\mathcal{G}(e^u-1)}{e^u}+|\nabla
u|^2 \quad \text{in }\Omega \\
        u  >  0 \quad \text{in }\Omega\\
        u  =  0 \quad \text{on }\partial\Omega\,,
\end{gather*}
for $s$ is small enough.
In comparison with this result, Theorems \ref{williams}  and
\ref{multsupliq} give results on the existence of two classical
solutions for $\zeta\neq 2$. This indicates a complex relation
between the convection term, the
function $f(x)$ and the domain $\Omega$.

For dimension $n=1$ results on multiplicity can be found, for
example, in  Agarwal and  O'Reagan
\cite{ao}.

To prove Theorems \ref{bono}, \ref{williams} and \ref{multsupliq},
we apply an ''inverse function'' strategy. We use that problem
$-\Delta u=u^{-\alpha}+f(x)$ in $\Omega$, $u=0$ on
$\partial\Omega$, $u>0$ on $\Omega$ (see Theorem 3.1 in
\cite{ag1}) has a unique solution for $f(x)\geq 0$. Moreover the
solution operator defined by $H(f):=u$ is a continuous and compact
map from $P$ into $P$, where $P$ is the positive cone in
$C^1(\overline\Omega)$ (see Lemma \ref{concorde} and Lemma
\ref{l}). Therefore, we may write the problem (\ref{maroon}) as
$u=H\big(s\mathcal{G}(x,u,\nabla u)+f(x)\big)$.

 Properties of $H$ and a classical theorem on nonlinear eigenvalue problems
stated in \cite{am}, give existence of an unbounded connected set
of solution pairs $(s,u)$, in an appropriate norm, to problem
(\ref{maroon}). Estimates on this solution set, combined with
nonexistence results, give a bifurcation from infinity at $s=0$.
We use similar ideas  to establish Theorems  \ref{williams} and
\ref{multsupliq}.

 \section{Statement of the main results}

Let us consider the weighted eigenvalue problem
\begin{equation}\label{autovalor}
\begin{gathered}
-\Delta u  =  \lambda m(x)u \quad\text{in }\Omega\\
        u  =  0 \quad\text{on }\partial\Omega\,,
\end{gathered}
\end{equation}
where $\Omega$ is a bounded domain in $\mathbb{R}^n$. Suppose
$m=m^+-m^-$ in $L^\infty(\Omega)$,
where $m^+=\max(m,0)$, $m^-=-\min(m,0)$. Denote
\[
\Omega_+=\{x\in \Omega:  m(x)> 0\}, \quad
\Omega_-=\{x\in \Omega:  m(x)< 0\}
\]
and $|\Omega_+|$, $|\Omega_-|$ its
Lebesgue measures.  It is well known (see
\cite{f} for a nice survey) that if
$|\Omega_+|>0$ and $|\Omega_-|>0$,
then (\ref{autovalor}) has a double sequence of
eigenvalues
\[
\dots\leq\lambda_{-2}<\lambda_{-1}<0<\lambda_{1}<\lambda_{2}\leq\dots,
\]
where $\lambda_1$ and $\lambda_{-1}$ are simple and the associated
eigenfunctions  $\varphi_1\in C(\overline\Omega)$,
$\varphi_{-1}\in C(\overline\Omega)$ can be taken  $\varphi_1>0$ on
$\Omega$, $\varphi_{-1}>0$ on $\Omega$.
Where $\lambda_1$ and
$\lambda_{-1}$ are the principal eigenvalues of (\ref{autovalor})
$\varphi_1$ and $\varphi_{-1}$ are the associated principal
eigenfunctions. Our first
result is as follows.


\begin{lemma}\label{yo}
Suppose $m=m^+-m^-$ in $L^\infty(\Omega)$ such that
$|\Omega^+|>0$, $|\Omega^-|>0$. Then the principal
eigenfunctions $\varphi_1>0$, $\varphi_{-1}>0$ of
(\ref{autovalor}) satisfy
\begin{equation}
\begin{gathered}
\| \varphi_{1}\|_{L^\infty(\Omega)}=\|
\varphi_{1}\|_{L^\infty(\mathop{rm supp} m^+,\; m^+dx)} \\
\| \varphi_{-1}\|_{L^\infty(\Omega)}=\|
\varphi_{-1}\|_{L^\infty(\mathop{rm supp}m^-, \;m^-dx)}
\end{gathered}
\end{equation}
where $\|\varphi_{1}\|_{L^\infty(\mathop{rm supp} m^+, \;m^+dx)}$
(respectively $\|\varphi_{-1}\|_{L^\infty(\mathop{rm supp} m^-, \;m^-dx)}$)
is the essential supremum on $\mathop{rm supp}m^+ $ with
respect to the measure $m^+dx$ (respectively on
$\mathop{rm supp}m^-$ w. r. t. $m^-dx$).
\end{lemma}

Here $\mathop{rm supp} m^+$  is the support of the
distribution $m^+$ in $\Omega$.
We take $s=0$ in (\ref{maroon}) or \eqref{amistades} and look for
multiple solutions of
\begin{equation}\label{mayonesa1}
\begin{gathered}
-u^\alpha\Delta u  =  K(x) \quad\text{in }\Omega\\
                u  =  0 \quad\text{on }\partial\Omega\,.
\end{gathered}
\end{equation}
We fix  $p>n$ and consider $K\in L^p(\Omega)$. It is shown in
\cite{ag1} that for $\alpha>0$, $0<K\in L^p(\Omega)$,
\eqref{mayonesa1} has a unique solution $u\in
W^{2,p}_{\rm loc}(\Omega)\cap C(\overline\Omega)$. On the other hand,
for $\alpha>0$ and $K<0$, we deduce from the Maximum Principle
that \eqref{mayonesa1} has no solution. Thus, if we want multiple
solutions,  $K$ should change
sign.

We give now two auxiliary results which will provide a family of
$\alpha$ and $K$'s giving multiple solutions to
\eqref{mayonesa1}
Let   $\lambda_{\pm j}((m))$ denote the eigenvalues of the
problem $-\Delta u=\lambda m(x)u$ in $\Omega$, $u=0$ on
$\partial\Omega$.

\begin{lemma}\label{cadillacs}
The function
\[
\alpha (t):=-\frac{\lambda_1((m^+-tm^-))}{\lambda_{-1}((m^+-tm^-))}
\]
is continuous on $(0,\infty)$ and satisfies
$\lim_{t\to 0^+}\alpha (t)=0$ and
$\lim_{t\to\infty}\alpha (t)=\infty$.
\end{lemma}

Our next lemma says that a weight $m$ with ``a positive and a negative bump'' gives  a bifurcation point to $F(u)$ for the proof
of Theorem \ref{basf}.

\begin{lemma}\label{hanson}
Let $y_+$, $y_-$ be fixed points of $\Omega$, let $\delta>0$ be such that the ball $B_{20\delta}\big(\frac{y_++y_-}{2}\big)$ with
radius $20\delta$ centered at $ \frac{y_++y_-}{2}$ is contained in $\Omega$, in such a way that the distance between $y_+$ and
$y_-$ is $8\delta$. If  $\varphi_{-1}$ is the principal positive eigenfunction associated to the principal negative eigenvalue
$\lambda_{-1}$ and $\varphi_1$ is the principal positive eigenfunction associated to the principal positive eigenvalue $\lambda_1$
of the problem
\begin{equation}\label{martin}
\begin{gathered}
-\Delta u  =  \lambda (m^+(x)-tm^-(x)) u \quad\text{in }\Omega\\
        u  =  0 \quad\text{on }\partial\Omega\,,
\end{gathered}
\end{equation}
where $m(x)=m^+(x)-m^-(x)\in C(\overline\Omega)$, is such that $\mathop{rm supp}m^+=\overline{B_\delta (y_+)}$, $\mathop{rm supp} m^-=
\overline{B_\delta (y_-)}$ and $m^-(x)>0$ in $B_\delta (y_-)$. Then there exists a positive constant $\epsilon(m^+,m^-)>0$
depending on $m^+$, $m^-$ such that for all $t\in (0,\epsilon(m^+,m^-))$
\begin{equation}\label{xet1}
\int_\Omega (m^+-tm^-)\varphi_{-1}^{-1}\varphi_1^3dx \neq 0\,.
\end{equation}
\end{lemma}

We give now a family of $\alpha$ and $K$ providing multiple
solutions to \eqref{mayonesa1}.


\begin{theorem}\label{basf}
Suppose  $m=m^+-m^-$ as in Lemma \ref{hanson}. For $t>0$, denote $m_t=m^+-tm^-$. Let $\lambda_1(m_t)>0$ in $\mathbb{R}$,
$\varphi_1(t)>0$ in $C(\overline\Omega)$, $\lambda_{-1}(m_t)<0$ in $\mathbb{R}$, $\varphi_{-1}(t)>0$ in $C(\overline\Omega)$, be
the principal eigenvalues and eigenfunctions of
\begin{gather*}
-\Delta u  =  \lambda m_t(x)u \quad\text{in }\Omega\\
                u  =  0 \quad\text{on }\partial\Omega\,.
\end{gather*}
Define
\[
\alpha(t)=-\frac{\lambda_1(m_t)}{\lambda_{-1}(m_t)}, \quad t>0\,.
\]
If $\alpha=\alpha(t)$ in \eqref{mayonesa1} and
\[
K=K(t,\rho)=\lambda_{-1}(m_t)m_t\varphi_{-1}(t)^{\alpha(t)+1}+\rho\varphi_{-1}(t)
\]
Then \eqref{mayonesa1} has at least two solutions for $t>0$ and
$\rho>0$ small enough.
\end{theorem}

\begin{remark} \label{rmk2} \rm
 The first term in $K$ is a negative function on $\Omega^+$, the second a
positive one.
\end{remark}

\begin{remark} \label{rmk3} \rm
For $\rho=0$, $(\alpha(t),\varphi_{-1}(t))\in\mathbb{R}^+\times C(\overline\Omega)^+$ could be a bifurcation pair for
\eqref{mayonesa1} since $u=\varphi_{-1}$ is a solution for $\alpha=\alpha(t)$ and $K=K(t,0)$.
\end{remark}

