\documentclass[reqno]{amsart} \AtBeginDocument{{\noindent\small {\em Electronic Journal of Differential Equations}, Vol. 2004(2004), No. 132, pp. 1--15.\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 2004 Texas State University - San Marcos.} \vspace{9mm}} \begin{document} \title[\hfilneg EJDE-2004/132\hfil Existence and multiplicity of positive solutions] {Existence and multiplicity of positive solutions for a singular problem associated to the p-Laplacian operator} \author[Carlos Aranda \& Tomas Godoy\hfil EJDE-2004/132\hfilneg] {Carlos Aranda, Tomas Godoy} % in alphabetical order \address{Carlos Aranda \hfill\break Departamento de Matematica de la Facultad de Ciencias, Universidad de Tarapaca, Av. General Velasquez 1775 Casilla 7-D, Arica, Chile} \email{caranda@uta.cl} \address{Tomas Godoy \hfill\break FaMAF, Universidad Nacional de Cordoba, Medina Allende y Haya de la Torre, Ciudad Universitaria, 5000 Cordoba, Argentina} \email{godoy@mate.uncor.edu} \date{} \thanks{Submitted June 18, 2004. Published November 16, 2004.} \thanks{Research partialy supported by ANPCYT, CONICET, SECYT-UNC, \hfill\break\indent Fundacion Antorchas, and Agencia Cordoba Ciencia} \subjclass[2000]{35J60, 35J65} \keywords{Singular problems; $p$-laplacian operator; \hfill\break\indent nonlinear eigenvalue problems} \begin{abstract} Consider the problem $$-\Delta_{p}u=g(u) +\lambda h(u)\quad\hbox{in }\Omega$$ with $u=0$ on the boundary, where $\lambda\in(0,\infty)$, $\Omega$ is a strictly convex bounded and $C^{2}$ domain in $\mathbb{R}^{N}$ with $N\geq2$, and $10\quad \text{in }\Omega. \end{gathered} Here$\lambda$is a nonnegative parameter,$\Delta_{p}$is the$p$-laplacian operator defined by$\Delta_{p}u:=\mathop{\rm div}(| \nabla u| ^{p-2}\nabla u) $with$10$in$\Omega$have been considered in \cite{cp} for the case where, for some$\alpha >0$and$p>0g(x,u)$and$h(x,\lambda u)$behave like$u^{-\alpha}$and$(\lambda u)^p $respectively. There, existence of positive solutions for$\lambda$nonnegative and small enough is obtained via a sub and supersolutions method and non existence of such solutions is also shown for large values of$\lambda$. From these results it seems a natural question to ask for similar results when the laplacian is replaced by the degenerated operator$\Delta_{p}$. Our aim in this paper is to study existence and (at least for the case$K=1$,$f=0$) multiplicity of positive solutions of (\ref{eeqq11}). Our approach to this problem is somewhat different from the followed in [4] and it is more in the line of fixed point theorems for nonlinear eigenvalue problems. We first study in section 2, for a nonnegative$F\in C(\overline{\Omega})$, the problem$-\Delta_{p}u=Kg(u)+F$in$\Omega$,$u=0$on$\partial\Omega$,$u>0$in$\Omega$. Lemma \ref{lem2.6} states that this problem has unique solution and Lemma \ref{lem2.10} says that the corresponding solution operator$S$for this problem, defined by$S(F):=u$is a compact, continuous and non decreasing map from$P\cup\{0\}$into$P$, where$P$is the positive cone in$C(\overline{\Omega})$. These results (Lemmas \ref{lem2.6} and \ref{lem2.10}) are suggested by the work of several authors in \cite{ag,cp,crandrabi,g,lazermck} where existence of positive solutions for this problem is obtained under different assumptions on$K$and$f$. In section 3 we consider problem (\ref{eeqq11}). We write it as$u=S(\lambda h(u)+f)$with$S$as above. The above stated properties of$S$allow us to apply a classical fixed point theorem for nonlinear eigenvalue problems to obtain in Theorem \ref{thm3.1} that for$\lambda$nonnegative and small enough there exists at least a (positive) solution of (\ref{eeqq11}) and that the solution set for this problem (i.e., the set of the pairs$(\lambda,u)$that solve it) contains an unbounded subcontinuum (i.e., an unbounded connected subset) emanating from$(0,u^{\ast})$, where$u^{\ast}$is the (unique) solution of the problem$-\Delta_{p}u=Kg(u)+f$in$\Omega$,$u=0$on$\partial\Omega$,$u>0$in$\Omega$. Concerning multiplicity of positive solutions of (\ref{eeqq11}), in section 4, Theorem \ref{thm4.6}, we prove that, if in addition, \begin{itemize} \item[(H5)]$\Omega$is a strictly convex domain in$R^{N}$\item[(H6)]$g$and$h$are locally Lipchitz on$(0,\infty) $and$[ 0,\infty) $respectively \item[(H7)]$10}h(s) /s^{p-1}>0$and$\lim_{s\to\infty}h(s) /s^{q}<\infty$for some$q\in(p-1,\frac{N(p-1) }{N-p}]$, \end{itemize} then the problem $$\label{eeqq12} \begin{gathered} -\Delta_{p}u =g(u)+\lambda h(u) \text{ in }\Omega ,\\ u =0\quad \text{on }\partial\Omega \\ u >0 \quad \text{in }\Omega \end{gathered}$$ 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. To see this in section 4 we study, for$j\in\mathbb{N}$, the problem $$\label{eeqq111tris} \begin{gathered} -\Delta_{p}u =g(u+\frac{1}{j})+\lambda h(u) \text{ in } \Omega,\\ u =0\quad \text{on }\partial\Omega \\ u >0\quad \text{in }\Omega. \end{gathered}$$ Lemma \ref{lem4.1} provides, for a given$\lambda_{0}>0$, an a priori bound for the$L^{\infty}$norm of the solutions$u$of (\ref{eeqq111tris}) corresponding to some$\lambda\geq \lambda_{0}$. On the other hand, from Theorem \ref{thm3.1} we have an unbounded subcontinuum$C_{j}$of the solution set for (\ref{eeqq111tris}) emanating now from$(0, u_{j}^{\ast})$where$u_{j}^{\ast}$is the (unique) solution of the problem$-\Delta_{p}v =g(v+\frac{1}{j})+\lambda h(v)$in$\Omega$,$v=0$on$\partial\Omega$,$v>0$in$\Omega$. Also (cf. Remark \ref{rmk3.2}, part ii))$C_{j} \subset [0,c)\times P$for some