\documentclass[reqno]{amsart} \usepackage{hyperref} \AtBeginDocument{{\noindent\small \emph{Electronic Journal of Differential Equations}, Vol. 2009(2009), No. 124, pp. 1--18.\newline ISSN: 1072-6691. URL: http://ejde.math.txstate.edu or http://ejde.math.unt.edu \newline ftp ejde.math.txstate.edu} \thanks{\copyright 2009 Texas State University - San Marcos.} \vspace{9mm}} \begin{document} \title[\hfilneg EJDE-2009/124\hfil Infinite multiplicity] {Infinite multiplicity of positive solutions for singular nonlinear elliptic equations with convection term and related supercritical problems} \author[C. C. Aranda \hfil EJDE-2009/124\hfilneg] {Carlos C. Aranda} % in alphabetical order \address{Carlos C. Aranda \newline Laboratorio de modelizaci\'on, c\'alculo num\'erico y diseno experimental\\ Facultad de Recursos Naturales, Universidad Nacional de Formosa, Argentina} \email{carloscesar.aranda@gmail.com} \thanks{Submitted August 24, 2009. Published October 1, 2009.} \thanks{Supported by Secretar\'{i}a de Ciencia y Tecnolog\'{i}a, UNaF} \subjclass[2000]{35J25, 35J60} \keywords{Bifurcation; degree theory, nonlinear eigenvalues and eigenfunctions} \begin{abstract} In this article, we consider the singular nonlinear elliptic problem \begin{gather*} -\Delta u = g(u)+h(\nabla u)+f(u) \quad\text{in }\Omega, \\ u = 0 \quad\text{on }\partial\Omega. \end{gather*} Under suitable assumptions on $g$ , $h$, $f$ and $\Omega$ that allow a singularity of $g$ at the origin, we obtain infinite multiplicity results. Moreover, we state infinite multiplicity results for related boundary blow up supercritical problems and for supercritical elliptic problems with Dirichlet boundary condition. \end{abstract} \maketitle \numberwithin{equation}{section} \newtheorem{theorem}{Theorem}[section] \newtheorem{lemma}[theorem]{Lemma} \newtheorem{remark}[theorem]{Remark} %\usepackage{amssymb,latexsym} \section{Introduction and statement of main results} In 1869, Lane \cite{l} introduced the equation $$\label{lane} -\Delta u=u^p$$ for $p$ a nonnegative real number and $u>0$ in a Ball of radius $R$ in $\mathbb{R}^3$, with Dirichlet boundary conditions. Lane was interested in computing both the temperature and the density of mass on the surface of the sun. Today the problem (\ref{lane}) is named Lane-Emden-Fowler equation \cite{e,f}. Singular Lane-Emden-Fowler equations ($p<0$) has been considered in a remarkable pioneering paper by Fulks and Maybe \cite{fm}. Nonlinear singular elliptic equations arise in applications, for example in glacial advance \cite{w}, ecology \cite{gl}, in transport of coal slurries down conveyor belts \cite{c}, micro-electromechanical system device \cite{egg} etc. Nonlinear singular elliptic equations have been studied intensively during the last 40 years, for a detailed review out of our scope in this article, see Hern\'andez and Mancebo \cite{hm}, and the recent book by Ghergu and R\u adulescu \cite{gr2}. Multiplicity is a question with few results. Apparently, the first multiplicity result for the problem $$\label{queso} \begin{gathered} -\Delta u = K(x)u^{-p}+u^q \quad\text{in } \Omega, \\ u = 0 \quad\text{on } \partial\Omega, \end{gathered}$$ where $\Omega$ is smooth bounded domain and $00}f(s) /s>0 and \lim_{s\to\infty}f(s) /s^{p}<\infty for some p\in(1,\frac{N}{N-2}]; \item \Omega is a strictly convex domain in \mathbb{R}^{N}. \end{enumerate} Then the problem \begin{gather*} -\Delta u = g(u)+\lambda f(u) \quad\text{in }\Omega,\\ u = 0 \text{ on }\partial\Omega, \end{gather*} has at least two positive solutions for \lambda positive and small enough and that \lambda=0 is a bifurcation