\documentclass[reqno]{amsart} \usepackage{hyperref} \AtBeginDocument{{\noindent\small \emph{Electronic Journal of Differential Equations}, Vol. 2012 (2012), No. 26, pp. 1--14.\newline ISSN: 1072-6691. URL: http://ejde.math.txstate.edu or http://ejde.math.unt.edu \newline ftp ejde.math.txstate.edu} \thanks{\copyright 2012 Texas State University - San Marcos.} \vspace{9mm}} \begin{document} \title[\hfilneg EJDE-2012/26\hfil Existence of solutions] {Existence of solutions to nonlocal elliptic equations with discontinuous terms} \author[F. J. S. A. Corr\^ea, R. G. Nascimento \hfil EJDE-2012/26\hfilneg] {Francisco Julio S. A. Corr\^ea, Rubia G. Nascimento} % in alphabetical order \address{Francisco Julio S. A. Corr\^ea \newline Universidade Federal de Campina Grande \\ Unidade Acad\^emica de Matem\'atica e Estat\'istica \\ CEP:58109-970, Campina Grande-PB, Brazil} \email{fjsacorrea@gmail.com} \address{Rubia G. Nascimento \newline Faculdade de Matem\'atica \\ Universidade Federal do Par\'a \\ CEP:66075-110, Bel\'em -PA, Brazil} \email{rubia@ufpa.br} \thanks{Submitted October 12, 2011. Published February 7, 2012.} \thanks{F.J.S.A.C. was supported by grants 620150/2008-4 and 303080/2009-4 from CNPq/Brazil. \hfill\break\indent R.G.N. was supported by grant 505407/2008-6 from CNPq/Brazil} \subjclass[2000]{35A15, 35J40, 34A36} \keywords{Variational methods; elliptic problem; discontinuous nonlinearity} \begin{abstract} In this article, we study the existence of nonnegative solutions for the elliptic partial differential equation \begin{gather*} -[M(\|u\|^{p}_{1,p})]^{1,p}\Delta_{p}u = f(x,u) \quad\text{in } \Omega , \\ u = 0 \quad\text{on } \partial\Omega , \end{gather*} where $\Omega \subset \mathbb{R}^N$ is a bounded smooth domain, $f:\overline{\Omega}\times \mathbb{R}^+\to \mathbb{R}$ is a discontinuous nonlinear function. \end{abstract} \maketitle \numberwithin{equation}{section} \newtheorem{theorem}{Theorem}[section] \newtheorem{lemma}[theorem]{Lemma} \newtheorem{claim}[theorem]{Claim} \newtheorem{remark}[theorem]{Remark} \newtheorem{definition}[theorem]{Definition} \allowdisplaybreaks \section{Introduction} This article concerns the existence of solution to the elliptic problem \begin{equation}\label{eq1} \begin{gathered} -[M(\|u\|^{p}_{1,p})]^{1,p}\Delta_{p}u = f(x,u) \quad\text{in } \Omega , \\ u = 0 \quad\text{on } \partial\Omega , \end{gathered} \end{equation} where $\Omega \subset \mathbb{R}^N$ is a bounded smooth domain, $f:\overline{\Omega} \times \mathbb{R}^{+}\to \mathbb{R}$ is a discontinuous function, $M:\mathbb{R}^{+}\to \mathbb{R}$, $\mathbb{R}^{+}=[0,\infty )$, $\Delta_{p}$ is the p-Laplacian $$ \Delta_{p}u=\text{div}(|\nabla u|^{p-2}\nabla u),\quad p>1, $$ and $\|\cdot \|_{1,p}$ is the usual norm $$ \|u\|^{p}_{1,p}=\int_{\Omega}|\nabla u|^{p} $$ in the Sobolev space $W^{1,p}_0(\Omega )$. The interest of the mathematicians on the so called nonlocal problems like \eqref{eq1} (nonlocal because of the