positive constant$c$. Since$C_{j}$is connected and unbounded, from these facts we obtain, for$\lambda$positive and small enough, two positive solutions of (\ref{eeqq111tris}) and then, going to the limit as$j$tends to infinity (perhaps after passing to a subsequence) we obtain two positive solutions for (\ref{eeqq111tris}). Lemmas \ref{lem4.2}, \ref{lem4.3}, \ref{lem4.5} and Remark \ref{rmk4.4} provide the necessary intermediate statements on order to do it. \section{Preliminaries} For this section, we assume that the conditions (H1), (H2), (H3) and (H4) stated at the introduction hold. Let us start with some preliminary remarks collecting some known facts about the$p$-Laplacian operator. \begin{remark} \label{rmk2.1} \rm Let us recall \cite{li,dibenedetto,vas} that for$v\in L^{\infty}(\Omega)$and$1v$somewhere and consider the non empty open set$V=\{ x\in U:u(x) >v(x)\} $. Since$-\Delta_{p}u+\Delta_{p}v\leq K(g(u) -g(v)) \leq0$in$V$and$u=v$on$\partial V$the comparison principle gives a contradiction. \end{remark} \begin{lemma} \label{lem2.3} For a nonnegative$F\in L^{\infty}(\Omega) $and for$j\in\mathbb{N}$, the problem $$\label{eeqq21} \begin{gathered} -\Delta_{p}u_{j} =Kg(u+\frac{1}{j}) +F\text{ in } \Omega,\; u_{j} \in W_{\rm loc}^{1,p}(\Omega) \cap C(\overline {\Omega}) , \\ u_{j} =0\quad \text{ on }\partial\Omega, \\ u_{j} >0\quad \text{ in }\Omega \end{gathered}$$ has a unique positive (weak) solution satisfying$u_{j}\in C^{1}(\overline{\Omega})$and$j\to\frac{1}{j}+u_{j}$is non increasing. Moreover,$u_{j}\geq c\,\delta$for some positive constant$c$independent of$j$. \end{lemma} \begin{proof} Let$g_{j}:\mathbb{R}\to\mathbb{R}$be defined by$g_{j}(s) =g(s) $for$s\geq\frac{1}{j}$and$g_{j}(s) =g(\frac{1}{j}) $for$s<\frac{1}{j}$, let$T_{j}:C(\overline{\Omega}) \to C(\overline{\Omega}) $be given by$T_{j}(v) =(-\Delta_{p}) ^{-1}(Kg(\frac{1}{j}+v) +F) $. Since for$v\in C(\overline{\Omega}) $we have $\| Kg(\frac{1}{j}+v) +F\| _{L^{\infty}( \Omega) }\| K\| _{L^{\infty}(\Omega) }g(\frac{1}{j}) +\| F\| _{L^{\infty}( \Omega) },$ it follows that$T_{j}$is a compact operator. Moreover, $0\leq T(v) \leq(-\Delta_{p}) ^{-1}\big(g( \frac{1}{j}) \| K\| _{L^{\infty}(\Omega) }+\| F\| _{L^{\infty}(\Omega) }\big) \quad \text{on }\Omega,$ and so the existence assertion of the lemma follows from the Schauder fixed point theorem (as stated in \cite[Corollary 11.2]{trudi}) applied to$T_{j}$on a closed ball (in$C(\overline{\Omega}) $) around$0$with radius large enough. If$v$and$w$are two different solutions of (\ref{eeqq21}) in$W_{\rm loc}^{1,p}(\Omega) \cap C(\overline{\Omega}) $, consider the open set$\Omega':=\{x\in\Omega:v(x) >w(x) \}$. If$\Omega'\neq\emptyset$then $$-\Delta_{p}v+\Delta_{p}w=K\big(g_{j}(\frac{1}{j}+v) -g_{j}(\frac{1}{j}+w) \big) \quad \text{in }\Omega '$$ and also$v=w$on$\partial\Omega'$, but, from our assumptions on$K$and$g$, the comparison principle gives$v\leq w$on$\Omega'$which is a contradiction. A similar contradiction is obtained if$v\frac{1}{j}+u_{j}\}$and observe that$-\Delta_{p}(\frac{1}{j+1}+u_{j+1}) +\Delta_{p}(\frac{1}{j} +u_{j}) =Kg(\frac{1}{j+1}+u_{j+1}) -Kg(\frac{1} {j}+u_{j}) \leq0$in$U$and$\frac{1}{j+1}+u_{j+1}\leq\frac{1} {j}+u_{j}$on$\partial U$, thus the comparison principle gives$U=\emptyset$. Finally,$-\Delta_{p}(u_{j}) =Kg(\frac{1}{j}+u_{j}) +F\geq Kg(1+u_{1}) $, so the strong positivity of$(-\Delta_{p}) ^{-1}$gives the last assertion of the lemma. \end{proof} \begin{remark}[Tolksdorf's estimates] \label{rmk2.4} \rm Let$\Omega'$and$\Omega''$be open subsets of$\Omega$such that$\Omega ''\subset\subset\Omega'\subset\subset\Omega$and suppose that$u\in W_{\rm loc}^{1,p}(\Omega)\cap C(\overline{\Omega})$satisfies$-\Delta_{p}u=v$on$\Omega$for some$v\in L^{\infty}(\Omega)$. Then there exist$\alpha\in(0,1) $such that$u\in C^{1,\alpha}(\overline{\Omega''}) $. Moreover, an upper bound of$\| u\| _{C^{1,\alpha}(\overline {\Omega''}) }$can be found depending only on$p$,$\Omega$,$\Omega'\Omega''$,$\| u\|_{L^{\infty}(\Omega') }$and$\| v\|_{L^{\infty}(\Omega') }$(cf. \cite[Theorem 1]{t}). \end{remark} The Tolksdorf's estimates imply the following result. \begin{remark} \label{rmk2.5} \rm Assume that the sequences$\{ F_{j}\} _{j\in\mathbb{N}}$and$\{ u_{j}\} _{j\in\mathbb{N}}$are in$L_{\rm loc}^{\infty}(\Omega) $and$W_{\rm loc}^{1,p}(\Omega)\cap C(\overline{\Omega}) $respectively with$10\quad \text{in }\Omega \end{gathered} has a unique positive solution in $W_{\rm loc}^{1,p}(\Omega) \cap C(\overline{\Omega})$ and this solution belongs to $C^{1}(\Omega) \cap C(\overline{\Omega})$. Moreover, $u\geq c\delta$ where $c$ is the positive constant given by Lemma \ref{lem2.3} and $u=\lim_{j\to\infty}u_{j}$ (in the pointwise sense) with $u_{j}$ as there. \end{lemma} \begin{proof} Let $u_{j}$ be as in Lemma \ref{lem2.3} and let $u=\lim_{j\to \infty}u_{j}$. Since $\frac{1}{j}+u_{j}\geq c\delta$ (with $c$ as there, and so independent of $j$) we have, for each subdomain $\Omega'\subset\subset\Omega$, $\| Kg(\frac{1}{j}+u_{j}) +F\| _{L^{\infty}( \Omega') }\leq\| K\| _{L^{\infty}(\Omega) }g(c\delta) +\| F\| _{L^{\infty }(\Omega) }.$ Also, $\| u_{j}\| _{L^{\infty}(\Omega') }% \leq\| \frac{1}{j}+u_{j}\| _{L^{\infty}(\Omega^{\prime }) }\leq1+\| u_{1}\| _{L^{\infty}(\Omega) }<\infty.