point from infinity for this problem. \end{theorem} Our first result in this article is as follows. \begin{theorem}\label{11} Let \Omega be a smooth bounded domain in \mathbb{R}^N, N\geq 3. Suppose the following conditions hold: \begin{enumerate} \item g:(0,\infty) \to(0,\infty) is non increasing locally H\"{o}lder continuous function (that may be singular at the origin); \item f is continuous, nonnegative and non decreasing function with f(0)=0; \item f(\xi_{i})\geq \beta\xi_{i}, f(\eta_{i})\leq \alpha\eta_{i} with \[ \xi_{1}<\eta_{1}<\dots<\xi_{i}<\eta_{i}<\xi_{i+1}<\dots<\xi_{m}, \quad m\leq\infty ;$ \item $\beta C(\Omega)(\int_{K}\varphi_{1})\varphi_{1}\geq 1$, on $K\subset\Omega$ compact 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; \item $v+\alpha\eta_{i}e\leq\eta_{i}$, where \begin{gather*} -\Delta v = g(v) \quad\text{in }\Omega, \\ v = 0 \quad\text{on }\partial\Omega, \end{gather*} and \begin{gather*} -\Delta e = 1 \quad\text{in }\Omega, \\ e = 0 \quad\text{on }\partial\Omega. \end{gather*} \end{enumerate} Then the problem $$\label{1} \begin{gathered} -\Delta u = g(u)+f(u) \quad\text{in }\Omega, \\ u = 0 \quad\text{on }\partial\Omega, \end{gathered}$$ has $m\leq\infty$ nonnegative classical solutions. Moreover the problem $$\label{sumo1} \begin{gathered} -\Delta u = g(u)+f(u) \quad\text{in }\Omega, \\ u = \epsilon \quad\text{on }\partial\Omega, \end{gathered}$$ has $2m-1\leq\infty$ nonnegative classical solutions for all $\epsilon >0$. \end{theorem} The behavior of the function $f$ in Theorem \ref{11} is closely related to a similar nonlinearity studied by Kielh\"ofer and Maier in \cite{km}. Under our best knowledge this is the first result on infinite multiplicity for nonlinear singular equations. Hern\'andez, Mancebo and Vega obtained the following theorem. \begin{theorem}[\cite{hmv}] Let $\Omega$ be a smooth bounded domain in $\mathbb{R}^N$, $N\geq 3$. Suppose the following conditions hold: $-10$. Then the problem $$\label{e5} \begin{gathered} -\Delta u = \lambda u^{-q}-u^{-p} \quad\text{in }\Omega, \\ u = 0 \quad\text{on }\partial\Omega, \end{gathered}$$ has a unique nonnegative classical solution. \end{theorem} Our second Theorem is related to multiplicity of a nonlinear eigenvalue problem. \begin{theorem}\label{drno5} Let $\Omega$ be a smooth bounded domain in $\mathbb{R}^N$, $N\geq 3$. Suppose the following conditions hold: \begin{enumerate} \item $00$ small enough. \end{theorem} Existence and nonexistence results for singular nonlinear elliptic equations with convection term have been stated by Zhang \cite{z1}, Zhang and Yu \cite{zy}, Ghergu and R\u adulescu \cite{gr,gr1}. Multiplicity for singular Lane-Emden-Fowler equation with convection term is a topic essentially open. A result was stated by Aranda and Lami Dozo in \cite{al}: \begin{theorem}[\cite{al}]\label{williams} Let $\Omega$ be a smooth bounded domain in $\mathbb{R}^N$, $N\geq 3$. Suppose the following conditions hold: \begin{enumerate} \item $00$; \item $0\leq\nu\lambda^*$. Furthermore there is bifurcation at infinity at $\lambda=0$. \end{theorem} Our third result in this article, it is concerned with infinite multiplicity for nonlinear elliptic equations with strong singularity and convection term. \begin{theorem}\label{122} Let $\Omega$ be a smooth bounded domain in $\mathbb{R}^N$, $N\geq 3$. Suppose the following conditions hold: \begin{enumerate} \item $g$ and $f$ satisfies conditions (1)--(4) of Theorem \ref{11}; \item $h$ is a locally H\"older continuous function on $\mathbb{R}^N$ and $0\leq h( \nabla u )\leq b_{1} | \nabla u |^{s}+b_{0}$, $0N$ with $\mathfrak{u}\geq \xi_i\kappa$; \item $v(x)+\alpha\eta_{i}e(x)<\eta_{i}$ for all $x\in\overline{\Omega}$, where \begin{gather*} -\Delta v = v^{-p} \quad\text{in }\Omega, \\ v = 0 \quad\text{on }\partial\Omega, \end{gather*} and \begin{gather*} -\Delta e = 1 \quad\text{in }\Omega, \\ e = 0 \quad\text{on }\partial\Omega. \end{gather*} \end{enumerate} Then the problem $$\label{alice} \begin{gathered} -\Delta u +u|\nabla u|^2 = u^{-p}+f(u) \quad\text{in } \Omega, \\ u = 0 \quad\text{on } \partial\Omega, \end{gathered}$$ has $m$ solutions in $H^{1,2}_0(\Omega)$. \end{theorem} \begin{remark} \label{rmk2} \rm Condition (3) indicates again a complex relation between domain, convection term and multiplicity. \end{remark} For large solutions Ghergu et al. stated the following result. \begin{theorem}[\cite{gnr}] Let $\Omega$ be a smooth bounded domain in $\mathbb{R}^N$, $N\geq 3$. Suppose the following conditions hold: \begin{enumerate} \item $f\in C^1[0,\infty)$, $f'\geq 0$, $f(0)=0$ and $f>0$ on $(0,\infty)$; \item $\int_1^\infty\left[F(t)\right]^{-2/a}dt<\infty$, where \item $\frac{F(t)}{f^{2/a}}\to 0$ as $t \to 0$; \item $\mathfrak{p}$, $\mathfrak{q}\in C^{0,\gamma}(\overline\Omega)$ are nonnegative functions such that for every $x_0\in\Omega$ with $\mathfrak{p}(x_0)=0$, there exists a domain $\Omega_0\ni x_0$ such that $\overline\Omega_0\subset\Omega$ and $\mathfrak{p}>0$ on $\partial\Omega_0$; \item $02$. Then the problem $$\label{e11} \begin{gathered} \Delta v = \frac{2}{v}|\nabla v|^2+v^p+\tilde{f}(u) \quad\text{in }\Omega, \\ u = \infty \quad\text{on }\partial\Omega, \end{gathered}$$ has $m\leq\infty$ nonnegative classical solutions. Moreover the problem $$\label{e12} \begin{gathered} \Delta v = \frac{2}{v}|\nabla v|^2+v^p+\tilde{f}(u) \quad\text{in }\Omega, \\ u = M \quad\text{on }\partial\Omega, \end{gathered}$$ has $2m-1\leq\infty$ nonnegative classical solutions for all $M >0$ big enough. \end{theorem} \begin{theorem}\label{supercritico} Let $\Omega$ be a smooth bounded domain in $\mathbb{R}^N$, $N\geq 3$. Suppose the following conditions hold: \begin{enumerate} \item $f(s)=s^2\tilde{f}(\frac{1}{s})$ satisfies {\rm (2)--(4)} of Theorem \ref{11}; \item $2 0 \quad\text{in } \Omega,\\ \varphi_{1} = 0 \quad\text{on } \partial\Omega. \end{gather*} \end{lemma} \begin{remark} \label{rmk3} \rm The proof of Lemma \ref{hopf} given by Brezis and Cabre in \cite{bc} relies on the superharmonicity of the laplacian operator. \end{remark} \begin{theorem}[\cite{ag}]\label{H} Let$P$be the positive cone in$L^\infty(\Omega)$. Let$S_{\epsilon}:P\to P$be the solution operator for the problem \begin{gather*} -\Delta u = g(u)+w \quad\text{in }\Omega, \\ u = \epsilon \quad\text{on }\partial\Omega, \end{gather*} gives$S_{\epsilon}(w)=u$where$\epsilon\geq 0$and$g:(0,\infty) \to(0,\infty)$is nonincreasing locally H\"{o}lder continuous function (that may be singular at the origin). Then$S_{\epsilon}:P\to P$is a continuous, non decreasing and compact map with$S_{\epsilon_0}(w)\leq S_{\epsilon_1}(w)$for$\epsilon_0<\epsilon_1$. \end{theorem} \begin{lemma}\label{comparacion} Let$\Omega$be a smooth bounded domain in$\mathbb{R}^N$,$N\geq 3$. Let$u,v\in C^{2}(\Omega)\cap C(\overline{\Omega})$be solutions of the problem \begin{gather*} -\Delta u -g(u)-h(\nabla u) \geq -\Delta v -g(v)-h(\nabla v) \quad\text{in }\Omega, \\ u \geq v\geq 0 \quad\text{on }\partial\Omega. \end{gather*} Then$u \geq v $on$\Omega$. \end{lemma} \begin{proof} Indeed suppose$v>u$somewhere and consider the non empty open set $\Omega_{\delta}=\{x\in\Omega |v(x)>u(x)+\delta, \; \delta >0\}.