Now we consider $K(x)\equiv 1$. Hence for $s=0$, (\ref{maroon})
has a unique solution.
 Our next theorem is related to the
topological nature of this nonlinear eigenvalue problem (\ref{maroon}). Let $P$ be the positive cone in $C^1(\overline\Omega)$
with its usual norm.


\begin{theorem}\label{bono}
Suppose $0<\alpha<1/n$, $K(x)\equiv 1$, $\mathcal{G}$ is nonnegative continuous and let $f(x)$ be a non-negative bounded
measurable function. Then, the set of pairs $(s,u)$ of solutions of $(\ref{maroon})$ is unbounded in $\mathbb{R}^+\times P$.
Moreover, if $\mathcal{G}(x,\eta,\xi)\geq g_0+|\xi|^2$ where $g_0>0$ in $\mathbb{R}$. Then, we have $s\leq 2n/
\sqrt{g_0}r(\Omega)$, where $r(\Omega)$ is the inner radius of $\Omega$. As a consequence, there is bifurcation at infinity for
some $s_*<\infty$.
\end{theorem}


Recall that the inner radius of $\Omega$ is given by $\sup\{r:
B_r(x)\subset\Omega \}$.

Finally, we obtain two results dealing with multiplicity for our
singular elliptic problem \eqref{amistades} with a convection
term, as in our title.

\begin{theorem}\label{williams} Suppose that
\begin{itemize}
\item[(i)] $0<\alpha<\frac{1}{n}$, $1<\beta <\frac{n+1}{n-1}$  and
$0<\zeta<\frac{2}{n}$.
\item[(ii)] $f\in L^\infty(\Omega)$, $f>0$.
\item[(iii)] $K(x)\equiv 1$.
\item[(iv)] $\mathcal{A}=1$ and
\[
0\leq\mathcal{B}<C\big\{ \frac{\int_\Omega
f\varphi_1dx\int_\Omega\varphi_1^2dx}{\int_\Omega\varphi_1dx}
\big\}^{\beta -1}
\]
where $\varphi_1$, $\lambda_1$ are the principal
eigenfunction an principal eigenvalue of the
operator $-\Delta$
($-\Delta\varphi_1=\lambda_1\varphi_1$) with
Dirichlet boundary conditions and $C$ is a
constant depending only in $\Omega$, $\beta$,
$\lambda_1$.

\end{itemize}
 Then there exist $0<s^{**}\leq
s^*<\infty$ such that for all  $s\in (0,s^{**})$
problem \eqref{amistades} admits at least two
solutions and no solutions for $s>s^*$.
Furthermore there is bifurcation at infinity at
$s=0$.
\end{theorem}

For a particular form of $f$ and for $K$ with
indefinite sign but in a more restricted class we
have the following result.

\begin{theorem}\label{multsupliq}
Suppose that
\begin{itemize}
\item[(i)] $0<\alpha<\frac{1}{n}$, $1<\beta
<\frac{n+1}{n-1}$, and $\zeta<\frac{2}{n}$.
\item[(ii)] $f=t\varphi_1$,
$t\geq B^{\frac{1}{1+\alpha}}\big[\lambda_1
(\frac{\alpha}{\lambda_1})^{\frac{1}{1+\alpha}}+(\frac{
\lambda_1}{\alpha})^{\frac{\alpha}{1+\alpha}}\big]$.
\item[(iii)] $| K(x)|\leq B\varphi_1^{1+\alpha}(x)$.
\item[(iv)] $\mathcal{A}=1$
and $0\leq\mathcal{B}<C$ where $C$ is a constant depending only in
$\lambda_1$, $\beta$, $B$.

\end{itemize}
Then there exists $0<s^{**}\leq s^*<\infty$ such that for all
$s\in (0,s^{**})$  problem
\eqref{amistades} has at least two solutions and no solutions for
$s>s^*$. Furthermore there is bifurcation at infinity for s=0.
\end{theorem}

We remark that estimate (ii) is needed  at the end of the following
section.

\begin{figure}[ht]
\begin{center}
\setlength{\unitlength}{0.3mm}
\begin{picture}(270,210)(-40,-5)
\put(-10,5){\vector(1,0){230}}
\put(-10,5){\vector(0,1){200}}
\put(-73,200){$\|u(s)\|_{C^1(\overline\Omega)}$}
\put(217,-8){$s$}
\put(195,-8){$s^{**}$}
\put(200,2){$|$}
\thicklines{\qbezier(200,70)(10,80)(0,210)}
\thicklines{\qbezier(-10,30)(175,25)(200,30)}
\end{picture}
\end{center}
\caption{ Behaviour of the two branches near $s=0$ in Theorem 2.9 }
\end{figure}

\section{Auxiliary Results}

It is our purpose in this section to prove some preliminary results.

\begin{proof}[Proof of Lemma \ref{yo}]
We set $\gamma>2$. Then from the identity
\[
-\Delta\varphi_{-1}^{\gamma}=\gamma\lambda_{-1}(m^+-m^-)\varphi_{-1}^{\gamma}-\gamma
(\gamma-1)\varphi_{-1}^{\gamma-2}| \nabla \varphi_{-1}| ^2
\]
and using that
\[
\int_{\Omega}\Delta\varphi_{-1}^\gamma dx
 = \int_\Omega\mathop{\rm div}\nabla\varphi_{-1}^{\gamma}dx
 = \int_{\partial\Omega}\langle\nabla\varphi_{-1}^{\gamma},n\rangle dx
 = \int_{\partial\Omega}\gamma\varphi_{-1}^{\gamma-1}\langle
\nabla\varphi_{-1}^{\gamma},n\rangle dx
 =  0,
\]
where the last equality holds because  $\varphi_{-1}^{\gamma -1}=0$ on $\partial\Omega$. So
\begin{align*}
-\gamma \lambda_{-1}\int_\Omega m^-\varphi_{-1}^{\gamma }dx
& = -\gamma\lambda_{-1}\int_\Omega m^+\varphi_{-1}^{\gamma
}dx+\gamma(\gamma -1)\int_\Omega\varphi_{-1}^{\gamma-2}|
\nabla\varphi_{-1}|^2dx\\
& \geq  \gamma(\gamma -1)\int_\Omega\varphi_{-1}^{\gamma-2}|
\nabla\varphi_{-1}|^2dx,
\end{align*}
and consequently
\[
\gamma^{1/\gamma}(-\lambda_{-1})^{1/\gamma}
\Big(\int_\Omega m^-\varphi_{-1}^{\gamma}dx\Big)^{1/\gamma}
\geq \gamma^{1/\gamma}(\gamma
-1)^{1/\gamma}\Big(\int_\Omega\varphi_{-1}^{\gamma
-2}| \nabla\varphi_{-1}|^2dx\Big)^{1/\gamma}\,.
\]
Letting $\gamma\to\infty$, we find
\[
\| \varphi_{-1}\|_{L^\infty(\mathop{rm supp} m^-,
 m^-dx)}\geq\| \varphi_{-1}\|_{L^\infty(\Omega,|\nabla\varphi_{-1}|^2dx)}
\]
where $\|\varphi_{-1}\|_{L^\infty(\Omega,|\nabla\varphi_{-1}|^2dx)}
={\rm ess\, sup\,}_\Omega|\varphi_{-1}|$ is taken
with respect the measure $|\nabla
\varphi_{-1}|^2dx$. We observe that
$-\Delta\varphi_{-1}=0$ in $\Omega-\{\mathop{rm supp}m^-\cup\text{supp }m^+\}$
to conclude that the
Lebesgue's measure of thee set
$\{x\in\Omega-\{\mathop{rm supp}m^-\cup\mathop{rm supp}m^+\}  :
\nabla\varphi_{-1}(x)=0 \}$ is zero.

 From $-\Delta\varphi_{-1}<0$ in $\mathop{rm supp}m^+$, we  infer that
\[
\sup _{\mathop{rm supp}m^+}\varphi_{-1}\leq\sup_{\partial\mathop{rm supp}m^+}
\varphi_{-1}
\]
and find that
\begin{align*}
\|\varphi_{-1}\|_{L^\infty(\Omega,|\nabla\varphi_{-1}|^2dx)}
& \geq  \|\varphi_{-1}\|_{L^\infty(\Omega-\{\mathop{rm supp}m^+
\cup\mathop{rm supp}m^-\},|\nabla\varphi_{-1}|^2dx)} \\
& =  \|\varphi_{-1}\|_{L^\infty(\Omega-\{\mathop{rm supp}m^+
 \cup\mathop{rm supp}m^-\})}\\
& = \| \varphi_{-1}\|_{L^\infty(\Omega-\{\mathop{rm supp}m^-\})};
\end{align*}
hence
\[
\|\varphi_{-1}\|_{L^\infty(\mathop{rm supp}m^-, \ m^-dx)}\geq
\| \varphi_{-1}\|_{L^\infty(\Omega-\{\mathop{rm supp}m^-\})}
\]
With the aid of this last expression, we arrive to the desired conclusion.
\end{proof}