$ After passing to a subsequence, from Remark \ref{rmk2.5} we can assume that $\{ u_{j}\}_{j\in\mathbb{N}}$ converges, in the $C^{1}$ norm, on each compact subset of $\Omega$, to a solution $u\in C^{1}(\Omega)$ of the problem $-\Delta_{p}u=Kg(u) +F$ in $\Omega$. Since (as shown in Lemma \ref{lem2.3}) $\frac{1}{j}+u_{j}$ is decreasing in $j$, we have $0\leq u\leq\frac{1}{j}+u_{j}$ for all $j$. Also, $u_{j}\in C( \overline{\Omega})$, $u_{j}=0$ on $\partial\Omega$ and so $u=0$ on $\partial\Omega$ and $u$ is continuous up to the boundary. Moreover, $\frac{1}{j}+u_{j}\geq c\delta$ gives, going to the limit, that $u\geq c\delta$. If $z\in W_{\rm loc}^{1,p}(\Omega) \cap C(\overline{\Omega})$ is another solution of (\ref{eeqq22}), consider the open set $U:=\{x\in\Omega:z(x)>u(x)\}$. From (\ref{eeqq22}) we have $-\Delta_{p}z\leq-\Delta_{p}(u)$ in $U$ and $z=u$ in $\partial U$, the comparison principle leads to $U=\emptyset$. Then $z\leq u$ in $\Omega$. Similarly we see that $u\leq z$. \end{proof} \begin{remark} \label{rmk2.7} \rm It is known \cite[section 4]{godoy} that if $m\in L^{\infty}(\Omega)$ and $| \{x\in \Omega:m(x) >0\}| >0$ then there exists a unique $\lambda=\lambda_{1}(-\Delta_{p},m,\Omega) \in( 0,\infty)$ such that the problem $-\Delta_{p}\Phi=\lambda m| \Phi| ^{p-2}\Phi$ in $\Omega$, $\Phi=0$ in $\partial\Omega$, $\Phi>0$ in $\Omega$ has a solution $\Phi\in W^{1,p}(\Omega) \cap C(\overline{\Omega})$. This solution is unique up to a multiplicative constant, belongs to $C^{1,\alpha}(\overline{\Omega })$ for some $\alpha\in(0,1)$, satisfies that $\nabla\Phi(x) \neq0$ for all $x\in\partial\Omega$ and there exists positive constants $c_{1}$ and $c_{2}$ such that $c_{1}\delta( x) \leq\Phi(x) \leq c_{2}\delta(x)$ for all $x\in\Omega$ (so $\Phi(x) >0$ for all $x\in\Omega$). For the case $m=1$ we will write $\lambda_{1}(-\Delta_{p},\Omega)$ instead of $\lambda_{1}(-\Delta_{p},m,\Omega)$. \end{remark} We recall also that if $0\leq h\in L^{\infty}(\Omega)$, $\lambda>0$ and if there exists a nonnegative and non identically zero solution $w\in W_{0}^{1,p}(\Omega)$ of the problem $-\Delta _{p}w=\lambda mw^{p-1}+h$ in $\Omega$ then $\lambda\geq\lambda_{1}( -\Delta_{p},m)$ \cite[Proposition 4.1]{godoy}. This implies the following result. \begin{remark} \label{rmk2.8} \rm Let $m\in L^{\infty}(\Omega)$ and let, as usual, $m^{+}=\max(m,0)$. Assume $m^{+}\neq0$ and let $\lambda\geq0$ such that there exists a nonnegative and non identically zero function $w\in W_{\rm loc}^{1,p}(\Omega) \cap C( \overline{\Omega})$ such that $-\Delta_{p}w\geq\lambda mw^{p-1}$ in $\Omega$. Then $\lambda\leq\lambda_{1}(-\Delta_{p},m,\Omega)$. Indeed, Let $v\in W_{0}^{1,p}(\Omega)$ be the (positive) solution of the problem $-\Delta_{p}v=\lambda mw^{p-1}$ in $\Omega$. Then $v\in C^{1}(\overline{\Omega})$ and $-\Delta_{p}w\geq -\Delta_{p}v$ in $\Omega$, $w=v$ on $\partial\Omega$, Thus the comparison principle gives $w\geq v$ in $\Omega$. Since $-\Delta_{p}v=\lambda mv^{p-1}+h$ with $h=\lambda m(w^{p-1}-v^{p-1})$, we have $0\leq h\in L^{\infty}(\Omega)$ and so Remark \ref{rmk2.7} applies to give that $\lambda\leq\lambda_{1}(-\Delta_{p},m,\Omega)$. \end{remark} \begin{remark} \label{rmk2.9} \rm This remark concerns to the behavior near the boundary of the solution of problem (\ref{eeqq22}). We will say that two functions $v_{1}$, $v_{2}:\Omega\to(0,\infty)$ are comparable if there exist positive constants $c_{1}$, $c_{2}$ such that $c_{1}v_{1}\leq v_{2}\leq c_{2}v_{1}$. Consider in Lemma \ref{lem2.6} the case $F=0$ and assume that $K$ is comparable with $\Phi^{\gamma}$ for some $\gamma\geq0$ and that $0<\liminf_{s\to0^{+}}s^{\alpha}g(s) \leq\limsup _{s\to0^{+}}s^{\alpha}g(s) <\infty$ for some $\alpha>\gamma+1$. Then the solution $u$ given there is comparable with $\Phi^{\frac{\gamma+p}{\alpha+p-1}}$ (and so with $\delta^{\frac{\gamma +p}{\alpha+p-1}}$) where $\Phi$ is a positive principal eigenfunction for $-\Delta_{p}$ in $\Omega$ with homogeneous Dirichlet boundary condition associated to the weight $m\equiv1$. Indeed, let $\beta=(\gamma +p) /(\alpha+p-1)$ and let $v=\Phi^{\beta}$. Since $0<\beta<1$ it follows that $v\in C^{1}(\Omega) \cap C( \overline{\Omega})$. A computation shows that $-\Delta_{p}v=\widetilde{K}v^{-\alpha}$ on $\Omega$, where $\widetilde{K}=\beta^{p-1}((1-\beta) (p-1) | \nabla\Phi| ^{p}+\lambda_{1}\Phi^{p}) .$ Taking into account that $0<\beta<1$, the properties of $\Phi$ stated in Remark \ref{rmk2.7} imply that $\widetilde{K}$ is comparable with $1$ and so, from our assumptions on $g$, we can choose positive constants $c$ and $c'$ such that $-\Delta_{p}(cv) =c^{p-1}\widetilde{K}v^{-\alpha}\leq g(v)$ and $-\Delta_{p}(c'v) =( c') ^{p-1}\widetilde{K}v^{-\alpha}\geq g(v)$. Let $U=\{x\in\Omega:u(x) v_{1}(x)\}$. Thus $U$ is a non empty open set and, from our assumptions on $g$ and $K$, \begin{gather*} -\Delta_{p}v_{1} =Kg(v_{1}) +F_{1}\geq Kg( v_{2}) +F_{2}=-\Delta_{p}v_{2}\quad \text{in }U,\\ v_{1} =v_{2}\quad \text{on }\partial U. \end{gather*} Then the comparison principle gives $v_{1}\geq v_{2}$ on $U$ which is a contradiction. Then $S$ is non decreasing. To see that $S$ is continuous, consider $F\in P\cup\{0\}$ and a sequence $\{F_{j}\}_{j\in\mathbb{N}}$ in $P\cup\{ 0\}$ that converges to $F$ in $C(\overline{\Omega})$. Let $M$ be an upper bound of $\{F_{j}\}_{j\in\mathbb{N}}$. Then \label{eeqq23} 00$and let$\eta=\eta(\varepsilon) >0$such that$S(M) \leq\varepsilon$on$\Omega-\Omega_{\eta}$where $$\label{eeqq24} \Omega_{\eta}:=\{x\in\Omega:\mathop{\rm dist}(x,\partial\Omega) >\eta\}.