$ Since$u,v\in C^{2}(\Omega), we have \begin{align*} -\Delta (u+\delta) -h(\nabla (u+\delta)) & = g(u)+q\\ & \geq g(v)+r \\ & = -\Delta v -h(\nabla v) \quad\text{on }\Omega_{\delta}, \end{align*} withq,r\in C(\overline{\Omega_{\delta}})$and$\overline{\Omega_{\delta}}\subset\Omega$. Also$u+\delta=v $on$\partial\Omega_{\delta}$and so the comparison Theorem 10.1 \cite{gt} implies$u+\delta\geq v $on$\Omega_{\delta}$. It follows$\Omega_{\delta}=\emptyset$a contradiction. \end{proof} \begin{lemma}\label{m29} Let$\Omega$be a smooth bounded domain in$\mathbb{R}^N$,$N\geq 3$. Suppose the following conditions hold: \begin{enumerate} \item$g:(0,\infty) \to(0,\infty)$is non increasing locally H\"{o}lder continuous function (that may be singular at the origin); \item$h$is locally H\"older continuous function on$\mathbb{R}^N$with$0\leq h( \nabla u )\leq b_{1} | \nabla u |^{s}+b_{0}$,$ 00 \quad\text{in }\partial\Omega, \end{gather*} has a classical solution. From \begin{align*} \Delta u_{j-1} + g_{j}(u_{j-1})+h(\nabla u_{j-1})+w(x) & \geq \Delta u_{j-1} + g_{j-1}(u_{j-1})+h(\nabla u_{j-1})+w(x) \\ & = 0 \\ & = \Delta u_{j} + g_{j}(u_{j})+h(\nabla u_{j})+w(x) \quad\text{in }\Omega, \end{align*} $u_{j-1}=u_{j}=\epsilon$ on $\partial\Omega$, using \cite[Theorem 10.1]{gt}, we infer $u_{j-1}\leq u_{j}$ in $\Omega$. Therefore for $j$ big enough there exists an unique $u_{\epsilon}=u_{j}$ solution of \begin{gather*} -\Delta u_{\epsilon} = g(u_{\epsilon})+h(\nabla u_{\epsilon})+w(x) \quad\text{in }\Omega , \\ u_{\epsilon} = \epsilon \quad\text{on }\partial\Omega. \end{gather*} If $\epsilon_{0}<\epsilon_{1}$, for $j$ big enough, we have $\Delta u_{\epsilon_{0}} + g_{j}(u_{\epsilon_{0}}) +h(\nabla u_{\epsilon_{0}})+w(x) = \Delta u_{\epsilon_{1}} + g_{j}(u_{\epsilon_{1}}) +h(\nabla u_{\epsilon_{1}})+w(x),$ on $\Omega$, $u_{\epsilon_{0}}0=\mathcal{M} \quad\text{on }\partial\Omega, \end{gather*} we obtain$u_{\epsilon}>r$on$\Omega$. \cite[Theorem 15.8]{gt} implies $-\Delta u_{\epsilon} = g(u_{\epsilon})+h(\nabla u_{\epsilon})+w(x) \\ \leq g(\mathcal{M})+C,$ on$\overline{\Omega'}\subset\Omega$. Using \cite[Theorem 9.11]{gt}, we have $\| u_{\epsilon} \|_{\mathcal{W}^{2,r}(\Omega')} \leq C(\| u_{\epsilon} \|_{L^{r}(\Omega')} +C) \leq C(\| u_{1} \|_{L^{r}(\Omega')} +C),$ with$r>N$. By the Sobolev imbedding \cite[Theorem 7.26]{gt}$u_{\epsilon}\to u$, in$C^{1,\gamma}(\Omega')$. A standard bootstrap argument implies that$u\in C^{2}(\Omega)\cap C(\overline{\Omega})$is a classical solution of problem (\ref{100}). The unicity follows from Lemma \ref{comparacion}. \end{proof} \begin{lemma}\label{car} Let$\Omega$be a smooth bounded domain in$\mathbb{R}^N$,$N\geq 3$. Suppose the following conditions hold: \begin{enumerate} \item$g:(0,\infty) \to(0,\infty)$is non increasing locally H\"{o}lder continuous function (that may be singular at the origin); \item$h$is locally H\"older continuous function on$\mathbb{R}^N$with$0\leq h( \nabla u )\leq b_{1} | \nabla u |^{s}+b_{0}$,$ 00$. \end{proof} \begin{lemma}\label{adel} Let$\Omega$be a smooth bounded domain in$\mathbb{R}^N$,$N\geq 3$. Suppose that$01$, satisfying: \begin{itemize} \item[(a)] Let$P$be the positive cone in$L^\infty(\Omega)$. Let$\mathcal{S}_{\epsilon,\delta} :P\to P$be the solution operator for the problem (\ref{poison}), gives$\mathcal{S}_{\epsilon,\delta}(w)=u_{\epsilon,\delta}$. Then$\mathcal{S}_{\epsilon,\delta} :P\to P$is continuous, compact and non decreasing map. \item[(b)] If$\epsilon_0<\epsilon_1$, then$\mathcal{S}_{\epsilon_0,\delta}(w)\leq \mathcal{S}_{\epsilon_1, \delta}(w)$. \item[(c)] If$\delta_0<\delta_1$, then$\mathcal{S}_{\epsilon,\delta_0}(w)\leq \mathcal{S}_{\epsilon, \delta_1}(w)$. \item[(d)]$\inf_{\Omega'}u_{\epsilon,\delta}\geq