presence of the term $M(\|u\|^{p}_{1,p})$, has increased because they represent a variety of relevant physical and engineering situations and requires a nontrivial apparatus to solve them. More precisely, we study the existence of nonnegative nontrivial solutions of the problem \begin{equation} \label{P-lambda} \begin{gathered} -[M(\|u\|^{p}_{1,p})]^{p-1}\Delta_{p} u = \lambda H(u-a)u^{q}+ h(x)u^{s} \quad \text{in } \Omega ,\\ u = 0 \quad\text{on } \partial\Omega , \end{gathered} \end{equation} where $M:\mathbb{R}^{+}\to \mathbb{R}$ is a continuous function, $10$ and $\lambda >0$ are real parameters, $h:\Omega \to (0, \infty )$ is a positive measurable function, $h\in L^{\infty}(\Omega )$ and $H$ is the Heaviside function $$ H(t)=\begin{cases} 0 & \text{if } t\leq 0,\\ 1 & \text{if } t>0. \end{cases} $$ We assume the following conditions: \begin{itemize} \item[(H1)] There exist $m_1,t_1>0$ such that $M(t)\geq m_1$ if $0\leq t \leq t_1$; \item[(H2)] There exist $m_2,t_2>0$ such that $00$ for all $t>0$. \end{itemize} Problems involving discontinuous nonlinearity appears in several physical situations. Among these, we may cite electrical phenomena, plasma physics, free boundary value problems, etc. The reader may consult Ambrosetti-Calahorrano-Dobarro \cite{acd}, Ambrosetti-Turner \cite{at}, Arcoya-Calahorrano \cite{ac}, Arcoya-Diaz-Tello \cite{adt}, Badialle \cite{badi1}, \cite{badi2}, and the references therein. Some physical problems are related to discontinuous surface $$ \Gamma_a(u)=\{x\in \Omega ; u(x)=a \} $$ which causes difficulties in analyzing this kind of problems. When $M\equiv 1$, \eqref{P-lambda} becomes a local problem, and has been widely studied. In particular, Alves-Bertone \cite{alves-bertone} and Alves-Bertone-Goncalves \cite{alves-bertone-valdo} use variants of the Mountain Pass Theorem (for locally Lipschitz functionals), the Ekeland Variational Principle, and the Subdifferential Calculus. On the other hand, after the work by Alves, Correa and Matofu \cite{acma} several papers appeared dealing with nonlocal problems with variational techniques; see for example \cite{fcme,fgio1,fgio2,fgio3,fcrn,ma,mao,perera-zhang,ricceri}. This article maybe the the first study of a nonlocal problem with variational techniques for a non-differentiable functional. We consider the non-differentiable functional $$ I_{\lambda , a}(u)=\frac{1}{p}\widehat{M}(\|u\|^{p}_{1,p})-\lambda \psi (u)-\frac{1}{s+1}\int_{\Omega}h(x)(u^{+})^{s+1}, $$ defined on $W^{1,p}_0(\Omega )$, where \begin{gather*} \widehat{M}(t)=\int^{t}_0[M(s)]^{p-1}ds, \quad \psi (u)=\int_{\Omega}F(u), \\ F(u)=\int^{u}_0f(t)dt, \quad f(t)=H(t-a)(t^{+})^{q}, \quad t^{+}=\max \{0,t\}. \end{gather*} By a solution to \eqref{P-lambda} we mean a function $u\in W^{1,p}_0(\Omega)\cap W^{p,\frac{q+1}{q}}(\Omega)$ satisfying \begin{equation}\label{sol} -[M(\|u\|^{p}_{1,p})]^{p-1}\Delta_{p}u(x)-h(x)u(x)^{s}\in \lambda[\underline{f}(u(x)),\overline{f}(u(x))] \quad \text{a.e. in } \Omega, \end{equation} where $f(t)=H(t-a)(t^{+})^{q}$ is nondecreasing, $$ \overline{f}(t)=\lim_{\delta\to 0^{+}}f(t+\delta ), \quad \underline{f}(t)=\lim_{\delta\to 0^{+}}f(t-\delta ). $$ Let us consider the level set $$ \Gamma_a(u)=\{x\in \Omega ; u(x)=a\}. $$ Note that if $|\Gamma_a(u)|=0$, then $u$ satisfies \begin{equation}\label{sol2} -[M(\|u\|^{p}_{1,p})]^{p-1}\Delta_{p}u(x)=\lambda H(u(x)-a)u(x)^{q}+h(x)u(x)^{s} \quad \text{a.e. in } \Omega . \end{equation} Clearly, a solution in the sense of \eqref{sol} is also a solution in the sense of \eqref{sol2}. The main result of this work is as follows. \begin{theorem}\label{teo1} Suppose that $M$ satisfies assumptions {\rm(H1)--(H4)} and $h\in L^{\infty}(\Omega )$. Then there are $\lambda^{\ast}>0$ and $a^{\ast}>0$ such that for $\lambda \in (0,\lambda^{\ast})$ and $a\in (0,a^{\ast})$, Problem \eqref{P-lambda} possesses at least two nontrivial and nonnegative solutions $u_1$ and $u_2$ satisfying \begin{itemize} \item[(i)] $|\Gamma_a(u_i)|=0$, $i=1,2$; \item[(ii)] $I_{\lambda,a}(u_2)<0a\}|>0$, $i=1,2$. \end{itemize} \end{theorem} \section{Abstract Framework} In this section we establish some basic results on critical point theory for locally Lipschitz functionals, developed by Chang \cite{chang} based on Convex Analysis and on the Subdifferential Calculus by Clarke \cite{clarke}. \begin{definition} \rm Let $X$ be a Banach space. We say that the functional $I:X\to \mathbb{R}$ is locally Lipschitz ($I\in {\rm Lip}_{\rm loc}(X,\mathbb{R})$) if, given $u\in X$, there exist a neighborhood $V\equiv V_{u}\subset X, u\in V$ and a constant $k\equiv k_{V}>0$ such that $$ |I(v_2)-I(v_1)|\leq k\|v_2-v_1\|, \quad v_1,v_2\in V. $$ \end{definition} \begin{definition}\rm The directional derivative of the locally Lipschitz functional $I:X\to \mathbb{R}$ at $u\in X$ in the direction $v\in X$ is defined by $$ I^{0}(u;v)=\limsup_{h\to 0,\lambda \to 0^{+}}\frac{I(u+h+\lambda v)-I(u+h)}{\lambda}. $$ \end{definition} We may prove that $I^{0}(u;v)$ is subadictive and positively homogeneous; that is, $$ I^{0}(u;v_1+v_2)\leq I^{0}(u;v_1)+I^{0}(u;v_2) $$ and $$ I^{0}(u;\lambda v)=\lambda I^{0}(u;v) $$ for all $u,v_1,v_2\in X$ and $\lambda >0$. Using these properties, it follows that $$ |I^{0}(u,v_1)-I^{0}(u,v_2)|\leq K|v_1-v_2|, \quad K\equiv K_{u}>0. $$ Consequently, $I^{0}(u, \cdot )$ is continuous and, because it is also convex, we may consider its subdifferential at $z\in X$ which is given by $$ \partial I^{0}(u;z)=\{\mu \in X^{\ast}; I^{0}(u;v)\geq I^{0}(u;z) +\langle \mu , v-z\rangle , v\in X\}, $$ where $X^{\ast}$ is the topological dual of $X$ and $\langle\cdot,\cdot\rangle$ is the duality pairing between $X^{\ast}$ and $X$. \begin{definition}\rm The generalized gradient of $I\in {\rm Lip}_{\rm loc}(X,\mathbb{R})$ at $u\in X$ is defined as being the set $$ \partial I(u)=\{\mu \in X^{\ast}; I^{0}(u;v)\geq \langle\mu , v\rangle , \text{ for all } v\in