$$ We have$S(F_{j_{k}}) \leq S(M) \leq\varepsilon$on$\Omega-\Omega_{\eta}$for all$k$. Also$S(F) \leq \varepsilon$on$\Omega-\Omega_{\eta}$, thus$\| S(F_{j_{k}}) -S(F) \| _{L^{\infty}(\Omega -\Omega_{\eta}) }\leq2\varepsilon$for all$k$. On the other hand, since$\{S(F_{j_{k}}) \}$converges in$C^{1}(\overline{\Omega}_{\eta}) $to$S(F) $, for$k$large enough we have$\| S(F_{j_{k}}) -S( F) \| _{L^{\infty}(\Omega_{\eta}) }\leq \varepsilon$. Then$\{S(F_{j_{k}}) \}$converges in$C(\overline{\Omega}) $to$S(F) $. Then$S$is continuous. To prove that$S$is a compact map, consider a bounded sequence$\{ F_{j}\}$in$P\cup\{0\}$and let$M\in( 0,\infty) $be an upper bound of$\{F_{j}\}$. For$\varepsilon>0$let$\eta=\eta(\varepsilon) $be chosen as above. As before, Remark \ref{rmk2.5} gives a subsequence$\{F_{j_{k}}\} $that converges, in the$C^{1}$norm, on each compact subset of$\Omega$. Thus, for$k$and$slarge enough, $\| S(F_{j_{k}}) -S(F_{j_{s}}) \| _{C(\overline{\Omega}_{\eta}) }\leq\varepsilon$ and \begin{align*} \| S(F_{j_{k}}) -S(F_{j_{s}}) \| _{C(\Omega-\Omega_{\eta}) } & \leq\| S(F_{j_{k} }) \| _{C(\Omega-\Omega_{\eta}) }+\| S(F_{j_{s}}) \| _{C(\Omega-\Omega_{\eta})}\\ & \leq2\| S(F_{j_{s}}) \| _{C(\Omega -\Omega_{\eta}) }\leq2\varepsilon \end{align*} Then\{S(F_{j_{k}}) \}_{k\in\mathbb{N}}$is a Cauchy's sequence in$C(\overline{\Omega}) $and the compactness of$S$follows. Since for each$j$,$g(.+\frac 1j)$satisfies the assumptions made for on$g$, (i) holds for each$S_{j}$. Finally, (ii) is a direct consequence of the comparison principle and, since$S(u) =\lim_{j\to\in}S_{j}(u) $(by Lemma \ref{lem2.6}), (iii) follows from (ii). \end{proof} \section{An existence result} Our assumptions for this section are those stated at the beginning of the Section 2. Let us introduce some additional notations. Consider, for$j\in\mathbb{N}$and$\lambda\geq0$the problem $$\label{eeqq31} \begin{gathered} -\Delta_{p}u =Kg(u+\frac{1}{j}) +\lambda h(u) +f\quad \text{ in }\Omega,\; u \in W_{\rm loc}^{1,p}(\Omega) \cap C(\overline{\Omega }) , \\ u =0\quad \text{on }\partial\Omega, \\ u >0\quad \text{in }\Omega \end{gathered}$$ Let$\pi:[ 0,\infty) \times P\to[ 0,\infty) $be defined by$\pi(\lambda,u) =\lambda$and for$j$as above, let \begin{gather*} \Sigma_{j} =\{(\lambda,u) \in[ 0,\infty) \times P:u\in W_{\rm loc}^{1,p}(\Omega) \cap C(\overline {\Omega}) \text{ and$u$solves (\ref{eeqq31})}\},\\ \Lambda_{j} =\pi(\Sigma_{j}) ,\quad u_{j}^{\ast} =S_{j}(f) \end{gather*} and let$\Sigma_{\infty}$,$\Lambda_{\infty}$and$u_{\infty}^{\ast}$be the sets and the function analogously defined replacing (\ref{eeqq31}) by (\ref{eeqq11}). Finally, let$C_{j}$(respectively$C_{\infty}$) be the connected component of$\Sigma_{j}$containing$u_{j}^{\ast}$(respectively of$\Sigma_{\infty}$containing$u_{\infty}^{\ast}$). With this notation, we have the following theorem. \begin{theorem} \label{thm3.1} \begin{itemize} \item[(i)]$(\lambda,u) \in \Sigma_{\infty}$implies that$u\in C^{1}(\Omega)$. \item[(ii)] For$f\in P\cup\{0\}$it holds that$C_{\infty}$is unbounded in$[0,\infty)\times P$. \item[(iii)]$\Lambda_{\infty}$is an interval containing$0$. \item[(iv)] For$j\in N$, (i), (ii) and (iii) hold with$\Sigma_{\infty}$,$\Lambda_{\infty}$and$u_{\infty}^{\ast}$replaced by$\Sigma_{j}$,$\Lambda_{j}$and$u_{j}^{\ast}$respectively and with \eqref{eeqq31} replaced by \eqref{eeqq11}. \item[(v)] There exists$\widetilde{\lambda}>0$such that$[0,\widetilde{\lambda}] \subset\Lambda_{\infty}$and$[0,\widetilde{\lambda}] \subset\Lambda_{j}$for each$j$. \end{itemize} \end{theorem} \begin{proof} (i) is given by Lemma \ref{lem2.3}. To see (ii) and (iii), observe that (\ref{eeqq11}) is equivalent to$S(\lambda h(u)+f)=u$. Let$T:[ 0,\infty) \times(P\cup\{0\}) \to C(\overline{\Omega})$be defined by$T(\lambda,v) =S(\lambda h(v+u_{\infty}^{\ast}) +f) -u_{\infty}^{\ast}$(since$S$is non decreasing we have$T(\lambda,v) \geq0$for$v\geq0$). Lemma \ref{lem2.10} implies that$T$is a continuous, non decreasing and compact map. 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,.) $. For each$\sigma\geq1$and$\rho>0$, we have also that$T(0,u) \neq\sigma u$for$u\in P\cap\rho\partial B$, where$B$denotes the open unit ball centered at$0$in$C(\overline{\Omega})$. Since$u$solves (\ref{eeqq11}) if and only if$u=v+u_{\infty}^{\ast}$with$v$a fixed point for$T$, in \cite[Theorem 17.1]{amann}, applied to$T$, gives that$C_{\infty}$is unbounded in$[0,\infty) \times P$and that$\Lambda_{\infty}$is an interval. Thus (i), (ii) and (iii) hold for$S$and, replacing in the above argument$g$by$g(.+\frac{1}{j}) $, we see that the same is true for each$S_{j}$. To prove (v) one observes that, by Lemma \ref{lem2.3} the problem \begin{gather*} -\Delta_{p}u =Kg(1+u) +f\quad \text{in }\Omega,\; u \in W_{\rm loc}^{1,p}(\Omega) \cap C(\overline{\Omega}) \\ u =0\quad \text{on }\partial\Omega \end{gather*} has a unique solution$u=u_{1}$that belongs to$C^{1}(\overline{\Omega})$. Thus$0\in\Lambda_{1}$. Since, by ii),$C_{1}$is unbounded, it follows that there exists$\widetilde{\lambda}>0$such that$\widetilde{\lambda}\in\Lambda_{1}-\{0\}$. Thus, by (iii), for$0<\lambda<\widetilde{\lambda}$there exists a positive solution$u_{\lambda,1}$of \begin{gather*} -\Delta_{p}u_{\lambda,1} =Kg(1+u_{\lambda,1}) +\lambda h(u_{\lambda,1}) +f\quad \text{in }\Omega,\; u_{\lambda,1} \in W_{\rm loc}^{1,p}(\Omega) \cap C( \overline{\Omega}) \\ u_{\lambda,1} =0\quad \text{ on }\partial\Omega \end{gather*} Now, by Lemma \ref{lem2.10},$S(\lambda h(u_{\lambda,1}) +f) \leq S_{1}(\lambda h(1+u_{\lambda,1}) +f) $and since the operator$u\to U(u) :=S(\lambda h(u) +f) $is a positive, non decreasing, continuous and compact map. \cite[Theorem 17.1]{amann} implies that the sequence$\{U^{j}(0) \}_{j\in\mathbb{N}}$converges to a fixed point of$U$. Then$\lambda\in\Sigma_{\infty}$. Similarly, by considering$S_{j}$instead of$S$we get that$\lambda \in\Lambda_{j}$for all$j$. \end{proof} \begin{remark} \label{rmk3.2} \rm (i) If for some$\lambda_{0}>0$and$j\in\mathbb{N}$we know that an a priori estimate$\| u\| _{L^{\infty}(\Omega) }\leq c$holds for each positive solution of (\ref{eeqq31}) associated to each$\lambda\geq\lambda_{0}$then an upper bound for$\Lambda_{j}$can be given. Indeed, in this case we have $-\Delta_{p}u=g(\frac{1}{j}+u) +\lambda h(u) +f\geq\lambda c^{1-p}h(u) u^{p-1}\quad \text{in }\Omega.