\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 $$\label{maroon} \begin{gathered} -\Delta u = K(x)u^{-\alpha}+s\mathcal{G}(x,u,\nabla u)+f(x) \quad \text{in }\Omega \\ u > 0 \quad \text{in }\Omega\\ u = 0 \quad \text{on }\partial\Omega \end{gathered}$$ 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 $$\label{amistades} \begin{gathered} -\Delta u = K(x)u^{-\alpha}+s(\mathcal{A}u^\beta+\mathcal{B}|\nabla u|^\zeta)+f(x) \quad \text{in }\Omega\\ u > 0 \quad \text{in }\Omega\\ u = 0 \quad \text{on }\partial\Omega\,, \end{gathered}$$ 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 $F:\mathcal{C}^+\to\mathcal{D}$, $$F(u)=-u^\alpha\Delta u$$ 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 $$\label{amistades1} \begin{gathered} -\Delta u = g(u)+\lambda|\nabla u|^\zeta+\mu f(x,u) \quad \text{in }\Omega\\ u > 0 \quad \text{in }\Omega\\ u = 0 \quad \text{on }\partial\Omega\,, \end{gathered}$$ 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, $$\label{amistades3} \begin{gathered} -\Delta_p u = g(u)+s\mathcal{G}(u) \quad \text{in }\Omega\\ u > 0 \quad \text{in }\Omega\\ u = 0 \quad \text{on }\partial\Omega\,, \end{gathered}$$ 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)] $10}\mathcal{G}(s)/s^{p-1} >0$ 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 $$\label{autovalor} \begin{gathered} -\Delta u = \lambda m(x)u \quad\text{in }\Omega\\ u = 0 \quad\text{on }\partial\Omega\,, \end{gathered}$$ 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 $$\begin{gathered} \| \varphi_{1}\|_{L^\infty(\Omega)}=\| \varphi_{1}\|_{L^\infty(\mathop{rm supp} m^+,\; m^+dx)} \\ \| \varphi_{-1}\|_{L^\infty(\Omega)}=\| \varphi_{-1}\|_{L^\infty(\mathop{rm supp}m^-, \;m^-dx)} \end{gathered}$$ 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 $$\label{mayonesa1} \begin{gathered} -u^\alpha\Delta u = K(x) \quad\text{in }\Omega\\ u = 0 \quad\text{on }\partial\Omega\,. \end{gathered}$$ We fix $p>n$ and consider $K\in L^p(\Omega)$. It is shown in \cite{ag1} that for $\alpha>0$, $00$ 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 $$\label{martin} \begin{gathered} -\Delta u = \lambda (m^+(x)-tm^-(x)) u \quad\text{in }\Omega\\ u = 0 \quad\text{on }\partial\Omega\,, \end{gathered}$$ 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 $t\in (0,\epsilon(m^+,m^-))$ $$\label{xet1} \int_\Omega (m^+-tm^-)\varphi_{-1}^{-1}\varphi_1^3dx \neq 0\,.$$ \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}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}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^-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 \label{lazer} 0<-\lambda_{-1}((-m^-)) 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 $$\label{banach1} \mathcal{C}= W^{2,p}(\Omega)\cap C(\overline\Omega)_e$$ for the norm$\|\cdot\|_{\mathcal{C}}=\|\cdot\|_{W^{2,p}(\Omega)}+\|\cdot\|_e$. Hence, the cone of positive functions $$\label{banach2} \mathcal{C}^+= W^{2,p}(\Omega)\cap C(\overline\Omega)_e^+$$ has non empty interior${\mathaccent"7017 {\mathcal{C}}}^+$. We also need $$\label{banach3} \mathcal{D}=\{f : fe^{-\alpha}\in L^p(\Omega)\}$$ 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 $$\label{ariston} \omega (t) =\frac{ F(u+tv)-F(u)}{t}+\alpha u^{\alpha -1}v\Delta u+u^\alpha\Delta v$$ To prove Gateaux differentiability, we need to establish $$\label{awards} \lim_{t\to 0}\|\omega (t)\|_{\mathcal{C}}=0$$ 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 \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} 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 $d_GF:{\mathaccent"7017 {\mathcal{C}}}^+\to L(\mathcal{C},\mathcal{D})$ whereL(\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 $$\label{duke} \begin{gathered} -\Delta u = u^{-\alpha}+f \quad\text{in }\Omega \\ u = 0 \quad\text{on }\partial\Omega \end{gathered}$$ 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 $$\label{hopf1} \begin{gathered} -\Delta u = f \quad\text{in }\Omega \\ u = 0 \quad\text{on }\partial\Omega \end{gathered}$$ satisfies $$\label{hopf2} u\geq C\Big(\int_{\Omega}fe\Big)e\,.$$ 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 $$\label{aerosmith} u_j(x)^{-\alpha}\leq C(\Omega)^{-\alpha}\Big( \int_\Omega edx\Big)^{\alpha-1}\Big( \int_\Omega w_j^{\alpha^2}e\,dx\Big)e^{-\alpha}\,.