\begin{proof}[Proof of Lemma \ref{cadillacs}]
 Continuity follows from well known results (\cite{f}).
 Since $m^+-tm^-<m^+$ for all $t>0$, we conclude that
$\lambda_1((m^+-tm^-))>\lambda_1((m^+))$
(\cite{f}). Clearly
\[
\lim_{t\to\infty}\lambda_{-1}((m^+-tm^-))
=\lim_{t\to\infty}\frac{1}{t}\lambda_{-1}((\frac{m^+}{t}-m^-))=0.
\]
Then
$\lim_{t\to\infty}\alpha (t)=\infty$.
Using $m^+-tm^->-tm^-$, we deduce that
$\lambda_{-1}((m^+-tm^-))<\lambda_{-1}((-tm^-))=\frac{1}{t}\lambda_{-1}((-m^-))$
and therefore
\[
\lim_{t\to 0^+}\lambda_{-1}((m^+-tm^-))=-\infty\,.
\]
Finally, from $\lim_{t\to
0^+}\lambda_1((m^+-tm^-))=\lambda_1((m^+))$, we find
$\lim_{t\to 0^+} \alpha (t)=0$.
\end{proof}

\begin{proof}[Proof of Lemma \ref{hanson}]
To prove this lemma, we bound $t| \lambda_{-1}((m^+-tm^-))|$.
  From $m^+-tm^->-tm^-$, we deduce
$\lambda_{-1}((m^+-tm^-))<\lambda_{-1}((-tm))$ (\cite{f}) and
therefore
\[
-t\lambda_{-1}((m^+-tm^-))>-\lambda_{-1}((-m^-))>0\,.
\]
From the equation
\begin{gather*}
- \Delta\varphi_{-1}  =  \lambda_{-1}(m^+-tm^-)\varphi_{-1}
 \quad\text{in }\Omega\\
\varphi_{-1}  =  0 \quad\text{on }\partial\Omega\,,
\end{gather*}
we  see that
\begin{gather*}
- \Delta\varphi_{-1}  =  -\lambda_{-1}(tm^--m^+)\varphi_{-1}
\quad\text{in }\Omega\\
\varphi_{-1} =  0 \quad\text{on }\partial\Omega\,.
\end{gather*}
We conclude that
\[
-\lambda_{-1}((m^+-tm^-;\Omega))=\lambda_{1}((tm^--m^+;\Omega))\,.
\]
Using $\mathop{rm supp}m^-\subset\Omega$, it follows that
\[
 \lambda_{1}((tm^--m^+;\Omega))
\leq \lambda_{1}((tm^--m^+;\mathop{rm supp}m^-))
  =   \lambda_{1}((tm^-;\mathop{rm supp}m^-))
\]
Thus, we have
\begin{equation}\label{lazer}
0<-\lambda_{-1}((-m^-))<t|\lambda_{-1}((m^+-tm^-;\Omega))|
<\lambda_{1}((m^-;\mathop{rm supp}m^-))
\end{equation} Our next tool is Harnack inequality. It asserts that if $u\in W^{1,2}(\Omega)$ satisfies
\begin{gather*}
-\Delta u +mu  =  0 \quad\text{in }\Omega\\
            u  \geq  0 \quad\text{on }\Omega,
\end{gather*}
then for any ball $B_{4R}(y)\subset\Omega$, we have
\[
\sup _{B_{R}(y)}u\leq C(N)^{1+R\sqrt{\|
m\|_{L^\infty(\Omega)}}}\inf _{B_R(y)}u
\]
(see Theorem 8.20 \cite{gt}).

 Now we are ready to deal with (\ref{xet1}). We may suppose
$\|\varphi_{-1}\|_{L^\infty(\Omega)}=1$. From Harnack inequality and Lemma \ref{yo}, we find
\[
1\leq C(N)^{1+R\sqrt{t|\lambda_{-1}|}}\inf _{\mathop{rm supp}m^-}\varphi_{-1}\,.
\]
Then
\begin{equation}\label{xat}
t\int_\Omega m^-\varphi_{-1}^{-1}\varphi_1^3dx\leq
tC(N)^{1+R\sqrt{t|\lambda_{-1}|}}\int_\Omega
m^-\varphi_1^3dx\,.
\end{equation}
Assume the claim in this Lemma false, i. e.,
\[
\int_\Omega (m^+-tm^-)\varphi_{-1}^{-1}\varphi_1^3dx=0\,.
\]
Then
\begin{align*}
\int_\Omega m^+\varphi_1^3dx
& \leq  \int_\Omega m^+\varphi_{-1}^{-1}\varphi_1^3dx \\
& =  t\int_\Omega m^-\varphi_{-1}^{-1}\varphi_1^3dx \\
& \leq tC(N)^{1+R\sqrt{t|\lambda_{-1}|}}\int_\Omega
m^-\varphi_1^3dx\,.
\end{align*}
Thus
\begin{align*}
\big(\inf _{\mathop{rm supp}m^+}\varphi_1\big)^3\int_{ \mathop{rm supp}m^+}m^+dx
& \leq  tC(N)^{1+R\sqrt{t|\lambda_{-1}|}}\int_\Omega m^-\varphi_1^3dx \\
& \leq tC(N)^{1+R\sqrt{t|\lambda_{-1}|}}\big(\text{sup
}_{\mathop{rm supp}m^-}\varphi_1\big)^3
\int_{ \mathop{rm supp}m^-}m^-dx\,.
\end{align*}
Consequently,
\begin{align*}
\big(\inf _{B_{5R}(\frac{1}{2}(y_++y_-))}\varphi_1\big)^3
& \leq  tC(N)^{1+R\sqrt{t|\lambda_{-1}|}}\big(\text{sup
}_{B_{5R}(\frac{1}{2}(y_++y_-))}\varphi_1\big)^3
\frac{\int_{ \mathop{rm supp}m^-}m^-dx}{\int_{
\text{\scriptsize{supp} }m^+}m^+dx}
\end{align*}
Hence
\begin{equation}\label{mis bosques}
\frac{1}{C(N)^{(1+R\sqrt{t|\lambda_{-1}|)}+3+15R\sqrt{\max
(\lambda_1,t\lambda_1)}}}\frac{\int_{\mathop{rm supp}m^+}m^+dx}
{\int_{\mathop{rm supp} m^-}m^-dx}\leq t\,.
\end{equation}
For small $t$, using (\ref{lazer}), we deduce that (\ref{mis bosques}) is a contradiction.
\end{proof}

Recall that the vector space
\[
C(\bar{\Omega})_{e} = \{u \in C(\bar{\Omega}); -se \leq u \leq s e
\text{ for some $s
> 0$ in } \mathbb{R}\},
\]
where $e$ is the solution of $-\Delta e=1$ in $\Omega$, $e=0$ on $\partial\Omega$, endowed with the norm
\[
||u||_e = \inf \{ s > 0; -s e \leq u \leq s e\}
\]
is a Banach space \cite{am}.  We will use the Banach space
\begin{equation}\label{banach1}
\mathcal{C}= W^{2,p}(\Omega)\cap C(\overline\Omega)_e
\end{equation}
for the norm $\|\cdot\|_{\mathcal{C}}=\|\cdot\|_{W^{2,p}(\Omega)}+\|\cdot\|_e$. Hence, the cone of positive functions
\begin{equation}\label{banach2}
\mathcal{C}^+= W^{2,p}(\Omega)\cap C(\overline\Omega)_e^+
\end{equation}
 has non empty interior ${\mathaccent"7017 {\mathcal{C}}}^+$. We also need
\begin{equation}\label{banach3}
\mathcal{D}=\{f : fe^{-\alpha}\in L^p(\Omega)\}
\end{equation}
which is a Banach space for the norm
\[
\|
f\|_{\mathcal{D}}=\Big(\int_\Omega|f|^pe^{-p\alpha}dx\Big)^{1/p}
\]

Note that all principal eigenfunctions are 
in ${\mathaccent"7017 {\mathcal{C}}}^+$.

\begin{lemma}\label{francia}
The map
$F:{\mathaccent"7017 {\mathcal{C}}}^+\to
\mathcal{D}$,
\[
F(u)=-u^\alpha\Delta u,
\]
is regular and has first and second derivatives
\begin{gather*}
dF(u)v =-\alpha u^{\alpha -1}v\Delta u -u^{\alpha}\Delta v,\\
d^2F(u)[v,h]=-\alpha(\alpha -1)u^{\alpha-2}vh\Delta u-\alpha
u^{\alpha -1}v\Delta h-\alpha u^{\alpha-1}h\Delta v
\end{gather*}
\end{lemma}