C$,$\overline\Omega'\subset\Omega$, where$C=C(p,\Omega',w(x))$is a constant independent of$\epsilon$,$\delta$and$C(\alpha,\Omega',0)>0$. \item[(e)] For$0<\epsilon,\delta<1$, we have$\| u_{\epsilon,\delta} -\epsilon\|_{H^{1,2}_0(\Omega)}\leq C$, where$C$is a constant independent of$\epsilon$and$\delta$. \end{itemize} \end{lemma} \begin{remark} \label{rmk46} \rm Items (a)--(c) contain the monotone and compactness properties of approximate solutions. Item (d) is a uniform Harnack inequality. Item (e) contains a uniform bound necessary for the compensated compactness technique. \end{remark} \begin{proof}[Proof of Lemma \ref{adel}] Let$g_{j}:\mathbb{R}\to\mathbb{R}$be a non increasing and locally H\"older continuous function defined by $g_{j}(s) = \begin{cases} s^{-p} & \text{if } s\geq\frac{1}{j},\\ C_{j} & \text{if } s\leq \frac{1}{j+1}. \end{cases}$ Using a standard argument involving$L^r$estimates \cite[Theorem 9.10]{gt}, Sobolev imbedding \cite[Theorem 7.26]{gt}, \cite[Theorem 10.1]{gt} and the Schauder fixed point Theorem, we deduce that the problem \begin{gather*} -\Delta u_{j,\epsilon,\delta} +\frac{u_{j,\epsilon, \delta}|\nabla u_{j,\epsilon,\delta}|^2 }{1+\delta u_{j,\epsilon,\delta}| \nabla u_{j,\epsilon,\delta}|^2} = g_j(u_{j,\epsilon,\delta})+w(x) \quad\text{in } \Omega, \\ u_{j,\epsilon,\delta} = \epsilon \quad\text{on } \partial\Omega, \end{gather*} has a unique solution$u_{j,\epsilon,\delta}\in\mathcal{W}^{2,r}(\Omega)\cap C(\overline\Omega)$for all$r>1$. If$w\in L^r(\Omega)$,$r>N$, then by \cite[Theorem 7.26]{gt},$u_{j,\epsilon,\delta}\in\mathcal{W}^{2,r}(\Omega) \hookrightarrow C^{1,\gamma}(\overline\Omega)$for some$\gamma>0. Calling $b_\delta(u,\nabla u)=\frac{u|\nabla u|^2}{1+\delta u|\nabla u|^2},$ we deduce \begin{align*} &-\Delta u_{j,\epsilon,\delta}+b_\delta(u_{j,\epsilon,\delta}, \nabla u_{j,\epsilon,\delta})-g_{j+1}(u_{j,\epsilon,\delta}) \\ &\leq -\Delta u_{j,\epsilon,\delta}+b_\delta(u_{j,\epsilon,\delta}, \nabla u_{j,\epsilon,\delta})-g_{j}(u_{j,\epsilon,\delta}) \\ &= w(x)\\ &= -\Delta u_{j+1,\epsilon,\delta}+b_\delta(u_{j+1,\epsilon,\delta}, \nabla u_{j+1,\epsilon,\delta})-g_{j+1}(u_{j+1,\epsilon,\delta}) \end{align*} in\Omega$and$u_{j+1,\epsilon,\delta}=u_{j,\epsilon,\delta}=\epsilon$on$\partial\Omega$. Using Theorem 10.1 \cite{gt}, we obtain that$u_{j+1,\epsilon,\delta}>u_{j,\epsilon,\delta}$in$\Omega$. Moreover from $-\Delta u_{j,\epsilon,\delta} +b_\delta (u_{j+1,\epsilon,\delta}, \nabla u_{j+1,\epsilon,\delta}) = g_j(u_{j,\epsilon,\delta})+w(x) \geq -\Delta\epsilon+b_\delta(\epsilon,\nabla\epsilon)\quad \text{in }\Omega,$ and$u_{j,\epsilon,\delta}=\epsilon$on$\partial\Omega$, using again \cite[Theorem 10.1]{gt}, we conclude$u_{j,\epsilon,\delta}>\epsilon$on$\Omega$. Letting$u_{\epsilon,\delta}=\lim_{j\to\infty}u_{j,\epsilon,\delta}$, we have \begin{gather*} -\Delta u_{\epsilon,\delta} +b_\delta(u_{\epsilon,\delta}, \nabla u_{\epsilon,\delta}) = u_{\epsilon,\delta}^{-p}+w(x) \quad\text{in } \Omega, \\ u_{\epsilon,\delta} = \epsilon \quad\text{on } \partial\Omega. \end{gather*} Using standard Nemytskii mappings properties and Sobolev Imbedding Theorems, we demonstrate the continuity and compacity of the map$\mathcal{S}_{\delta,\epsilon}$. This states ($\mathfrak{a}$). Comparison \cite[Theorem 10.1]{gt} implies if$\epsilon_0<\epsilon_1$then$\mathcal{S}_{\epsilon_0,\delta}(w)=u_{\epsilon_0, \delta}\delta_0>0. It follows that \begin{align*} &-\Delta \mathcal{S}_{\epsilon,\delta_1}(w) +\frac{\mathcal{S}_{\epsilon,\delta_1}(w)|\nabla \mathcal{S}_{\epsilon,\delta_1}(w)|^2}{1+\delta_0 \mathcal{S}_{\epsilon,\delta_1}(w)|\nabla \mathcal{S}_{\epsilon,\delta_1}(w)|^2}\\ & \geq -\Delta \mathcal{S}_{\epsilon,\delta_1}(w) +\frac{\mathcal{S}_{\epsilon,\delta_1}(w)|\nabla \mathcal{S}_{\epsilon,\delta_1}(w)|^2} {1+\delta_1\mathcal{S}_{\epsilon,\delta_1}(w)| \nabla \mathcal{S}_{\epsilon,\delta_1}(w)|^2} \\ & = (\mathcal{S}_{\epsilon,\delta_1}(w))^{-p}+w(x) \\ & \geq (\mathcal{S}_{\epsilon,\delta_0}(w))^{-p}+w(x)\\ & = -\Delta \mathcal{S}_{\epsilon,\delta_0}(w) +\frac{\mathcal{S}_{\epsilon,\delta_0}(w)|\nabla \mathcal{S}_{\epsilon,\delta_0}(w)|^2} {1+\delta_0\mathcal{S}_{\epsilon,\delta_0}(w)|\nabla \mathcal{S}_{\epsilon,\delta_0}(w)|^2}\quad\text{on }\widehat{\Omega}, \end{align*} and\mathcal{S}_{\epsilon,\delta_1}(w(x)) =\mathcal{S}_{\epsilon,\delta_0}(w(x))$on$\partial\widehat{\Omega}$. Using Theorem 10.1 \cite{gt}, we infer$\mathcal{S}_{\epsilon,\delta_1}(w)>\mathcal{S}_{\epsilon,\delta_0}(w)$on$\widehat{\Omega}$. This contradiction implies$\mathcal{S}_{\epsilon,\delta_1}(w)\leq\mathcal{S}_{\epsilon,\delta_0}(w). We also have \begin{align*} -\Delta u_{\epsilon,\delta} +b_\delta (u_{\epsilon,\delta}, \nabla u_{\epsilon,\delta}) & \geq -\Delta u_{\epsilon,\delta} +b_\delta (u_{\epsilon,\delta},\nabla u_{\epsilon,\delta})-w(x)\\ & = u_{\epsilon,\delta}^{-p}\\ & \geq u_{1,\delta}^{-p}\\ & = -\Delta \omega_\delta +b_\delta(\omega_\delta, \nabla\omega_\delta)\quad\text{in }\Omega, \end{align*} andu_{\epsilon,\delta}=\epsilon>0=\omega_\delta$on$\partial\Omega$. Therefore$u_{\epsilon,\delta}>\omega_\delta$on$\Omega$. By definition \begin{gather*} -\Delta u_{1,\delta} +b_\delta(u_{1,\delta},\nabla u_{1,\delta}) = u_{1,\delta}^{-p}+w(x) \quad\text{in } \Omega \\ u_{1,\delta} = 1 \quad\text{on } \partial\Omega. \end{gather*} So, we have \begin{gather*} -\Delta u_{1,\delta} -u_{1,\delta}^{-p}\leq w(x) = -\Delta u_1-u_1^{-p} \quad\text{in } \Omega \\ u_{1,\delta} = 1 =u_1 \quad\text{on } \partial\Omega. \end{gather*} Therefore,$u_{1,\delta}\leq u_1$in$\Omega. Similarly \begin{align*} -\Delta\omega_\delta +b_\delta (\omega_{\delta},\nabla \omega_{\delta}) & = u_{1,\delta}^{-p} \\ & \geq u_1^{-p}\\ & = -\Delta \mathcal{O}_\delta +b_\delta( \mathcal{O}_\delta, \nabla\mathcal{O}_\delta)\quad\text{in }\Omega, \end{align*} and\omega_\delta\geq\mathcal{O}_\delta$in$\partial\Omega$. Then, we obtain$\omega_\delta\geq\mathcal{O}_\delta$. For$a\in\Omega$, we define $\mathcal{V}(x)=C(C-| x-a|^2).$ It follows that, for$Csmall enough \begin{align*} -\Delta \mathcal{O}_\delta +b_\delta ( \mathcal{O}_\delta,\nabla \mathcal{O}_\delta) & = g(u_1) \\ & \geq C_1 \\ & \geq -\Delta \mathcal{V}+ \mathcal{V}|\nabla\mathcal{V}|^2\\ & \geq -\Delta \mathcal{V}+ \frac{\mathcal{V}|\nabla\mathcal{V} |^2}{1+\delta\mathcal{V}|\nabla\mathcal{V}|^2}\\ & = -\Delta\mathcal{V}+b_\delta(\mathcal{V},\nabla\mathcal{V}) \quad\text{in }B_{\sqrt{C}}(a)\subset\Omega, \end{align*} and\mathcal{O}_\delta\geq 0=\mathcal{V}$on$\partial B_{\sqrt{C}}(a)$. Therefore, we deduce$\mathcal{O}_\delta\geq\mathcal{V}$in$B_{\sqrt{C}}(a). We conclude $u_{\epsilon,\delta}\geq\omega_\delta\geq\mathcal{O}_\delta \geq\mathcal{V} \quad\text{in }B_{\sqrt{C}}(a)\subset\Omega.