\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 $00$,
$\mathcal{A}\equiv0$, $\mathcal{B}\equiv1$,
$0<\zeta\leq 2$ 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)] $1
0}\mathcal{G}(s)/s^{p-1} >0$ 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 $\alpha>0$, $0 u_{2}$ 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)}