\begin{proof} Consider
\begin{equation}\label{ariston}
\omega (t) =\frac{ F(u+tv)-F(u)}{t}+\alpha u^{\alpha -1}v\Delta
u+u^\alpha\Delta v
\end{equation}
To prove  Gateaux differentiability, we need to establish
\begin{equation}\label{awards}
\lim_{t\to 0}\|\omega (t)\|_{\mathcal{C}}=0
\end{equation}
 From the Mean-Value Theorem one has (at almost every $x\in\Omega$)
\begin{align*}
F(u+tv)-F(u) & =  -\int_0^1\frac{d}{d\xi}\left\{ (u+\xi
tv)^\alpha\Delta (u+\xi tv)\right\}d\xi \\
& = -t\int_0^1\left\{ \alpha(u+\xi tv)^{\alpha-1}v\Delta (u+\xi
tv)+(u+\xi tv)^\alpha\Delta v\right\}d\xi\,.
\end{align*}
Thus
\begin{equation}\label{bus}
\begin{aligned}
\|\omega (t)\|_{\mathcal{D}}
& \leq \|\int_0^1\alpha v\left\{ u^{\alpha -1}\Delta u-(u+\xi
tv)^{\alpha-1}\Delta (u+\xi tv)\right\}d\xi\|_{\mathcal{D}} \\
&\quad +  \|\int_0^1\Delta v\left\{u^\alpha-(u+\xi
tv)^\alpha\right\}d\xi\|_{\mathcal{D}}\,.
\end{aligned}
\end{equation}
Using the definition of
$\|\cdot\|_{\mathcal{D}}$, Jensen inequality
and Fubini Theorem, we obtain
\begin{align*}
\|\int_0^1\Delta v\{u^\alpha-(u+\xi tv)^\alpha\}d\xi\|_{\mathcal{D}}^p
& = \int_\Omega|\int_0^1\Delta v\{u^\alpha-(u+\xi tv)^\alpha\}
d\xi|^p\ e^{-p\alpha}dx\\
& \leq \int_0^1d\xi\int_\Omega|\Delta v\{u^\alpha-(u+\xi tv)^\alpha\}|^p
 e^{-p\alpha}dx\,.
\end{align*}
A similar estimate is valid for the second term in (\ref{bus}) and
consequently, the Lebesgue Dominated-Convergence Theorem implies
(\ref{awards}). Next we prove continuity of the map
\[
d_GF:{\mathaccent"7017 {\mathcal{C}}}^+\to L(\mathcal{C},\mathcal{D})
\]
where $L(\mathcal{C},\mathcal{D})$ is provided with the operator norm. Recall that
\[
\|d_GF(u_j)-d_GF(u)\|_{L(\mathcal{C},\mathcal{D})}
=\sup_{v\in\mathcal{C},\|v\|_{\mathcal{C}}\leq 1}\|d_GF(u_j)v-d_GF(u)v
\|_{\mathcal{D}}\,.
\]
Furthermore,
\begin{align*}
\|d_GF(u_j)v-d_GF(u)v\|_{\mathcal{D}}
& = \|-\alpha u_j^{\alpha -1}v\Delta u_j-u_j^\alpha\Delta v+\alpha
u^{\alpha -1}v\Delta u+u^\alpha\Delta
v\|_{\mathcal{D}} \\
& \leq  \|\alpha v(u^{\alpha -1}\Delta u -u_j^{\alpha -1}\Delta
u_j)\|_{\mathcal{D}}  +\|(u^\alpha
-u_j^\alpha)\Delta v\|_{\mathcal{D}} \\
&\leq  \|\alpha v\Delta u(u^{\alpha -1}-u_j^{\alpha
-1})\|_{\mathcal{D}}+\|\alpha v u_j^{\alpha
-1}(\Delta u-\Delta u_j)\|_{\mathcal{D}}\\
& \quad +\|(u^\alpha -u_j^\alpha)\Delta v\|_{\mathcal{D}}\,.
\end{align*}
If $\| u-u_j\|_{\mathcal{C}}$, that is $| u-u_j|\leq\frac{1}{j}\ e$ in
$\Omega$, we prove  now that each of
these last three terms tends to zero. From
\begin{align*}
| u(x)^{\alpha-1}-u_j(x)^{\alpha-1} |
& =  |(\alpha-1)\int_0^1(\xi u_j(x)+(1-\xi)u(x))^{\alpha-2}d\xi
(u(x)-u_j(x))| \\
& \leq  \frac{| 1-\alpha|}{j} C\ e(x)^{\alpha-1}
\end{align*}
and using $| v|\leq\varphi_{-1}$, we get
\[
\|\alpha v\Delta u(u^{\alpha -1}-u_j^{\alpha -1})\|_{\mathcal{D}}
 \leq  C\frac{\alpha| 1-\alpha|}{j}\| \ e^\alpha\Delta u\|_{\mathcal{D}}
=  C\frac{\alpha| 1-\alpha|}{j}\|\Delta u\|_{L^p(\Omega)}\,.
\]
Similarly,
\begin{gather*}
\|\alpha v u_j^{\alpha -1}(\Delta u-\Delta
u_j)\|_{\mathcal{D}} \leq C\|\Delta u-\Delta
u_j\|_{L^p(\Omega)}, \\
\|(u^\alpha -u_j^\alpha)\Delta
v\|_{\mathcal{D}}\leq C\frac{\alpha}{j}\,.
\end{gather*}
This proves continuity of the Gateaux derivative and hence $F$ is
Fr\'echet differentiable. For the second derivative we proceed
similarly.
\end{proof}

In \cite[Theorem 3.1]{ag1} it is stated that
\begin{equation}\label{duke}
\begin{gathered}
-\Delta u  =  u^{-\alpha}+f  \quad\text{in }\Omega \\
        u  =  0 \quad\text{on }\partial\Omega
\end{gathered}
\end{equation}
with non-negative $f\in L^p(\Omega)$ ($p>n$), has a unique
solution $u\in W^{2,p}_{\rm loc}(\Omega)\cap C(\overline\Omega)$.

\begin{lemma}\label{concorde}
Suppose $0<\alpha <\frac{1}{n}$. Then the solution map of problem
\eqref{duke} $f\to u$, denoted $H$ is well defined from
$\{f\in C(\overline\Omega): f(x)\geq 0\text{, $x\in\Omega$}\}$ into
$\{u\in C^1(\overline\Omega): u(x)\geq 0\text{, $x\in\Omega$},
\ u(x)=0\text{ and } \frac{\partial u} {\partial n}(x)<0
\text{, $x\in\partial\Omega$}\}$. Moreover $H$ is a continuous and compact
map.
\end{lemma}

\begin{proof}  $0<\alpha <\frac{1}{n}$ allow us to fix  $p>n$ such that
$\alpha p<1$. In the proof of this Lemma we will
use this $p$. From the proof in
\cite[Theorem 1]{ag1}, we know that $u_j=Hf_j\geq w$, where
$w$ satisfies
\begin{gather*}
-\Delta w  =  u_1^{-\alpha} \quad\text{in }\Omega\\
        w  =  0 \quad\text{on }\partial\Omega
\end{gather*}
and $u_1\in W^{2,p}(\Omega)$ is the unique solution of the problem
\begin{gather*}
-\Delta u_1  =  u_1^{-\alpha} +f_j \quad\text{in }\Omega\\
        u_1  =  1 \quad\text{on }\partial\Omega\,.
\end{gather*}
Using the Maximum Principle, we have $u_1^{-\alpha}\leq
w_1^{-\alpha}$, where $w_1$ is the solution of the problem
\begin{gather*}
-\Delta w_1  =  f_j \quad\text{in }\Omega\\
        w_1  =  1 \quad\text{on }\partial\Omega\,.
\end{gather*}
Using again the Maximum Principle we see that $u_1^{-\alpha}\leq 1$ on $x\in\overline\Omega$. We recall a Uniform Hopf Principle
as it is formulated in  Diaz-Morel-Oswald \cite{dmo}. It asserts that there exists a constant $C$, depending only on $\Omega$,
such that for all $f\geq 0$, $f\in L^1(\Omega)$, each weak solution $u$ of
\begin{equation}\label{hopf1}
\begin{gathered}
-\Delta u  =  f \quad\text{in }\Omega \\
        u  =  0 \quad\text{on }\partial\Omega
\end{gathered}
\end{equation}
satisfies
\begin{equation}\label{hopf2}
u\geq C\Big(\int_{\Omega}fe\Big)e\,.
\end{equation}
Applying this Uniform Hopf Principle, we get
\[
w(x)\geq C(\Omega)\Big( \int_\Omega u_1^{-\alpha}edx\Big)e(x)\,.
\]
Jensen inequality implies
\[
\Big( \int_\Omega u_1^{-\alpha}edx\Big)^{-\alpha}
\leq  \Big(\int_\Omega e\,dx\Big)^{\alpha-1}\Big( \int_\Omega
u_1^{\alpha^2}edx\Big)\,.
\]
As before, we have $u_1\leq w_j$ where $w_j$ is the unique
solution of
\begin{gather*}
-\Delta w_j  =  1+f_j \quad\text{in }\Omega\\
        w_j  =  1 \quad\text{on }\partial\Omega\,.
\end{gather*}
Thus
\begin{equation}\label{aerosmith}
u_j(x)^{-\alpha}\leq C(\Omega)^{-\alpha}\Big( \int_\Omega
edx\Big)^{\alpha-1}\Big( \int_\Omega
w_j^{\alpha^2}e\,dx\Big)e^{-\alpha}\,.
\end{equation}
If $f_j\to f$  in $C(\overline\Omega)$, then there exist a constant $C$, independent of $j$, such that
\[
\| u_j^{-\alpha}\|_{L^p(\Omega)}<C\,.
\]
Then $\| u_j\|_{W^{2,p}(\Omega)}<C$, so
Rellich-Kondrachov Theorem implies $u_j\to u$ strongly in
$C^1(\overline\Omega)$. Using (\ref{aerosmith}) we conclude that
$u_j^{-\alpha}\to u^{-\alpha}$ strongly in $L^p(\Omega)$,
and therefore $u$ is a solution of the problem
\begin{gather*}
-\Delta u  = u^{-\alpha}+f  \quad\text{in }\Omega \\
        u  =  0 \quad\text{on }\partial\Omega\,.
\end{gather*}
Compactness is deduced  from (\ref{aerosmith}).
\end{proof}