$ This states (d). Now we consider (e): \begin{align*} \| u_{\epsilon,\delta}-\epsilon\|_{H_0^{1,2}}^2 & = \int_{\Omega}|\nabla (u_{\epsilon,\delta}-\epsilon)|^2dx\\ & \leq \int_{\Omega}u_{\epsilon,\delta}^{-p}(u_{\epsilon,\delta} -\epsilon)dx+\int_{\Omega}w(u_{\epsilon,\delta}-\epsilon)dx \\ & \leq \int_{\Omega}(u_{\epsilon,\delta}-\epsilon)^{-p} (u_{\epsilon,\delta}-\epsilon)dx+\| w\|_{H^{-1}}\| u_{\epsilon,\delta} -\epsilon\|_{H^{1,2}_0}\\ &\leq \int_{\Omega}(\mathcal{S}_{1,1}(w) -\epsilon)^{1-p}dx +\| w\|_{H^{-1}}\| u_{\epsilon,\delta}-\epsilon\|_{H^{1,2}_0}. \end{align*} This states (e). \end{proof} \section{Proofs of mains results} \begin{proof}[Proof of Theorem \ref{11}] LetP$be the positive cone in$L^\infty(\Omega)$. Let$A:P\to P$be the solution operator for the problem \begin{gather*} -\Delta z = g(z)+f(u) \quad\text{in }\Omega ,\\ z = 0 \quad\text{on }\partial\Omega, \end{gather*} gives$A(u)=z$. Using Theorem \ref{H}, we infer that$A:P\to P$is a well defined, continuous, non decreasing and compact map. Let us to denote with$\kappa$the indicator function of the set$K$. Using hypothesis$(3^\circ)$, we get $-\Delta A(\xi_i\kappa) = g(A(\xi_i\kappa))+f(\xi_i\kappa) \geq \beta\xi_i\kappa.$ We will denote by$u=(-\Delta)^{-1}h$the solution operator of the problem$-\Delta u=h$in$\Omega$and$u=0$on$\partial\Omega$. It follows that \begin{gather*} -\Delta ( A(\xi_i\kappa) -(-\Delta)^{-1}\beta\xi_i\kappa ) \geq 0 \quad\text{in }\Omega, \\ A(\xi_i\kappa) -(-\Delta)^{-1}\beta\xi_i\kappa = 0 \quad\text{on }\partial\Omega. \end{gather*} From the maximum principle, Lemma \ref{hopf} and (4), we infer $A(\xi_i\kappa) \geq (-\Delta)^{-1}\beta\xi_i\kappa \geq \beta\xi_i C(\Omega)\Big(\int_{\Omega}\kappa\varphi_{1}\Big)\varphi_{1} \geq \xi_i\kappa.$ Now$A(0)N$. By the Sobolev imbedding \cite[Theorem 7.26]{gt},$u_k\to u$in$C^{1,\gamma}(\Omega')$. A standard bootstrap argument implies that$u\in C^2(\Omega)\cap C(\overline\Omega)$is a classical solution of problem (\ref{a}) in the interval$[A^{2}(\xi_{i}\kappa),\eta_{i}]$. \end{proof} \begin{proof}[Proof of Theorem \ref{inner}] Let$P$the positive cone in the space$L^\infty(\Omega)$. Let$\mathcal{H}_\epsilon:P\to P$be the solution operator of the problem \begin{gather*} -\Delta z = z^{-p}+f(u) \quad\text{in }\Omega,\\ z = \epsilon \quad\text{on }\partial\Omega, \end{gather*} gives$\mathcal{H}_\epsilon(u)=z$. Using Theorem \ref{H}, we deduce that$\mathcal{H}_\epsilon:P\to P$is a well defined, continuous, non increasing and compact map with$\mathcal{H}_{\epsilon_0}(u)\leq\mathcal{H}_{\epsilon_1}(u)$for$\epsilon_0<\epsilon_1$. Let$\mathcal{T}_\epsilon:P\to P$be the solution operator of the problem \begin{gather*} -\Delta z +\frac{z|\nabla z|^2}{1+\epsilon z|\nabla z|^2} = z^{-p}+f(u) \quad\text{in }\Omega,\\ z = \epsilon \quad\text{on }\partial\Omega, \end{gather*} gives$\mathcal{T}_\epsilon(u)=z$. By Lemma \ref{adel}, we infer that$\mathcal{T}_\epsilon:P\to P$is a well defined, continuous, non increasing and compact map with$\mathcal{T}_{\epsilon_0}(u)\leq\mathcal{T}_{\epsilon_1}(u)$for$\epsilon_0<\epsilon_1. From \begin{align*} -\Delta \mathcal{T}_{\epsilon}(u) & \leq -\Delta \mathcal{T}{_\epsilon(u)}+\frac{\mathcal{T}_\epsilon(u)|\nabla \mathcal{T}_\epsilon(u)|^2}{1+\epsilon \mathcal{T}_\epsilon(u)|\nabla \mathcal{T}_\epsilon(u)|^2} \\ & = u^{-p}+f(u)\\ & = -\Delta\mathcal{H}_\epsilon (u)\quad\text{in }\Omega , \end{align*} and\mathcal{T}_{\epsilon}(u)=\mathcal{H}_\epsilon (u)$on$\partial\Omega$implies$\mathcal{T}_{\epsilon}(u)\leq\mathcal{H}_\epsilon (u)$in$P. \begin{align*} -\Delta (\mathcal{H}_\epsilon (\eta_i)) & = (\mathcal{H}_\epsilon (\eta_i))^{-p}+f(\eta_i) \\ & \leq (\mathcal{H}_\epsilon (0))^{-p}+\alpha\eta_i \\ & = -\Delta( \mathcal{H}_\epsilon (0)+\alpha\eta_i(-\Delta)^{-1}1) \quad\text{in }\Omega. \end{align*} From-\Delta(-\Delta)^{-1}1=1$in$\Omega$and$(-\Delta)^{-1}1=0$on$\partial\Omega$, we implies$\mathcal{H}_\epsilon (\eta_i)=\mathcal{H}_\epsilon (0) +\alpha\eta_i(-\Delta)^{-1}1$on$\partial\Omega$. Therefore,$\mathcal{H}_\epsilon (\eta_i)\leq \mathcal{H}_\epsilon (0) +\alpha\eta_i(-\Delta)^{-1}1$in$\Omega$. By