$ Also,$u=S_{j}(\lambda h(u) +f) \geq S(0) \geq c\delta$for some positive constant$c$(cf. Lemmas \ref{lem2.10} and \ref{lem2.6}), then$h(u) \geq h(c\delta) $and so, by Remark \ref{rmk2.8},$\lambda$does not exceed the principal eigenvalue for$-\Delta_{p}$associated to the weight function$c^{1-p}h(c\delta) $. (ii) On the other hand if$\inf_{s>0}\frac{h(s) }{s^{p-1}}>0$a similar result holds. Indeed, in this case from (\ref{eeqq31}) we have$-\Delta_{p}u\geq\lambda\inf_{s>0}\frac{h(s) }{s^{p-1}}u^{p-1}$in$\Omega$,$u=0$on$\partial\Omega$and so, again by Remark \ref{rmk2.8},$\lambda\inf_{s>0}\frac{h(s) }{s^{p-1}}\leq\lambda_{1}(-\Delta_{p},\Omega)$. \end{remark} The following proposition gives some additional information about the regularity of the solutions of (\ref{eeqq11}). \begin{proposition} \label{prop3.3} Assume that$\sup_{s>0}s^{\alpha}g(s) <\infty$for some$\alpha\in[0,\frac{2p-1}{p-1})$. Then$u\in W_{0}^{1,p}(\Omega)$for all positive weak solution$u\in W_{\rm loc}^{1,p}(\Omega)\cap C(\overline{\Omega})$of \eqref{eeqq11} with$\lambda>0$. \end{proposition} \begin{proof} We have $$\label{eeqq32} \int_{\Omega}| \nabla u| ^{p-2}\langle \nabla u,\nabla \varphi\rangle =\int_{\Omega}(Kg(u) +\lambda h(u) +f) \varphi$$ for all$\varphi\in C_{c}^{\infty}(\Omega)$and so, since$u\in W_{\rm loc}^{1,p}(\Omega)$this equality holds also for all$\varphi\in W_{0}^{1,p}(\Omega)$such that$\operatorname{\rm supp}\varphi\subset\Omega$. For$\varepsilon>0$, let$u_{\varepsilon}:=\max(u,\varepsilon) -\varepsilon$. Since$u\in C(\overline{\Omega}) $and$u=0$on$\partial\Omega$, we have$\operatorname*{supp}u_{\varepsilon}\subset\Omega$. So we can take$\varphi=u_{\varepsilon}as test function in (\ref{eeqq11}) to obtain \label{eeqq33} \begin{aligned} \int_{\Omega}\chi_{\{u>\varepsilon\}}| \nabla u|^{p} &=\int_{\Omega}(\lambda h(u) +Kg(u) +f) u_{\varepsilon}\\ &\leq \int_{u>\varepsilon}(\lambda h(u) +Kg(u) +f) u\\ &\leq\int_{\Omega}(\lambda h(u)+Kg(u) +f) u \end{aligned} We claim that the last integral is finite. Indeed, it is enough to show thatug(u) \in L^{1}(\Omega)$and clearly this holds if$\alpha\leq1$. Suppose now$\alpha>1. We have \label{eeqq34} \begin{aligned} -\Delta_{p}u & =\lambda h(u) +Kg(u)+f \\ & \leq \Big (\big(\lambda h(\| u\| _{L^{\infty }(\Omega) }) +\| f\| _{L^{\infty}( \Omega) }\big) \| u\| _{L^{\infty}( \Omega) }^{\alpha}+c_{1}\| K\| _{L^{\infty}( \Omega) }\Big) u^{-\alpha}\\ &=cu^{-\alpha} \end{aligned} wherec=c_{\lambda,u}=(\lambda h(\| u\| _{L^{\infty}(\Omega) }+\| f\| _{L^{\infty}( \Omega) }) ) \| u\| _{L^{\infty}( \Omega) }+c_{1}\| K\| _{L^{\infty}(\Omega) }$. Let$w\in W_{\rm loc}^{1,p}(\Omega) \cap C(\overline{\Omega }) $be the solution (provided by Lemma \ref{lem2.6}) of the problem$-\Delta_{p}w=cw^{-\alpha}$in$\Omega$,$w=0$on$\partial\Omega$. Then, from (\ref{eeqq34}),$-\Delta_{p}u-cu^{-\alpha}\leq-\Delta_{p}w-cw^{-\alpha}$in$\Omega$, also$u=w=0$on$\partial\Omega$and so Remark \ref{rmk2.1} gives$u\leq w$in$\Omega$. On the other hand, Remark \ref{rmk2.8} gives$w\leq c'\Phi^{\frac{p}{\alpha+p-1}}$for some constant$c'$where$\Phi$is a positive principal eigenfuntion for$-\Delta_{p}$on$\Omega$. Then $$0\leq ug(u) \leq c''\Phi^{\frac{p}{\alpha+p-1}} \Phi^{-\frac{\alpha p}{\alpha+p-1}}=c'' \Phi^{-\frac{p(\alpha-1) }{\alpha+p-1}}.$$ Since$0\leq \alpha<\frac{2p-1}{p-1}$implies$\frac{p(\alpha-1) }{\alpha+p-1}<1$our claim holds. By Lemma \ref{lem2.6},$u(x) >0$for all$x\in\Omega$and so, from \eqref{eeqq34} and from the monotone convergence Theorem, we get$|\nabla u|^{p}\in L^{1}(\Omega)$. \end{proof} \section{A multiplicity result} In this section we assume that in addition to the conditions (H1) (H2) and (H3) stated at the introduction, the conditions (H5), (H6) and (H7) also hold. In \cite[Proposition 4.1]{azizieh} it is proved that if$\Omega$is a strictly convex and bounded domain with$C^{2}$boundary and if$G:\mathbb{R}% \to\mathbb{R}$is a locally Lipchitz function, then there exists$\rho>0$, with$\rho$depending only on$\Omega$and$N$, such that if$10}s^{-p+1}h(s) >0$and$0<\lim_{s\to\infty}s^{-q}h(s) <\infty$for some$q\in(p-1,\frac {N(p-1)}{N-p}]$. Then for each$\lambda_{0}>0$there exists a positive constant$c_{\lambda_{0}}$such that for all$j$and for all positive solution$u$of the problem $$\label{eeqq35} \begin{gathered} -\Delta_{p}u =g(u+\frac{1}{j}) +\lambda h(u)\quad \text{in }\Omega,\; u \in W_{\rm loc}^{1.p}(\Omega) \cap C(\overline{\Omega}) , \\ u =0\quad \text{on }\partial\Omega, \\ u >0\quad \text{in }\Omega, \end{gathered}$$ with$\lambda\geq\lambda_{0}$it holds that$\|u\| _{L^{\infty}(\Omega) }0}(h(s) /s^{p-1}) $and let$u$be a positive solution of (\ref{eeqq35}) corresponding to some$\lambda>0$. We have$-\Delta_{p}u=c\lambda u^{p-1}+H$in$\Omega$with$H:=g(u+\varepsilon) +\lambda(h(u)-cu^{p-1}) $. Since$H\in