$$ 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)}0 in } \mathbb{R}, 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 $$\label{2} \begin{gathered} -\mathcal{L} u+K(x)u^{-\alpha} = f(x) \quad \text{in } \Omega \\ u > 0 \quad \text{in } \Omega \\ u = 0 \quad \text{on } \Omega \end{gathered}$$ 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 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 $$\label{boca} \begin{gathered} -\Delta v = \alpha \frac{\Delta u}{u}v \quad\text{in }\Omega\\ v = 0 \quad\text{on }\partial\Omega\,. \end{gathered}$$ 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 $$\label{boca12} \begin{gathered} -\Delta v = \lambda_1mv \quad\text{in }\Omega \\ v = 0 \quad\text{on }\partial\Omega \end{gathered}$$ which implies$\ker dF(\varphi_{-1})=\langle \varphi_1\rangle$. The equation$dF(\varphi_{-1})v=f$is equivalent to $$\label{fredholm} \begin{gathered} -\Delta v = \lambda_1mv+\varphi_{-1}^{-\alpha}f \quad\text{in }\Omega\\ v = 0 \quad\text{on }\partial\Omega \end{gathered}$$ 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 $$\label{ecu1} F(\varphi_{-1}+\widehat{v})=F(\varphi_{-1})+\rho\varphi_{-1}$$ Inserting Taylor formula in (\ref{ecu1}), $F(\varphi_{-1}+\widehat{v})=F(\varphi_{-1})+dF(\varphi_{-1})\widehat{v}+\Psi (\widehat{v})$ we find $$\label{pintura} dF(\varphi_{-1})\widehat{v}+\Psi(\widehat{v})=\rho\varphi_{-1}$$ 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 $$\label{quemado} W'(0)=r\varphi_1-\varphi_{-1}$$ 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*} Fromd\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 $$\label{xet} \int_\Omega m\varphi_{-1}^{-1}\varphi_1^3dx\not=0\,.$$ 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 $$\label{cd251} F(s,v+u_*)-u_*=v$$ where$u_*$is the unique solution of the problem $$\label{duke21} \begin{gathered} -\Delta u_* = u_*^{-\alpha}+f \quad\text{in }\Omega \\ u_* = 0 \quad\text{on }\partial\Omega \end{gathered}$$ satisfies the equation $$\begin{gathered} -\Delta (v+u_*) = (v+u_*)^{-\alpha}+ s\mathcal{G}(x,v+u_*, \nabla( v+u_*))+f \quad\text{in }\Omega\\ v+u_* > 0 \quad\text{in }\Omega\\ v+u_* = 0 \quad\text{on }\partial\Omega\,. \end{gathered}$$ 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 $$\label{cd25} F(s,v+u_*)-u_*=v$$ where$u_*$is the unique solution of $$\label{duke2} \begin{gathered} -\Delta u_* = u_*^{-\alpha}+f \quad\text{in }\Omega \\ u_* = 0 \quad\text{on }\partial\Omega\,. \end{gathered}$$ 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 $$\label{secondary12} \int_\Omega u\varphi_1dx\leq \frac{C(\beta,\lambda_1)s^{-\frac{1}{\beta-1}}}{\lambda_1(\beta-1)}\int_\Omega \varphi_1dx\,.$$ From \eqref{amistades}, we have$-\Delta u\geq f$. Using the Uniform Hopf Principle (\ref{hopf1}), (\ref{hopf2}) and (\ref{secondary12}), it follows that $$\label{isat} s\leq \big\{ \frac{C(\beta,\lambda_1)\int_\Omega\varphi_1dx}{ \lambda_1(\beta-1)C(\Omega)\int_\Omega f\varphi_1dx\int_\Omega\varphi_1^2dx} \big\}^{\beta -1}$$ 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 $$\label{gun} s\int_\Omega u^\beta\varphi_1dx\leq \frac{\lambda_1 C(\beta,\lambda_1)s^{-\frac{1}{\beta-1}}}{\lambda_1(\beta-1)}\int_\Omega \varphi_1dx\,.$$ 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 $$\label{convection} |\nabla u|^\zeta u \leq \frac{|\nabla u|^{\zeta q}}{q}+\frac{u^{\vartheta+1}}{\vartheta+1} \leq |\nabla u|^2+1+u^{\vartheta}u\,.$$ 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 $$\label{james12} C_1\int_\Omega|\nabla u|^2dx\leq s\int_\Omega u^\beta u\,dx+sC_2\int_\Omega u^\vartheta u\,dx+ C_3\| u\|_{H^1_0(\Omega)}+C_4\,,$$ 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 $\frac{1}{2\frac{n+1}{n-1}}=\frac{1}{2}-\frac{1}{n}+\frac{\frac{1}{n+1}}{n}, \quad \frac{1}{q}=\frac{1}{2}-\frac{1}{n}+\frac{\frac{2}{n+1}}{n},$ withq>\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 $$\label{tasi12} \frac{C_1}{2}\| \nabla u\|_{L^2(\Omega)}^2 \leq C\{1+C_\epsilon^{1-\gamma} s^{1-\gamma-\frac{\gamma}{\beta -1}}\}\| \nabla u\|_{L^2(\Omega)} +C(\delta)\,.$$ Finally if$u$is a solution of the problem \eqref{amistades} with$s>\delta>0$, there exists a constant$C(\delta)>0$such that$\| u\|_{H_0^{1,2}(\Omega)}0$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 $$\label{spath} F(s,u+u_*)-u_*=u\,,$$ 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} \begin{thebibliography}{00} \bibitem{ao} R. 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