condition (4), for$\epsilonsmall enough $$\mathcal{H}_\epsilon (\eta_i)<\eta_i$$ By hypothesis (3), we have \begin{align*} -\Delta \mathfrak{u} +\mathfrak{u}|\nabla \mathfrak{u}|^2 & = \beta\xi_i\kappa\\ & = -\Delta \mathfrak{v}+ \frac{\mathfrak{v}|\nabla\mathfrak{v}|^2} {1+\epsilon\mathfrak{v}|\nabla\mathfrak{ v}|^2}\\ & \leq -\Delta \mathfrak{v}+\mathfrak{v}|\nabla\mathfrak{v}|^2 \quad\text{in }\Omega, \end{align*} and\mathfrak{u}=0<\epsilon=\mathfrak{v}$on$\partial\Omega$. Therefore \cite[Theorem 10.1]{gt} implies$\mathfrak{v}\geq\mathfrak{u}\geq \xi_i\kappa. From \begin{align*} -\Delta\mathcal{T}_\epsilon(\xi_i\kappa) + \frac{\mathcal{T}_\epsilon(\xi_i\kappa)| \nabla\mathcal{T}_\epsilon(\xi_i\kappa)|^2} {1+\epsilon\mathcal{T}_\epsilon(\xi_i\kappa) |\nabla\mathcal{T}_\epsilon(\xi_i\kappa)|^2 } &= (\mathcal{T}_\epsilon(\xi_i\kappa))^{-p} +f(\xi_i\kappa)\\ & \geq \beta\xi_i\kappa \\ & = -\Delta\mathfrak{v}+ \frac{\mathfrak{v}|\nabla \mathfrak{v}|^2}{1+\epsilon\mathfrak{v}|\nabla \mathfrak{v}|^2} \quad\text{in }\Omega, \end{align*} and\mathcal{T}_\epsilon(\xi_i\kappa)=\epsilon=\mathfrak{v}$on$\partial\Omega$implies$\mathcal{T}_\epsilon(\xi_i\kappa)\geq \mathfrak{v}\geq \mathfrak{u} \geq \xi_i\kappa$. It follows that$\mathcal{T}_\epsilon[\xi_i\kappa,\eta_i]\subset [\xi_i\kappa,\eta_i]$, and so there exists a fixed point$u_{i,\epsilon}$of$\mathcal{T}_\epsilon$in$[\xi_i\kappa,\eta_i]$for$\epsilon$small enough. By a compensated compactness method, the Murat's lemma'', the Fatou lemma technique'' of Freshe (see \cite[Theorem 3.4]{s}) and Lemma \ref{adel}, letting$u_i=\lim_{\epsilon\searrow 0}u_{i,\epsilon}$, we have$u_i\in[\xi_i\kappa,\eta_i]$belongs to$H^{1,2}_0(\Omega)$. Moreover,$u_i$solves problem ($\ref{alice}$). \end{proof} \begin{proof}[Proof of Theorem \ref{blowup}] Using the identity $\Delta (\frac{1}{u})=\frac{2}{u^3}|\nabla u|^2-\frac{1}{u^2}\Delta u$ If$u$solves the equation \begin{gather*} -\Delta u = u^{2-p}+f(u) \quad\text{in }\Omega, \\ u = 0 \quad\text{on } \partial\Omega. \end{gather*} Then $\Delta(\frac{1}{u})=u\frac{2}{u^4}|\nabla u|^2+u^{-p}+\frac{1}{u^2}f(u)$ Calling$z=\frac{1}{u}$, we conclude \begin{gather*} -\Delta z = \frac{2}{z}|\nabla z|^2+z^p+z^2f(\frac{1}{z}) \quad\text{in }\Omega, \\ z = \infty \quad\text{on } \partial\Omega. \end{gather*} Using the hypothesis of Theorem \ref{11}, we conclude the proof. \end{proof} \begin{proof}[Proof of Theorem \ref{supercritico}] Let us to define the function $g_0(s) = \begin{cases} s^{2-q}-s^{2-p} & \text{if }s>(\frac{q-2}{p-2})^{\frac{1}{q-p}}\\ \big(\frac{q-2}{p-2}\big)^{2-q}-\big(\frac{q-2}{p-2}\big)^{2-p} & \text{if } s\leq(\frac{q-2}{p-2})^{\frac{1}{q-p}} \end{cases}$ Using Lemma \ref{H} with$g(u)=g_0(u+\frac{1}{\epsilon})$, we can define the solution operator$z=H_{1/\epsilon}(h)$of the problem \begin{gather*} -\Delta z = g_0(z)+h \quad\text{in }\Omega, \\ z = \frac{1}{\epsilon} \quad\text{on }\partial\Omega. \end{gather*} Moreover, this operator is well defined$H_{1/\epsilon}:\{h\in L^\infty(\Omega)| h\geq0\}\to\{z\in C(\overline{\Omega})| z\geq 0\}$, it is continuous, non decreasing and compact. Therefore, we can define$z=A_{1/\epsilon}(u):\{u\in L^\infty(\Omega)| h\geq0\} \to\{z\in C(\overline{\Omega})| z\geq0\}$, the continuous, increasing and compact solution operator of the problem \begin{gather*} -\Delta z = g_0(z)+f(u) \quad\text{in }\Omega \\ z = \frac{1}{\epsilon} \quad\text{on }\partial\Omega. \end{gather*} If$\kappa$is the indicator function of the set$K$, as in the proof of Theorem \ref{11}, we deduce $A_{1/\epsilon}(\xi_i\kappa)\geq\xi_i\kappa\,.$ From$A_{1/\epsilon}(0)