\begin{lemma}\label{l}
Suppose $\mathcal{L}=\Delta+c(x)$ satisfies the maximum principle
and suppose
\begin{equation}\label{3}
| K(x)|\leq B\varphi_1^{1+\alpha}(x) \quad
\text{for some $B>0$ in }  \mathbb{R},
\end{equation}
where $\varphi_1$ is the principal eigenfunction corresponding to
the principal positive eigenvalue of the problem
$-\mathcal{L}u=\lambda u$ in $\Omega$, $u=0$ on $\partial\Omega$.
If $f\in L^p(\Omega)$, $p>n$, satisfies
\[
f\geq t_0\varphi_1 \quad \text{p. p.}
\]
 where $t_0=B^{\frac{1}{1+\alpha}} \big[\lambda_{1}
(\frac{\alpha}{\lambda_{1}})^{\frac{1}{1+\alpha}}+
(\frac{\lambda_{1}}{\alpha})^{\frac{\alpha}{1+\alpha}}\big]$.
Then
\begin{equation}\label{2}
\begin{gathered}
-\mathcal{L} u+K(x)u^{-\alpha}  =  f(x)
  \quad \text{in } \Omega \\
  u  >  0   \quad \text{in }  \Omega \\
  u  =  0   \quad \text{on }  \Omega
\end{gathered}
\end{equation}
 has a strong  solution $u\in W^{2,p}(\Omega)$. Moreover,
if $f>t_0\varphi_1$ then
$u>(\frac{\alpha B}{\lambda_1})^{\frac{1}{1+\alpha}}\varphi_1$ and
it is unique within the set
$\{v>(\frac{\alpha B}{\lambda_1})^{\frac{1}{1+\alpha}}\varphi_1\}$.
If instead of $f$ we consider
$f_1>f_2\geq t\varphi_1$ in $C(\overline\Omega)$ with $t>t_0$, then
 corresponding solutions $u_1, \ u_2$ in
$\{u\in C(\overline\Omega): u\geq C(t)\varphi_1\}$ satisfy $u_1>u_2$.
\end{lemma}

\begin{proof} Let us consider, for $g\in L^{\infty}(\Omega)$,
 the solution operator
$h=(-\mathcal{L})^{-1}g $ defined by
$-\mathcal{L}h=g$ in $\Omega$, $h=0$ on $\partial\Omega$.
Then $h$ lies in $W^{2,p}( \Omega)\cap W_{0}^{1,p}
(\Omega)$ for all $ 1<p<\infty$. We define
\[
G_{C}=\{u\in C(\overline{\Omega}): u\geq C\varphi_{1} \}
\]
If $t \geq t_0$, then there exists a unique $C(t) \geq (\frac{\alpha B}{\lambda_{1}})^{\frac{1}{1+\alpha}}$ satisfying
$t=\lambda_{1}C(t)+\frac{B}{C(t)^{\alpha}}$. We prove now that for $f \in G_{t}$, $u \in G_{C(t)}$ the operator
\[
F(u)=(-\mathcal{L})^{-1}(f-Ku^{-\alpha})
\]
is well defined from $G_{C(t)}$ into $G_{C(t)}$. Moreover,
 it is continuous for the usual topology on $C(\overline{\Omega})$.
Indeed, if $u \in G_{C(t)}$ then $-Ku^{-\alpha} \geq -C(t)^{-\alpha}B\varphi_{1}$ and consequently $f-Ku^{-\alpha} \geq
\lambda_{1}C(t)\varphi_{1}$. Now positivity of $\mathcal{L}^{-1}$ implies $(-\mathcal{L})^{-1} (f-Ku^{-\alpha}) \geq
C(t)\varphi_{1}$.

To see that $F$ is a continuous map, let $(u_{n})\in
G_{C(t)}$ be a sequence such that $u_{n}\to u$ in
$C(\overline{\Omega})$ , then $K(x)u_{n}(x)^{-\alpha} \to
K(x)u(x)^{-\alpha}$,
pointwise on $\Omega$. Since $| K(x)u_{n}^{-\alpha}
(x)|\leq C(t)^{-\alpha}B\varphi_{1}(x)$, Lebesgue's Dominated
Convergence Theorem gives $f-Ku_{n}^{-\alpha}\to
f-Ku^{-\alpha}$ in $ L^{p}(\Omega)$, $ 1<p<\infty$.
Then the classical $L^{p}$ theory for elliptic operators implies
\[
(-\mathcal{L})^{-1}(f-Ku_{n}^{-\alpha})
\to(-\mathcal{L})^{-1}(f-Ku^{-\alpha})
 \]
in $W^{2,p}(\Omega)$ for all $ 1<p<\infty$ and then
$F(u_{n})\to F(u)$ in $C(\overline{\Omega})$. Moreover
$\overline{F(G_{C(t)})}$ is a compact set in
$C(\overline{\Omega})$. In fact, we have
\[
\|(-\mathcal{L})^{-1} (f-Ku^{-\alpha})
\|_{W^{2,p}(\Omega)} \leq
C_{0}\|f-Ku^{-\alpha}\|_{L^{p}(\Omega)}\leq C,
\]
for all $u\in G_{C(t)}, \ 1<p<\infty$, then it is clear that
$\overline{F(G_{C})}$ is compact in $C(\overline{\Omega})$. Since
$G_{C(t)}$ is a convex closed set, Schauder Fixed Point Theorem
provides a fixed point for $F$ in $G_{C(t)}$, so a solution to
(\ref{2}).


Suppose now that  for $f \in G_{t}$ there exist two different
solutions, $u$ and $v$ of (\ref{2}), then
\begin{align*}
-\mathcal{L}(u-v) &=  -K(u^{-\alpha}-v^{-\alpha})\\
 &= \alpha K ( \int_0^1 (ru+(1-r)v)^{-\alpha-1}\, dr)(u-v).
\end{align*}
We define $m=K\int_0^1 (ru+(1-r)v)^{-\alpha-1}\, dr$. Thus,
 we can write, recalling that $\mathcal{L}=\Delta +c(x)$,
\begin{gather*}
\Delta(u-v)+(c+\alpha m)(u-v) =0 \quad \text{in }\Omega \\
u-v= 0 \quad \text{on }  \partial\Omega\,.
\end{gather*}
Since $u\not\equiv v$ we may suppose $u-v$ is positive somewhere
in $\Omega$. Now, \cite[Corollary 1.1]{bnv} implies that the
principal eigenvalue $\lambda_1((\Delta+c+\alpha m))$
of the problem
\begin{gather*}
\Delta h+(c+\alpha m)h =\lambda h \quad \text{in } \Omega \\
 h= 0 \quad \text{on } \partial\Omega,
\end{gather*}
is a nonpositive number. We recall Lipschitz continuity of this
eigenvalue with respect to $L^{\infty}$-norm of the coefficient
function $c+\alpha m$ (see for example \cite[Proposition 2.1]{bnv})
and the estimate $| m|\leq BC(t)^{-1-\alpha}$ to infer that
\[
|\lambda_1((\Delta+c+\alpha m))
-\lambda_1((\Delta+c))|\leq \|c+\alpha m-c\|_{L^{\infty}(\Omega)}
\leq\frac{\alpha B}{C(t)^{1+\alpha}}
\]
Considering the choice of $C(t)$, we find
\[
0<\lambda_{1}-\frac{\alpha B}{C(t)^{1+\alpha}}
\leq\lambda_1((\Delta+c+\alpha m)),
\]
and this is a contradiction.

If $u_{1}\not> u_{2}$ in our last assertion, then there exists
$x_{0}\in\Omega$ such that $u_{2}(x_{0})\geq u_{1}(x_{0})$, and
$u_2-u_1$ is a nontrivial solution of
\begin{gather*}
\mathcal{L}(u_{2}-u_{1})+\alpha \tilde{m} (u_{2}-u_{1})
\geq 0 \quad \text{in } \Omega \\
 u_{2}-u_{1}=0 \quad \text{on } \partial\Omega,
\end{gather*}
where $\tilde{m}$ is similar to $m$. From
\cite[Corollary 1.1]{bnv} we obtain
$\lambda_1((\Delta+c+\alpha \tilde{m}))\leq 0$ and this is a contradiction,
because $0\leq \tilde{m}\leq BC(t)^{-1-\alpha}$ and as before,
we have $\lambda_1((\Delta +c+\alpha \tilde{m}))> 0$.
\end{proof}


\begin{remark} \label{rmk7} \rm
When $\mathcal{L}=\Delta$, $t_0$ is sharp under condition (\ref{3})
for $K=B\varphi_1^{1+\alpha}$ and $f\in\{t\varphi_1:t>0\}$. Indeed
\begin{gather*}
-\Delta u+B\varphi_1^{1+\alpha}u^{-\alpha}
 =  t\varphi_1 \quad\text{in }\Omega \\
u  =  0 \quad\text{on }\partial\Omega
\end{gather*}
implies
\[
t_0\int_\Omega\varphi_1^2dx\leq\int_\Omega
\Big(\lambda_1\frac{u}{\varphi_1}
+B(\frac{u}{\varphi_1})^{-\alpha}\Big)\varphi_1^2dx
=t\int_\Omega\varphi_1^2dx.
\]
\end{remark}


\section{Proofs}

\begin{proof}[Proof of Theorem \ref{basf}]
 Consider the map $F :{\mathaccent"7017 {\mathcal{C}}}^+\to
\mathcal{D}$ given by $F(u)=-u^\alpha\Delta u$. According
to Lemma \ref{francia}, $dF(u)v=0$ if and only if $v$ satisfies
\begin{equation}\label{boca}
\begin{gathered}
-\Delta v   =  \alpha \frac{\Delta u}{u}v \quad\text{in }\Omega\\
 v  =  0 \quad\text{on }\partial\Omega\,.
\end{gathered}
\end{equation}
Suppose $m$ is as in Lemma \ref{yo} and consider the eigenvalue
problem
\begin{gather*}
-\Delta u   =  \lambda mu \quad\text{in }\Omega \\
 u =  0 \quad\text{on }\partial\Omega\,.
\end{gather*}
At $u=\varphi_{-1}$ and for $\alpha =
-\frac{\lambda_1}{\lambda_{-1}}$ in (\ref{boca}),
$dF(\varphi_{-1})v=0$ is equivalent to
\begin{equation}\label{boca12}
\begin{gathered}
-\Delta v   =  \lambda_1mv \quad\text{in }\Omega \\
 v  =  0 \quad\text{on }\partial\Omega
\end{gathered}
\end{equation}
which implies $\ker  dF(\varphi_{-1})=\langle
\varphi_1\rangle$. The equation $dF(\varphi_{-1})v=f$ is
equivalent to
\begin{equation}\label{fredholm}
\begin{gathered}
-\Delta v   =  \lambda_1mv+\varphi_{-1}^{-\alpha}f \quad\text{in }\Omega\\
        v   =  0  \quad\text{on }\partial\Omega
\end{gathered}
\end{equation}
By hypothesis $f\varphi_{-1}^{-\alpha}\in L^p(\Omega)$ with $p>n$,
hence the Fredholm alternative yields that (\ref{fredholm}) has a
solution $v\in H^{1,2}_0(\Omega)$ if and only if $\int_\Omega
\varphi_{-1}^{-\alpha}f\varphi_1dx=0$. If we have a solution $v$
since $m\in L^\infty(\Omega)$ a Brezis-Kato result (see for
example  Struwe appendix B [14]) implies that $v\in
\mathcal{C}$.