L^{\infty}(\Omega)$and$H>0$in$\Omega$, Remark \ref{rmk2.7} gives that$\lambda\leq c^{-1}\lambda_{1}( -\Delta_{p},\Omega) $. To prove the Lemma we proceed by contradiction. Assume that there exists a sequence$\{u_{n},\lambda_{n},\varepsilon_{n}\}_{n\in \mathbb{N}}$such that$j_{n}\in\mathbb{N}$,$\lambda_{n}\geq\lambda_{n}$, with$u_{n}$satisfying (\ref{eeqq35}) for$\lambda=\lambda_{n}$and such that$\| u_{n}\| _{L^{\infty}(\Omega) }\geq n$and let$G_{n}:\mathbb{R}\to\mathbb{R}$be defined by $G_{n}(s)=\begin{cases} g(s+\frac{1}{j_{n}}) +\lambda _{n}h(s) & \text{for }s>0,\\ g(\frac{1}{j_{n}}) +\lambda_{n}h(0) & \text{for }s\leq 0. \end{cases}$ Thus each$G_{n}$is locally Lipchitz and so, by \cite[Proposition 4.1]{azizieh}, there exists$x_{n}\in\Omega$such that$\| u_{n}\|_{L^{\infty}(\Omega) }=u_{n}(x_{n}) $and$\mathop{\rm dist}(x_{n},\partial\Omega) \geq\rho$with$\rho$as described at the beginning of the section. Let$\alpha_{n}=\|u_{n}\| _{L^{\infty}(\Omega) }$and let$\Omega_{n}=\alpha_{n}^{k}(\Omega-x_{n}) $where$\Omega_{n}:=\{x-x_{n}:x\in\Omega\}$and$k=\frac{q-p+1}{p}$. Observe that, since$q>p-1$, we have$k>0$. Let$w_{n}:\Omega_{n}\to\mathbb{R}$be defined by $w_{n}(y) =\frac{1}{\alpha_{n}}u_{n}\big(\frac{1}{\alpha _{n}^{k}}y+x_{n}\big) .$ Lemma \ref{lem2.3} applied to$F:=\lambda_{n}h(u_{n}) \in C( \overline{\Omega})$gives$u_{n}\in C^{1}(\overline{\Omega })$and so$w_{n}C^{1}(\overline{\Omega}_{n}) $. Let$v\in C^{1}(\overline{\Omega}) $be the solution of the problem \begin{gather*} -\Delta_{p}v =\lambda_{0}h(v) \quad \text{in }\Omega,\\ v =0\quad \text{on }\partial\Omega. \end{gather*} We have, for some positive constant$c_{1}$that$v\geq c_{1}\delta$in$\Omega$. Since$-\Delta_{p}u_{n}\geq\lambda_{0}h(u_{n}) =-\Delta_{p}v$in$\Omega$and$u_{n}=v$on$\partial\Omega$we get that$u_{n}\geq c_{1}\delta$and so$w_{n}(y) \neq0$for$y\in \Omega_{n}$. A computation gives that $$\label{eeqq36} \begin{gathered} -\Delta_{p}w_{n}(y) =\frac{1}{\alpha_{n}^{q}}g(\alpha_{n} w_{n}+\frac{1}{j_{n}})+\lambda_{n}w_{n}^{q}\frac{h(\alpha_{n} w_{n}) }{(\alpha_{n}w_{n}) ^{q}}\quad \text{in }\Omega_{n}\\ w_{n}=0\quad \text{on }\partial\Omega_{n}. \end{gathered}$$ For$r>0$, let$\overline{B}_{r}(0) $be the closed ball in$\mathbb{R}^{n}$centered at$0$with radius$r$. Since (by our contradiction hypothesis)$\lim_{n\to\infty}\alpha_{n}^{k}=\infty$, by our choice of$x_{n}$there exists$n_{0}=n_{0}(r) $such that$\overline {B}_{r}(0) \subset\Omega_{n}$for all$n\geq n_{0}$. For$c_{1}$as above and for$n$large enough we have $u_{n}(\frac{1}{\alpha_{n}^{k}}y+x_{n}) \geq c_{1}\delta( \frac{1}{\alpha_{n}^{k}}y+x_{n}) \geq c_{1}\delta(\frac{\rho}% {2})$ for all$y\in\Omega_{n}$. Then (recalling that, by Remark \ref{rmk3.2},$\lambda _{n}\leq c_{2}^{-1}\lambda_{1}(-\Delta_{p},\Omega) $with$c_{2}=\inf_{s>0}(h(s) /s^{p-1}) $) we get that, for$y\in\overline{B}_{r}(0) ,% \begin{align*} 0 & \leq\alpha_{n}^{-q}1g(\alpha_{n}w_{n}(y) +\frac{1}{j_{n} })+\lambda_{n}h(\alpha_{n}w_{n}(y) ) \\ & \leq\alpha_{n}^{-q}g(c_{1}\frac{\rho}{2})+\frac{1}{c_{2}}\lambda_{1}( -\Delta_{p},\Omega) \alpha_{n}^{q}u_{n}^{q}(\alpha_{n} ^{-k}1y+x_{n}) \frac{h(u_{n}(\alpha_{n}^{-k} y+x_{n}) ) }{u_{n}^{q}(\alpha_{n}^{-k}y+x_{n}) }\\ & \leq\alpha_{n}^{-q}g(c_{1}\frac{\rho}{2})+\frac{1}{c_{2}}\lambda_{1}( -\Delta_{p},\Omega) \sup_{s>c_{1}\frac{\rho}{2}}\frac{h( s) }{s^{q}} \end{align*} Thus, from (\ref{eeqq36}) and Remark \ref{rmk2.4}, there exist positive constantsc_{2}$and$\alpha\in(0,1)$such that$\| w_{n}\| _{C^{1,\alpha }(\overline{B}_{r/2}(0) )}\leq c_{2}$for all$n$large enough. Then we can find a subsequence$\{w_{n_{j}}\}_{j\in\mathbb{N}}$that converges in$C^{1}(\overline{B}_{r/2}(0) ) $to some nonnegative$w\in C^{1}(\overline{B}_{r}(0) ) $with$\| w\| _{L^{\infty}(\overline{B}% _{r}(0) ) }=1$. After passing to a furthermore subsequence we can assume that$\lambda_{n_{j}}$converges to some$\lambda^{\ast}\in[ \lambda_{0},\lambda_{1}(-\Delta_{p}% ,\Omega)] $. We take test functions in$C_{c}^{\infty}( \overline{B}_{r/2}(0) ) $in (\ref{eeqq36}) and taking the limit as$n$tends to$\infty$we get$-\Delta_{p}w\geq\lambda^{\ast }Bw^{q}$on$\overline{B}_{r/2}(0) $, where$B=\lim _{s\to\infty}(s^{-q}h(s) ) $. Since$w\not \equiv0$, Remark \ref{rmk2.4} gives$w(x) >0$for all$x\in\overline{B}_{r/2}(0) $and so, again taking test functions in$C_{c}^{\infty}(\overline{B}_{r/2}(0) ) $in (\ref{eeqq36}) and going to the limit as$n$tends to$\infty$, we obtain now $$-\Delta_{p}w=B\lambda^{\ast}w^{q}\text{ on }\overline{B}_{r/2}( 0) . \label{eeqq37}%$$ Taking a sequence of balls$\overline{B}_{r_{i}}(0) $with radius increasing to$\infty$and repeating the above argument on the subsequence$w_{n_{j}}$obtained in the previous step, we can obtain a Cantor diagonal subsequence, still denoted by$w_{n_{j}}$, which converges in the$C^{1}$norm on each compact set in$\mathbb{R}^{N}$to a function$\widetilde{w}\in C^{1}(\mathbb{R}^{N}) $satisfying$-\Delta_{p}\widetilde{w}=B\lambda^{\ast}\widetilde{w}^{q}$on$\mathbb{R}% ^{N}$. Since, under our assumptions on$p$and$q$, this problem has no solution \cite{mp2} we obtain a contradiction. \end{proof} \begin{lemma} \label{lem4.2} For$\sigma>\| S(0) \|_{L^{\infty}(\Omega) }$there exists$\lambda_{\sigma}$and$j_{\sigma}\in\mathbb{N}$such that for$j>j_{\sigma}$and$0\leq\lambda<\lambda_{\sigma}$, problem \eqref{eeqq35} has no positive solution$u$satisfying$\| u\| _{L^{\infty}(\Omega)}=\sigma$. \end{lemma} \begin{proof} We proceed by contradiction. Suppose that there exists a sequence \break$\{j_{n},u_{n},\lambda_{n}\}_{n=1}^{\infty}$with$\lim_{n\to\infty}j_{n}=\infty$,$\lambda_{n}>0$,$\lim_{n\to \infty}\lambda_{n}=0$, and where$u_{n}$is a positive solution of (\ref{eeqq35}) for$\lambda=\lambda_{n}$and$j=j_{n}$satisfying$\| u_{n}\| _{L^{\infty}(\Omega) }=\sigma$. Let$M>0$be an upper bound of$\{\lambda_{n}\}$. By Lemma \ref{lem2.10} we have $0 0, let \eta =\eta(\varepsilon) >0 such that S_{1}(Mh(\sigma) ) <\varepsilon on \Omega-\Omega_{\eta} (with \Omega_{\eta} defined by (\ref{eeqq24})). Proceeding as in the proof of the continuity of S in Lemma \ref{lem2.10}, we get that \| u_{n_{k}}-u\| _{L^{\infty}(\Omega_{\eta}) }<\varepsilon for k large enough and that \| u_{n_{k}}-u\| _{L^{\infty}( \Omega-\Omega_{\eta}) }<2\varepsilon for all k. Then \{u_{n_{k}}\}_{k\in\mathbb{N}} converges to u in C(\overline{\Omega}). Since \|S(0)\|_{L^{\infty}(\Omega)}<\sigma=\| u_{n}\| _{L^{\infty}(\Omega) } for all n, we get a contradiction. \end{proof} \begin{lemma} \label{lem4.3} Let \widetilde{\lambda} be as in Theorem \ref{thm3.1} and let \widetilde{u} be a positive solution of \eqref{eeqq31} corresponding to j=1 and \lambda=\widetilde{\lambda} (taking there K=1 and f=0). Then for \sigma>\| \widetilde{u}\| _{L^{\infty}(\Omega)}, 0\leq\lambda\leq\widetilde{\lambda} and j\in\mathbb{N} there exists a positive solution u of (\ref{eeqq35}) satisfying u\in C^{1}(\Omega) \cap C(\overline{\Omega}), u\geq S(0) and \| u\| _{L^{\infty}(\Omega)}\leq\sigma. \end{lemma} \begin{proof} For 0<\lambda\leq \widetilde{\lambda}, j\in\mathbb{N}, Lemma \ref{lem2.10} gives \label{eeqq38} 0\| \widetilde{u}\| _{L^{\infty}( \Omega) } there exists \lambda_{\sigma}>0 such that for 0\leq\lambda\leq\widetilde{\lambda} (\ref{eeqq12}) has a positive solution u satisfying u\in C^{1}(\Omega) \cap C(\overline{\Omega}) and \| u\| _{L^{\infty}(\Omega)}=\sigma. \item[(ii)] For \sigma>\| \widetilde{u}\| _{L^{\infty}( \Omega) } and for 0\leq\lambda\leq\widetilde{\lambda} there exists a positive solution u of (\ref{eeqq12}) satisfying u\in C^{1}( \Omega) \cap C(\overline{\Omega}) , u\geq S(0) and \| u\| _{L^{\infty}(\Omega) } \leq\sigma. \end{itemize} Indeed, the proofs are the same, replacing there S_{j} by S and g(\frac{1}{j}+.) by g whenever they appear and using Lemma \ref{lem2.6} instead of Lemma \ref{lem2.3}. \end{remark} \begin{lemma} \label{lem4.5} For \sigma\geq\| \widetilde{u}\|_{L^{\infty}(\Omega)} +\| S(0) \|_{L^{\infty}(\Omega)} we have \begin{itemize} \item[(i)] There exist \eta>0 and j_{\sigma}\in N such that for 0<\lambda<\eta and j\geq j_{\sigma}, problem \eqref{eeqq35} has a positive solution u_{j} satisfying \|u_{j}\| _{L^{\infty}(\Omega) }\geq\sigma. \item[(ii)] There exist \eta>0 such that for 0<\lambda<\eta problem (\ref{eeqq12}) has a positive solution u satisfying \| u\|_{L^{\infty}(\Omega)}\geq\sigma. \end{itemize} \end{lemma} \begin{proof} Let \Sigma_{j}, C_{j}, u_{j}^{\ast}, and \widetilde {\lambda} be as in Theorem \ref{thm3.1} and let \widetilde{u} be as in Lemma \ref{lem4.3}. Let \sigma, j_{\sigma}, \lambda_{\sigma} be as in Lemma \ref{lem4.2} and let \eta=\min(\lambda_{\sigma,}\widetilde{\lambda}) . For 0<\lambda<\eta and j\geq j_{\sigma} let \lambda_{0}\in( 0,\lambda) and let c_{\lambda_{0}} be the constant provided by Lemma \ref{lem4.1}. Clearly we can assume that c_{\lambda_{0}}\geq\sigma. Let O_{1}=O_{11}\cup O_{12}\cup O_{13} with \begin{gather*} O_{11} =\{(\overline{\lambda},\overline{u}) \in \Sigma_{j}:0\leq\overline{\lambda}<\lambda\text{ and }\| \overline {u}\| _{L^{\infty}(\Omega) }<\sigma\},\\ O_{12} =\{(\overline{\lambda},\overline{u}) \in \Sigma_{j}:\lambda<\overline{\lambda}<\lambda_{1}(-\Delta_{p}% ,\Omega) \text{ and }\| \overline{u}\| _{L^{\infty}( \Omega) }\sigma\}\,.$ Suppose, by contradiction, that there not exists a positive solution$u_{j}$of problem (\ref{eeqq35}) such that$\| u_{j}\| _{L^{\infty}(\Omega) }\geq\sigma$. Clearly, this assumption implies that$O_{1}$and$O_{2}$are disjoint relative open sets in$\Sigma_{j}$. Moreover,$\Sigma_{j}\subset O_{1}\cup O_{2}$. Indeed, suppose that$( \overline{\lambda},\overline{u}) \in\Sigma_{j}$and consider the case$\overline{\lambda}<\lambda$. Then$\overline{\lambda}<\lambda_{\sigma}$and so, by Lemma \ref{lem4.2},$\| \overline{u}\| _{L^{\infty}( \Omega) }\neq\sigma$. Thus$(\overline{\lambda},\overline {u}) \in O_{11}\cup O_{2}$In the case$\overline{\lambda}=\lambda$, taking into account that$\lambda>\lambda_{0}$and Lemma \ref{lem4.1} we get that$(\overline{\lambda},\overline{u}) \in O_{13}$and in the case$\overline{\lambda}>\lambda$, again by Lemma \ref{lem4.1} we get that$( \overline{\lambda},\overline{u}) \in O_{12}$. Then$\Sigma_{j}\subset O_{1}\cup O_{2}$. Let$C_{j}$be the unbounded connected component of$\Sigma_{j}$containing$(0,u_{j}^{\ast}) $. Thus$C_{j}\subset O_{1}\cup O_{2}$. Since, by Theorem \ref{thm3.1},$C_{j}$is unbounded and since$O_{1}$is bounded, we get that$C_{j}\cap O_{2}\neq\emptyset$. Since$C_{j}$is connected this implies that$C_{j}\cap O_{1}=\emptyset$. But, since$(0,u_{j}^{\ast}) \in C_{j}$and $\| u_{j}^{\ast}\| _{L^{\infty}(\Omega) } =\| S_{j}(0) \| _{L^{\infty}(\Omega)} \leq\| S_{1}(0) \| _{L^{\infty}(\Omega) } \leq\| S_{1}(\widetilde{\lambda}h(\widetilde{u}) ) \| _{L^{\infty}(\Omega) } =\| \widetilde {u}\| _{L^{\infty}(\Omega) }<\sigma$ and so we get that$u_{j}^{\ast}\in O_{1}$. Then$C_{j}\cap O_{1}\neq \emptyset$which is a contradiction. Thus i) holds. To prove (ii), consider for$j\geq j_{\sigma}$the solution$u_{j}$given by the part (i) and observe that $u_{j}=S_{j}(\lambda h(u_{j}) ) \geq S_{j}( 0) \geq S(0) \geq\widetilde{c}\delta$ where the constant$\widetilde{c}$is independent of$j$(these inequalities follow from Lemma \ref{lem2.10} part (i) and from Lemma \ref{lem2.6} applied with$K=1$and$f=0$). Also, by Lemma \ref{lem4.1},$u_{j}\leq c_{\frac{\lambda}{2}}$and so $$\label{eeqq39} \begin{gathered} -\Delta_{p}u_{j}=g(\frac{1}{j}+u_{j}) +\lambda h( u_{j}) \leq g(\widetilde{c}\delta) +\lambda_{1}( -\Delta_{p},m,\Omega) h(c_{\frac{\lambda}{2}}) \quad \text{in }\Omega\\ u_{j}=0\quad \text{ on }\partial\Omega \end{gathered}$$ Since$0\leq u_{j}\leq c_{\frac{\lambda}{2}}$, from Remark \ref{rmk2.5}, after passing to some subsequence, we can assume that$\{u_{j}\}_{j\in\mathbb{N}}$converges, in the$C^{1}$norm, on each compact subset of$\Omega$, to some function$u\in C^{1}(\Omega) $satisfying$u\geq\widetilde{c}\delta$(and so$u(x) >0$for$x\in\Omega$) which is a solution of the problem$-\Delta_{p}u=g(u) +\lambda h(u) $in$\Omega$. Let$w=(-\Delta_{p}) ^{-1}(h(c_{\frac{\lambda}{2}}) ) $. From(\ref{eeqq39}) we have$0\leq u_{j}\leq w$. Since$w\in C( \overline{\Omega}) $and$w=0$on$\partial\Omega$we obtain that$u$is continuous up to the boundary and that$u=0$on$\partial\Omega$. Finally, let$\rho=\rho(\Omega,N) $be as at the beginning of this section. Thus$\| u_{j}\| _{L^{\infty}(\Omega) }=\| u_{j}\| _{L^{\infty}(\overline{\Omega_{\rho}}) }$for all$j$and since$\{u_{j}\}_{j\in\mathbb{N}}$converges in$C^{1}(\overline{\Omega_{\rho}}) $to$u$we get that$\| u\| _{L^{\infty}(\Omega) }\geq\| u\| _{L^{\infty}(\overline{\Omega_{\rho}}) }% =\lim_{j\to\infty}\| u_{j}\| _{L^{\infty}( \overline{\Omega_{\rho}}) }=\lim_{j\to\infty}\| u_{j}\| _{L^{\infty}(\Omega) }\geq\sigma$and the proof of the lemma is complete. \end{proof} \begin{theorem} \label{thm4.6} Assume the conditions (H1), (H2), (H3), (H5), (H6) and (H7) are satisfied. Then \begin{itemize} \item[(i)] For$\lambda$positive and small enough there exist at least two positive solutions of the problem \eqref{eeqq12}. \item[(ii)]$\lambda=0$is a bifurcation point from infinity. \end{itemize} \end{theorem} \begin{proof} To prove (i) observe that for$\lambda$positive and small enough, taking into account Lemma \ref{lem4.5} (ii) we have a solution$u\in C(\overline{\Omega}) \cap C^{1}(\Omega) $of \eqref{eeqq12} which satisfies$\| u\| _{L^{\infty}(\Omega)} \geq\sigma+1$and, by Remark \ref{rmk4.4} (ii), a solution$v\inC(\overline{\Omega}) \cap C^{1}(\Omega) $such that$\| v\| _{L^{\infty}(\Omega) }\leq\sigma$. To prove (ii) note that, proceeding as in Remark \ref{rmk3.2}, we have$\Lambda_{\infty}\subset[ 0,c^{-1}\lambda_{1}(-\Delta_{p},\Omega)]$with$c=1/\inf_{s>0}(h(s) /s^{p-1})$. Since by Theorem \ref{thm3.1}$C_{\infty}$is unbounded, (ii) follows). \end{proof} \subsection*{Acknowledgements} We wish to thank Djairo G. De Figueiredo for his useful suggestions and to the referee for his/her careful reading and interesting comments. \begin{thebibliography}{00} \bibitem{amann} H. Amann, \emph{Fixed point equations and nonlinear eigenvalue problems in ordered Banach spaces}, SIAM Review 18 (1976), 620-709. \bibitem{ag} C. Aranda - T. Godoy, \emph{On a nonlinear Dirichlet problem with singularity along the boundary}, Diff. and Int. Equat., Vol 15, No. 11 (2002) 1313-1322. \bibitem{azizieh} C. Azizieh - P. Cl\'ement, \emph{A priori estimates and continuation methods for positive solutions of p-Laplace equations}, J. Differ Equations 179, (2002), 213-245. \bibitem{cp} M. M. Coclite - G. Palmieri, \emph{On a singular nonlinear Dirichlet problem}, Commun. Partial Differential Equations 14(10), (1989), 1315-1327 \bibitem{crandrabi} M. G. Crandall - P. H. Rabinowitz - L. Tartar, \emph{On a Dirichlet problem with a singular nonlinearity}, Commun. Partial Differential Equations, 2,2 (1977) 193-222. \bibitem{dibenedetto} E. Dibenedetto, \emph{$C^{1,\alpha}$local regularity of weak solutions of degenerate elliptic equations}, Nonlinear Anal. 7, (1998), 827-850. \bibitem{gs} B. Gidas - J Spruck, \emph{A priori bounds for positive solutions of nonlinear elliptic equations}, Commun Partial Differential Equations 6, (8), (1981), 883-901. \bibitem{trudi} D. Gilbarg - N. S. Trudinger, \textit{Elliptic partial differential equations of second order}, Springer-Verlag, Berlin-New York, (1983) Second edition. \bibitem{godoy} T. Godoy - J.P. Gossez - S. Paczka \textit{On the antimaximum principle for the$p$-laplacian with indefinite weight}, Nonlinear Anal., TMA, 51 (2002), 449-467. \bibitem{g} S. M. Gomes, \emph{On a singular nonlinear elliptic problem}, SIAM J. Math. Anal., 17 (6) (1986), 1359-1369. \bibitem{lazermck} A. C. Lazer - P. J. McKenna, \emph{On a singular nonlinear elliptic boundary value problem}, Proc. Amer. Math. Soc. 111 (1991) No. 3, 721-730. \bibitem{li} G. M. Liebermann, \emph{Boundary regularity for solutions of degenerate elliptic equations}, Nonlinear Anal., TMA, 12, 1203-1219. \bibitem{mp2} E. Mitidieri - S. I. Pohozaev, \emph{Nonexistence of positive solutions for quasilinear elliptic problems on$\mathbb{R}^{N}\$}, Proc. of the Steklov Math. Inst., 227, (1999), 186-216. \bibitem{t} P. Tolskdorf, \emph{Regularity for a more general class of quasilinear elliptic equations}, J. Diff. Equations 51, (1984) 126-150 \bibitem{vas} J. L. Vasquez, \emph{A strong maximum principle for some quasilinear elliptic equations}, Appl. Math. Optimization 12, (1984) 191-202. \end{thebibliography} \end{document}