We want to solve the equation
\begin{equation}\label{ecu1}
F(\varphi_{-1}+\widehat{v})=F(\varphi_{-1})+\rho\varphi_{-1}
\end{equation}
Inserting Taylor formula in (\ref{ecu1}),
\[
F(\varphi_{-1}+\widehat{v})=F(\varphi_{-1})+dF(\varphi_{-1})\widehat{v}+\Psi
(\widehat{v})
\]
we find
\begin{equation}\label{pintura}
dF(\varphi_{-1})\widehat{v}+\Psi(\widehat{v})=\rho\varphi_{-1}
\end{equation}
We use now the well known Lyapunov-Schmidt method.  First we
denote
\begin{gather*}
\langle \varphi_{-1}^{-\alpha}\varphi_1\rangle^\perp_{\mathcal{C}}
=\{w\in\mathcal{C}:\int_\Omega
w\varphi_{-1}^{-\alpha}\varphi_1dx=0\}, \\
\langle \varphi_{-1}^{-\alpha}\varphi_1\rangle^\perp_{\mathcal{D}}
=\{w\in\mathcal{D}:\int_\Omega
w\varphi_{-1}^{-\alpha}\varphi_1dx=0\}\,.
\end{gather*}
Observe that $\int_\Omega
\varphi_{-1}\varphi_{-1}^{-\alpha}\varphi_1dx\neq  0$, thus we
have the decompositions as direct sums
\[
\mathcal{C}=\langle\varphi_{-1}\rangle\oplus\langle
\varphi_{-1}^{-\alpha}\varphi_1\rangle^\perp_{\mathcal{C}}, \quad
\mathcal{D}=\langle\varphi_{-1}\rangle\oplus\langle
\varphi_{-1}^{-\alpha}\varphi_1\rangle^\perp_{\mathcal{D}}
\]
and consequently if  $\widehat{v}\in\mathcal{D}$, we get
the unique decomposition
\[
\widehat{v}=\widehat{s}\varphi_{-1}+w
\]
with $w\in \langle
\varphi_{-1}^{-\alpha}\varphi_1\rangle^\perp_{\mathcal{D}}$.
Let us denote
\[
P:\mathcal{D}\to\langle\varphi_{-1}\rangle ,\quad
Q:\mathcal{D}\to\langle
\varphi_{-1}^{-\alpha}\varphi_1\rangle^\perp_{\mathcal{D}}
\]
linear operators such that $P\widehat{v}=\widehat{s}\varphi_{-1}$
and $Q\widehat{v}=w$. We can replace (\ref{pintura}) by the
equivalent system
\begin{gather}\label{ecu2}
QdF(\varphi_{-1})\widehat{v}+Q\Psi(\widehat{v})=0,\\
\label{ecu3}
P\Psi(\widehat{v})=\rho\varphi_{-1}\,.
\end{gather}
To solve (\ref{ecu2}), we define the function
\begin{gather*}
\Gamma : \mathbb{R}\times
\langle\varphi_{-1}^{-\alpha}\varphi_1\rangle^\perp_{\mathcal{C}}
\to\langle\varphi_{-1}^{-\alpha}\varphi_1\rangle^\perp_{\mathcal{D}}, \\
\Gamma (\widehat{s},w)=
QdF(\varphi_{-1})(\widehat{s}\varphi_{-1}+w)+Q\Psi(\widehat{s}
\varphi_{-1}+w)\,.
\end{gather*}
This function satisfies
\begin{gather}\label{ecu4}
\Gamma(0,0)=0, \\
\label{ecu5}
d_w\Gamma(0,0)w_0=QdF(\varphi_{-1})w_0, \\
\label{ecu6}
d_{\widehat{s}}\Gamma(0,0)=QdF(\varphi_{-1})\varphi_{-1}\,.
\end{gather}
The operator $d_w\Gamma(0,0)$ has inverse from
$\langle\varphi_{-1}^{-\alpha}\varphi_1\rangle^\perp_{\mathcal{C}}$
to
$\langle\varphi_{-1}^{-\alpha}\varphi_1\rangle^\perp_{\mathcal{D}}$.
The Implicit Function Theorem applies to $\Gamma$: there exist an
interval $(-s^*,s^*)$ and a function
\[
W:(-s^*,s^*)\to\langle\varphi_{-1}^{-\alpha}\varphi_1\rangle^\perp_{\mathcal{C}}
\]
 such that
$\widehat{v}=s\varphi_{-1}+W(s)$ solves  (\ref{ecu2}), with
\[
W(0)=0 \quad\text{and}\quad
W'(0)=-[QdF(\varphi_{-1})]^{-1}QdF(\varphi_{-1})\varphi_{-1}\,.
\]
Using  $\mathop{\rm Im}dF(\varphi_{-1})=\langle
\varphi_{-1}^{-\alpha}\varphi_1\rangle^\perp_{\mathcal{D}}$
and $W'(0)\in\langle
\varphi_{-1}^{-\alpha}\varphi_1\rangle^\perp_{\mathcal{C}}$,
we conclude
\[
dF(\varphi_{-1})W'(0)=-dF(\varphi_{-1})\varphi_{-1}\,.
\]
Hence $W'(0)+\varphi_{-1}\in \text{Ker} dF(\varphi_{-1})
=\langle\varphi_1\rangle$. Thus
\begin{equation}\label{quemado}
W'(0)=r\varphi_1-\varphi_{-1}
\end{equation}
with $r\neq 0$ because $\varphi_{-1}\not\in \langle
\varphi_{-1}^{\alpha}\varphi_1\rangle^\perp$. From (\ref{ecu3}),
we find
\[
\rho= \int_\Omega
\varphi_{-1}P\Psi(s\varphi_{-1}+W(s))dx
=\langle\varphi_{-1},P\Psi(s\varphi_{-1}+W(s))\rangle\,.
\]
The function
\[
\chi(s)=\langle\varphi_{-1},P\Psi(s\varphi_{-1}+W(s))\rangle
\]
is regular and has first and second derivatives given by
\[
\chi'(s)=\langle\varphi_{-1},Pd\Psi(s\varphi_{-1}+W(s))[\varphi_{-1}
+W'(s)]\rangle\,,
\]
\begin{align*}
\chi''(s) & =
\langle\varphi_{-1},Pd^2\Psi(s\varphi_{-1}+W(s))[\varphi_{-1}
+W'(s),\varphi_{-1}+W'(s)]\rangle \\
&\quad +\langle\varphi_{-1},Pd\Psi(s\varphi_{-1}+W(s))[W''(s)]\rangle\,.
\end{align*}
 From  $d\Psi (0)=0$ and  $d^2\Psi(0)=d^2F(\varphi_{-1})$, we
obtain
\begin{gather*}
\chi '(0)=0, \\
\chi''(0)=\langle \varphi_{-1},Pd^2F(\varphi_{-1})
[r\varphi_1,r\varphi_1]\rangle\,.
\end{gather*}
Direct calculations show that
\[
d^2F(\varphi_{-1})[\varphi_1,\varphi_1]  =
\lambda_1(1-\frac{\lambda_1}{\lambda_{-1}})
\varphi_{-1}^{\alpha-1}\varphi_1^2m\,.
\]
Using the decomposition
$d^2F(\varphi_{-1})[r\varphi,r\varphi]=s\varphi_{-1}+w $ with
$w\in \langle \varphi_{-1}^{-\alpha}\varphi_1\rangle^\perp_{\mathcal{D}}$,
we find
\[
s=r^2\lambda_1(1-\frac{\lambda_1}{\lambda_{-1}})\frac{\int_\Omega
m\varphi_{-1}^{-1}\varphi_1^3
dx}{\int_\Omega\varphi_{-1}^{1-\alpha}\varphi_1dx}\,.
\]
Then $\chi''(0)\not =0$ is equivalent to
\begin{equation}\label{xet}
\int_\Omega m\varphi_{-1}^{-1}\varphi_1^3dx\not=0\,.
\end{equation}
If (\ref{xet}) is true, then there exist an nonempty open interval such that the equation (\ref{ecu3}) has at least two solutions.
Lemma \ref{hanson} states the existence of a class $m$'s satisfying (\ref{xet}).
 \end{proof}



\begin{proof}[Proof of Theorem \ref{bono}]
 From Lemma \ref{concorde} the operator
\[
F(s,u):=H(s\mathcal{G}(x,u,\nabla u)+f)
\]
is well defined and is continuous, compact from  $\mathbb{R}_{\geq
0}\times P^+$ to $P$ where $P$ is the cone of positive functions
in $C^1(\overline\Omega)$ with the usual norm. Furthermore a
solution $v$ of the equation
\begin{equation}\label{cd251}
F(s,v+u_*)-u_*=v
\end{equation}
where $u_*$ is the unique solution of the problem
\begin{equation}\label{duke21}
\begin{gathered}
-\Delta u_*  =  u_*^{-\alpha}+f  \quad\text{in }\Omega \\
        u_*  =  0 \quad\text{on }\partial\Omega
\end{gathered}
\end{equation}
satisfies the equation
\begin{equation}
\begin{gathered}
-\Delta (v+u_*)  =  (v+u_*)^{-\alpha}+ s\mathcal{G}(x,v+u_*,
\nabla( v+u_*))+f \quad\text{in }\Omega\\
        v+u_*  >  0 \quad\text{in }\Omega\\
        v+u_*  =  0 \quad\text{on }\partial\Omega\,.
\end{gathered}
\end{equation}
The operator
$T(s,v):=F(s,v+u_*)-u_*$
is well defined from  $\mathbb{R}_{\geq 0}\times P$ to $P$ and is a
continuous compact operator, moreover $T(0,0)=0$ and since
$T(0,v)=0$ for all $v\in P\cup \{0\}$, $v=0$ is the unique fixed
point of $T(0,\cdot)$. For each $\sigma\geq1$ and $\rho>0$, we
have also that $T(0,v)\not =\sigma v$ for $v\in P\cap\rho\partial
B$ where $B$ denotes the open unit ball centered at $0$ in
$C^1(\overline\Omega)$. Using Theorem 17.1 in Amman's article
\cite{am} there exist a nonempty  set $\Sigma$ of pairs $(s,v)$ in
$\mathbb{R}_{\geq 0}\times P$ that solves the equation
(\ref{cd25}). Moreover $\Sigma$ is a closed, connected and
unbounded subset of $\mathbb{R}_{\geq 0}\times P$ containing
$(0,0)$. The nonexistence Corollary 1.1 in \cite{z} implies the
last affirmation.
\end{proof}

\begin{proof}[Proof of Theorem \ref{williams}]
We start as in the proof of Theorem \ref{bono}. Hence, from Lemma
\ref{concorde}, the operator
\[
F(s,u):=H(s(\mathcal{A}u^\beta+\mathcal{B}|\nabla u|^\zeta)+f)
\]
is well defined, continuous and  compact from
$\mathbb{R}_{\geq 0}\times P^+$ to $P$ where $P$ is the cone of
positive functions in $C^1(\overline\Omega)$ with the usual norm.
We study the fixed point equation
\begin{equation}\label{cd25}
F(s,v+u_*)-u_*=v
\end{equation}
where $u_*$ is the unique solution of
\begin{equation}\label{duke2}
\begin{gathered}
-\Delta u_*  =  u_*^{-\alpha}+f  \quad\text{in }\Omega \\
        u_*  =  0 \quad\text{on }\partial\Omega\,.
\end{gathered}
\end{equation}
Moreover if $v$ is a solution of (\ref{cd25}), $v+u_*$ is a
solution of problem \eqref{amistades}. Using
Amman's article \cite[Theorem 17.1]{am}, we obtain the existence
of a nonempty, closed, connected and unbounded set $\Sigma$ of
pairs $(s,v)$ in
$\mathbb{R}_{\geq 0}\times P$ that solves (\ref{cd25}).

To prove existence of two solutions we obtain a
constant $C_1$ and a estimate $C(\delta)>0$ for
$\delta>0$ such that:
\begin{itemize}
\item[(a)] If $(s,u)$ solves equation
\eqref{amistades} then $s\leq C_1$.
\item[(b)] If $(s,u)$ solves \eqref{amistades} then
$\|u\|_{L^\infty(\Omega)}\leq C(\delta)$ for
  all $s\geq\delta$.

\end{itemize}
Using that $\Sigma$ is unbounded,  the conclusion
of Theorem \ref{williams} follows.

First we prove (a).
The function $Q(u)=\lambda_1\beta u-su^{\beta}$ where  and
$1<\beta <\infty$, has a global maximum on the set of positive
real numbers at
$u=(\frac{\lambda_1}{s})^{\frac{1}{\beta -1}}$,
furthermore
\[
Q\big((\frac{\lambda_1}{s})^{\frac{1}{\beta
-1}}\big)=C(\beta,\lambda_1)s^{-\frac{1}{\beta-1}}
\]
where $C(\beta,\lambda_1)$ is a strictly positive constant
depending only on $\beta$ and $\lambda_1$. From the inequality
\[
\lambda_1\beta u-su^\beta\leq
C(\beta,\lambda_1)s^{-\frac{1}{\beta-1}}\,.
\]
Using  equation \eqref{amistades}, we deduce
\[
-\Delta u\geq\lambda_1\beta
u-C(\beta,\lambda_1)s^{-\frac{1}{\beta-1}}
\]
and therefore
\[
\lambda_1\int_{\Omega}u\varphi_1dx\geq\lambda_1\beta
\int_{\Omega}u\varphi_1dx-C(\beta,\lambda_1)
s^{-\frac{1}{\beta-1}}\int_{\Omega}\varphi_1dx\,.
\]
Finally
\begin{equation}\label{secondary12}
\int_\Omega u\varphi_1dx\leq
\frac{C(\beta,\lambda_1)s^{-\frac{1}{\beta-1}}}{\lambda_1(\beta-1)}\int_\Omega
\varphi_1dx\,.
\end{equation}
 From \eqref{amistades}, we have
$-\Delta u\geq f$.
 Using the Uniform Hopf Principle (\ref{hopf1}), (\ref{hopf2}) and
(\ref{secondary12}), it follows that
\begin{equation}\label{isat}
s\leq \big\{ \frac{C(\beta,\lambda_1)\int_\Omega\varphi_1dx}{
\lambda_1(\beta-1)C(\Omega)\int_\Omega
f\varphi_1dx\int_\Omega\varphi_1^2dx} \big\}^{\beta -1}
\end{equation}
This is the constant $C_1$ and (a) is proved.

Now we prove (b).
We establish a priori bounds for solutions of  problem
\eqref{amistades} using a Brezis-Turner technique (see \cite{bt}).
Multiplying \eqref{amistades} by $\varphi_1$ and integrating, we
find
\[
\lambda_1\int_\Omega u\varphi_1dx= s\int_\Omega
u^\beta\varphi_1dx+s\mathcal{B}\int_\Omega |\nabla
u|^\zeta\varphi_1dx+\int_\Omega u^{-\alpha}\varphi_1dx
+\int_\Omega f\varphi_1dx\,.
\]
 From (\ref{secondary12}) it follows that
\begin{equation}\label{gun}
s\int_\Omega u^\beta\varphi_1dx\leq \frac{\lambda_1
C(\beta,\lambda_1)s^{-\frac{1}{\beta-1}}}{\lambda_1(\beta-1)}\int_\Omega
\varphi_1dx\,.
\end{equation}
Using the hypothesis $\zeta<\frac{2}{n}$ and Young inequality, we
obtain a $q\geq 1$ such that $0<\zeta q\leq 2$,
$\frac{1}{q}+\frac{1}{\vartheta+1}=1$,
$0\leq\vartheta<\frac{n+1}{n-1}$ and
\begin{equation}\label{convection}
|\nabla u|^\zeta u
 \leq  \frac{|\nabla u|^{\zeta q}}{q}+\frac{u^{\vartheta+1}}{\vartheta+1}
 \leq  |\nabla u|^2+1+u^{\vartheta}u\,.
\end{equation}
Using the assumption
\[
\mathcal{B}<\big\{ \frac{\lambda_1(\beta-1)C(\Omega)\int_\Omega
f\varphi_1dx\int_\Omega\varphi_1^2dx}{C(\beta,\lambda_1)\int_\Omega\varphi_1dx}
\big\}^{\beta -1},
\]
inequalities (\ref{isat}), (\ref{convection}), and multiplying
\eqref{amistades} by $u$ and then integrating, we find
\begin{equation}\label{james12}
C_1\int_\Omega|\nabla u|^2dx\leq s\int_\Omega u^\beta
u\,dx+sC_2\int_\Omega u^\vartheta u\,dx+ C_3\|
u\|_{H^1_0(\Omega)}+C_4\,,
\end{equation}
where $C_i$ for $i=1,\dots 4$ are positive constants independent of
$s$. Using  H\"{o}lder inequality, (\ref{gun}) and the fact that
if $1<\beta<\frac{n+1}{n-1}$ then for all $\epsilon >0$ there
exist a positive constant $C_\epsilon$ such that for all $s>0$
holds $s^\beta\leq \epsilon s^{\frac{n+1}{n-1}}+C_\epsilon$, we
deduce
\begin{align*}
\int_\Omega u^\beta u\,dx
& =  \int_\Omega u^{\gamma\beta}\varphi_1^\gamma u^{(1-\gamma)
 \beta}\varphi_1^{-\gamma}u \,dx\\
& \leq \Big(\int_\Omega u^\beta\varphi_1 dx\Big)^\gamma
 \Big(\int_\Omega u^\beta\varphi_1^{\frac{-\gamma}{1-\gamma}}
 u^{\frac{1}{1-\gamma}}dx\Big)^{1-\gamma}\\
& \leq  \big(Cs^{-1-\frac{1}{\beta-1}}\big)^\gamma
 \Big(\int_\Omega u^\beta(\frac{u}{\varphi_1^\gamma})
 ^{\frac{1}{1-\gamma}}dx\Big)^{1-\gamma}\\
& \leq Cs^{-\gamma-\frac{\gamma}{\beta-1}}\Big\{\epsilon^{1-\gamma}
\Big( \int_\Omega \frac{ u^{\frac{n+1}{n-1}+\frac{1}{1-\gamma} }}
{ \varphi_1^{ \frac{\gamma}{1-\gamma}} } dx \Big)^{1-\gamma}\\
&\quad +C_\epsilon^{1-\gamma}\Big(\int_\Omega(\frac{u}{\varphi_1^\gamma})
^{\frac{1}{1-\gamma}}dx\Big)^{1-\gamma}\Big\}\,.
\end{align*}
For $\gamma =2/(n+1)$, we find
\begin{align*}
\int_\Omega u^\beta u\,dx
& \leq Cs^{-\gamma-\frac{\gamma}{\beta-1}}\epsilon^{1-\gamma}
\Big(\int_\Omega \big(\frac{ u}{\varphi_1^{1/(n+1)}
}\big)^{2\frac{n+1}{n-1}} dx \Big)^{\frac{n-1}{2(n+1)}2}\\
& \quad + Cs^{-\gamma-\frac{\gamma}{\beta-1}}C_\epsilon^{1-\gamma}
\Big(\int_\Omega\big(\frac{u}{\varphi_1^{2/(n+1)}}\big)^{\frac{n+1}{n-1}}dx
\Big)^{\frac{n-1}{n+1}}\,.
\end{align*}
Since
\[
\frac{1}{2\frac{n+1}{n-1}}=\frac{1}{2}-\frac{1}{n}+\frac{\frac{1}{n+1}}{n},
\quad
\frac{1}{q}=\frac{1}{2}-\frac{1}{n}+\frac{\frac{2}{n+1}}{n},
\]
with $q>\frac{n+1}{n-1}$, we apply  Hardy-Sobolev inequality
in \cite[Lemma 2.2]{bt},
\[
\|\frac{v}{\varphi_1^\tau}\|_{L^q(\Omega)}
\leq C\| v\|_{H^1_0(\Omega)}\quad \text{for all $v$ in }H^1_0(\Omega)
\]
where $C$ is a non-negative constant,
$0\leq\tau\leq 1$,
$\frac{1}{q}=\frac{1}{2}-\frac{1}{n}+\frac{\tau}{n}$,
$\varphi_1$ is the principal eigenfunction of the
operator $-\Delta$
($-\Delta\varphi_1=\lambda_1\varphi_1$) with
Dirichlet boundary condition, and the H\"{o}lder
inequality to obtain
\[
\int_\Omega u^\beta u\,dx  \leq
Cs^{-\gamma-\frac{\gamma}{\beta-1}}\big\{\epsilon^{1-\gamma}\|\nabla
u\|_{L^2(\Omega)}^2+C_\epsilon^{1-\gamma}\|\nabla
u\|_{L^2(\Omega)}\big\}\,.
\]
 From (\ref{james12}), we  conclude  that
\begin{eqnarray}\label{tasi}
C_1\| \nabla u\|_{L^2(\Omega)}^2 & \leq & C
 s^{1-\gamma-\frac{\gamma}{\beta
-1}}\left\{\epsilon^{1-\gamma}\| \nabla
u\|_{L^2(\Omega)}^2+C_\epsilon^{1-\gamma}\| \nabla
u\|_{L^2(\Omega)}\right\}\nonumber \\ &  &  + C\|
\nabla u\|_{L^2(\Omega)}+C(\delta)\,,
\end{eqnarray}
where $C$ is a non-negative constant independent of $s$. The
condition $\beta <\frac{n+1}{n-1}$ implies
\[
1-\gamma-\frac{\gamma}{\beta
-1}=\frac{n-1}{n+1}-\frac{2}{(n+1)(\beta -1)} < 0\,.
\]
Therefore if $s\geq\delta$, we can choose $\epsilon>0$ such that
\[
C s^{1-\gamma-\frac{\gamma}{\beta -1}}\epsilon^{1-\gamma}\leq
\frac{C_1}{2}\,.
\]
It now follows from (\ref{tasi}) that
\begin{equation}\label{tasi12}
\frac{C_1}{2}\| \nabla u\|_{L^2(\Omega)}^2 \leq
C\{1+C_\epsilon^{1-\gamma}
 s^{1-\gamma-\frac{\gamma}{\beta -1}}\}\| \nabla
u\|_{L^2(\Omega)} +C(\delta)\,.
\end{equation}
Finally if $u$ is a solution of the problem \eqref{amistades} with
$s>\delta>0$, there exists a constant $C(\delta)>0$ such that
$\| u\|_{H_0^{1,2}(\Omega)}<C(\delta)$ and using
classical H\"{o}lder estimates for weak solutions (see \cite{gt})
and Sobolev imbedding theorem  we conclude the proof of (b).
The proof is complete.
 \end{proof}

\begin{proof}[Proof of Theorem \ref{multsupliq}]
 From Lemma \ref{l}, the problem
\begin{gather*}
-\Delta u =  K(x)u^{-\alpha} + f \quad\text{in }\Omega \\
  u  =  0 \quad\text{on }\partial\Omega
\end{gather*}
under the conditions $| K(x)| \leq B\varphi_1^{1+\alpha}(x)$
for some $B>0$ in $\mathbb{R}$, $f>t_0\varphi_1$ where
$t_0=B^{\frac{1}{1+\alpha}}\big[\lambda_1(\frac{\alpha}{\lambda_1})^{\frac{1}{1+\alpha}}+(\frac{
\lambda_1}{\alpha})^{\frac{\alpha}{1+\alpha}}\big]$, has a
unique strong solution $u\in W^{2,p}(\Omega)$ within the set
$\{v>(\frac{\alpha
B}{\lambda_1})^{\frac{1}{1+\alpha}}\varphi_1\}$. Furthermore if we
denote $H$ the solution map $f\to u$, it is a continuous
and compact map from the set $\{f\in
C^1(\overline\Omega):f>t_0\varphi_1\}$ to $\{u\in
C^1(\overline\Omega): u>(\frac{\alpha
B}{\lambda_1})^{\frac{1}{1+\alpha}}\varphi_1\}$
(see Lemma \ref{l}). Hence the map
\[
F(s,u)=H(s(u^{\beta}+|\nabla u|^\zeta)+t\varphi_{1}). \ \
\]
with $t\geq t_0$ is well  from $\mathbb{R}_{\geq 0} \times P$ to
$P$, where $P$ is the cone of positive functions in
$C^1(\overline\Omega)$. Like in the proof of previous theorems, we
study the fixed point equation
\begin{equation}\label{spath}
F(s,u+u_*)-u_*=u\,,
\end{equation}
where $u_*$ is the unique solution in in the set
$\{v>(\frac{\alpha B}{\lambda_1})\varphi_1\}$ (see Lemma \ref{l})
\begin{gather*}
-\Delta u_*  =  Ku_*^{-\alpha}+ t\varphi_{1} \quad \text{in } \Omega \\
 u_* = 0 \quad \text{on } \partial\Omega\,.
\end{gather*}

If $(s,u)$ solves (\ref{spath}) then $(s,u+u_*)$ solves equation
\eqref{amistades}. Now using again the Corollary 17.2 in
\cite{am}, we find a connected, closed unbounded in $\mathbb{R}\times P$
and emanating from $(0,0)$ set $\Sigma$ of pairs
$(s,u)$ satisfying the equation (\ref{spath}). Since the obtained
solution $u$ of problem  \eqref{amistades} satisfies
$u\geq (\frac{\alpha B}{\lambda_1})^{\frac{1}{1+\alpha}}\varphi_1$,
 we deduce
\[
| K| u^{-\alpha}\leq
B^{\frac{1}{1+\alpha}}\big(\frac{\lambda_1}{\alpha
}\big)^{\frac{\alpha}{1+\alpha}}\varphi_1
\]
and from  \eqref{amistades}, we have
\[
-\Delta u  \geq  su^{\beta}  \geq  \lambda_1\beta
u-C(\beta,\lambda_1)s^{-\frac{1}{\beta-1}}\,.
\]
Multiplying by $\varphi_1$ and integrating, we find
\[
\lambda_1\int_{\Omega}u\varphi_1dx\geq
\lambda_1\beta\int_{\Omega}u\varphi_1dx
-C(\beta,\lambda_1)s^{-\frac{1}{\beta-1}}\int_{\Omega}\varphi_1dx\,.
\]
Thus
\[
 (\frac{\alpha B}{\lambda_1})^{\frac{1}{1+\alpha}}\int_\Omega\varphi_1^2dx
 \leq  \int_\Omega u\varphi_1dx
 \leq \frac{C(\beta,\lambda_1)s^{-\frac{1}{\beta-1}}}{\lambda_1(\beta-1)}
 \int_\Omega \varphi_1dx\,.
\]
Consequently,
\[
s\leq \big\{ \frac{C(\beta,\lambda_1)}{\lambda_1(\beta-1)}
(\frac{\lambda_1}{\alpha
B})^{\frac{1}{1+\alpha}}\frac{\int_\Omega\varphi_1dx}
{\int_\Omega\varphi_1^2dx}\big\}^{\beta -1}\,.
\]
Recalling that
\[
\lambda_1\int_\Omega u\varphi_1dx= s\int_\Omega
u^\beta\varphi_1dx+t\int_\Omega\varphi_1^2dx-\int_\Omega
K(x)u^{-\alpha}\varphi_1dx\,,
\]
we see that
\[
s\int_\Omega u^\beta\varphi_1dx\leq \frac{
C(\beta,\lambda_1)s^{-\frac{1}{\beta-1}}}{\beta-1}\int_\Omega
\varphi_1dx\,.
\]
The rest of the proof is similar to that one of Theorem \ref{williams}.
\end{proof}

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