X\}. $$ \end{definition} Since $I^{0}(u;0)=0$, it follows that $\partial I(u)=\partial I^{0}(u;0)$. Furthermore, for all $v\in X$, we have $$ I^{0}(u;v)=\max\{\langle\mu , v\rangle ;v\in \partial I(u)\}. $$ An important property of the generalized gradient is as follows: if $u\in X$, then $\partial I(u)$ is a convex, nonempty and $\text{weak}^{\ast}-\text{compact}$. In particular, there is $\omega \in \partial I(u)$ such that $$ m(u)=\min\{\|\omega \|_{\ast};\omega \in \partial I(u)\}. $$ The reader may find more properties on this subject in \cite{clarke} and \cite{grossinho}. We note that $\partial I(u)=\{I'(u)\}$ when $I\in C^{1}(X,\mathbb{R})$. \begin{definition}\rm A sequence $(u_n)\subset X$ is a Palais-Smale sequence at the level $c$ $((PS)_{c})$, if $$ I(u_n)\to c, \quad m(u_n)\to 0. $$ \end{definition} \begin{definition}\rm We say that the functional $I\in {\rm Lip}_{\rm loc}(X,\mathbb{R})$ satisfies the Palais-Smale condition at the level $c$, if any $(PS)_{c}$ sequence possesses a strongly convergent subsequence. \end{definition} The proof of our main result rests heavily on the following version of the Mountain Pass Theorem for ${\rm Lip}_{\rm loc}$ functionals whose proof may be found in Chang \cite{chang}. Its proof uses an appropriate version of the Deformation Lemma whose proof is found in \cite{grossinho}. We say that $u_0\in X$ is a critical point of $I$ if $0\in \partial I(u_0)$. Clearly, every local minimum (maximum) point is a critical point. \begin{theorem}\label{Mountais Pass}\rm Let $I\in {\rm Lip}_{\rm loc}(X,\mathbb{R})$ be a functional such that $I(0)=0$ and suppose that: \begin{itemize} \item[(i)] There are constants $\eta >0$ and $\rho >0$ such that $I(u)>\eta$, for $\|u\|=\rho , u\in X$; \item[(ii)] There is $e\in X$, with $\|e\|>\rho$, such that $I(e)<0$. \end{itemize} If, in addition, $I$ satisfies the Palais-Smale condition at the level $$ c=\inf_{\gamma \in \Gamma}\max_{t\in [0,1]}I(\gamma (t)),$$ where $$ \Gamma =\{\gamma \in C([0,1],X), \gamma (0)=0 \text{ and } \gamma (1)=e\}, $$ then $c>0$ is a critical value of $I$. \end{theorem} \section{Preliminary Results} In this section we establish some results for proving the main result of this article. \begin{lemma}\label{lem1} There is $\lambda^{\ast}>0$ such that the functional $I_{\lambda ,a}$ satisfies the geometric conditions $(i)$ and $(ii)$ of Mountain Pass Theorem \ref{Mountais Pass}, for all $a>0$ \end{lemma} \begin{proof} (i) Let us consider $u\in W^{1,p}_0(\Omega )$ such that $0<\|u\|_{1,p}^{p}=r 0$ sufficiently small, there exists $$ \lambda^{\ast}=\frac{m^{p-1}_1(q+1)}{4pC_1r^{(q+1)-p}} $$ such that $I_{\lambda ,a}(u)\geq \eta >0$ for $\|u\|_{1,p}=r$, for all $\lambda \in (0,\lambda^{\ast})$, and for some $\eta > 0$. (ii) Let $\varphi$ be a function in $C^{\infty}_0(\Omega ), \varphi >0$ in $\Omega$. Hence, for $t>0$ with $\|t\varphi \|^{p}_{1,p}>t_2$, it follows from (H2) and recalling that $H \geq 0$, we have $$ I_{\lambda ,a}(t\varphi )\leq \frac{1}{p}m^{p-1}_2t^{p}\|\varphi \|^{p}_{1,p}-\frac{1}{s+1}\int_{\Omega}h(x)(u^{+})^{s+1}+\widetilde{C} $$ and because $10$ such that $|I_{\lambda ,a}(u_n)|\leq C_2$ for all $n\in \mathbb{N}$. Because $$ \frac{1}{s+1}[-\langle \omega_n,u_n\rangle]\leq \frac{1}{s+1}|\langle \omega_n,u_n\rangle |\leq \frac{1}{s+1}\|\omega_n\|_{\ast}\|u_n\|_{1,p}\leq C_3\|u_n\|_{1,p}, $$ by \eqref{eq3}, we obtain \begin{align*} I_{\lambda ,a}(u_n)-\frac{1}{s+1}\langle \omega_n+\lambda\rho_n,u_n\rangle & \leq C_2+\frac{1}{s+1}|\langle \omega_n,u_n\rangle| +\frac{1}{s+1}|\langle \lambda \rho_n,u_n\rangle|\\ &\leq C_2+C_3\|u_n\|_{1,p}+\frac{\lambda}{s+1}|\langle \rho_n,u_n \rangle|. \end{align*} Since $(\rho_n)\subset \partial\psi (u_n)$, $\langle \rho_n,v\rangle \leq \psi^{0}(u_n,v)$ for all $v\in W^{1,p}_0(\Omega )$. Using arguments found in \cite{chang} and \cite{clarke}, we can show that \begin{align*} \langle \rho_n,u_n \rangle &\leq \psi^{0}(u_n;u_n)\\ &\leq \int_{\{u_n<0\}}\underline{f}(u_n)u_n +\int_{\{u_n>0\}}\overline{f}(u_n)u_n\\ & \leq \int_{\{u_n>0\}}|u_n|^{q+1} \leq \int_{\Omega}|u_n|^{q+1}\\ &\leq C_{4}\|u_n\|^{q+1}_{1,p}. \end{align*} and \begin{equation}\label{eq4} I_{\lambda,a}(u_n)-\frac{1}{s+1}l\angle \omega_n+\lambda\rho_n,u_n\rangle\leq C_2+C_3\|u_n\|_{1,p}+C_{5}\|u_n\|^{q+1}_{1,p}. \end{equation} From inequalities \eqref{eq3}-\eqref{eq4}, it follows that \begin{align*} &\frac{1}{p}\widehat{M}(\|u_n\|^{p}_{1,p}) -\frac{1}{s+1}[M(\|u_n\|^{p}_{1,p})]^{p-1}\|u_n\|^{p}_{1,p}\\ & \leq C_2+C_3\|u_n\|_{1,p}+C_{5}\|u_n\|^{q+1}_{1,p} + \lambda\int_{\Omega}\int^{u_n}_0(t^{+})^{q}\,dt\,dx. \end{align*} Using the Sobolev immersions, \begin{equation} \begin{split} &\frac{1}{p}\int^{\|u_n\|^{p}_{1,p}}_0[M(s)]^{p-1}ds -\frac{1}{s+1}[M(\|u_n\|^{p}_{1,p})]^{p-1}\|u_n\|^{p}_{1,p} \\ & \leq C_2+C_3\|u_n\|_{1,p} + C_7\|u_n\|^{q+1}_{1,p}. \end{split} \end{equation} From the continuity of $M^{p-1}$ on $[0,\|u_n\|^{p}_{1,p}]$ and in view of Mean Value Theorem for integrals, there exists $\xi_n$, $0<\xi_n<\|u_n\|^{p}_{1,p}$, such that $$ \frac{1}{p}\int^{\|u_n\|^{p}_{1,p}}_0[M(s)]^{p-1}ds=[M(\xi_n)]^{p-1}\|u_n\|^{p}_{1,p}. $$ Since $M$ is a nonincreasing function, we obtain \begin{equation} [M(\xi_n)]^{p-1}\geq [M(\|u_n\|^{p}_{1,p})]^{p-1} \end{equation} and \begin{align*} &\frac{1}{p}[M(\|u_n\|_{1,p}^p)]^{p-1}\|u_n\|_{1,p}^p- \frac{1}{s+1}[M(\|u_n\|_{1,p}^p)]^{p-1}\|u_n\|_{1,p}^p\\ &\leq C_2 + C_3 \|u_n\|_{1,p} + C_7\|u_n\|_{1,p}^{q+1}, \end{align*} from which $$ \big(\frac{1}{p}-\frac{1}{s+1}\big)[M(\|u_n\|^{p}_{1,p})]^{p-1}\|u_n\|^{(p-1)-q}_{1,p} \leq \frac{C_2}{\|u_n\|^{q+1}_{1,p}}+\frac{C_3}{\|u_n\|^{q}_{1,p}}+C_7. $$ From (H3), we conclude that $(u_n)$ is bounded. Showing the claim. \medskip Since $\{u_n\}$ is $(P.S)$ sequence, using standard arguments, we can assume, without loss of generality, that $u_n\geq 0$ for all $x \in \Omega$. As $(u_n)$ is bounded and using the reflexivity of $W_0^{1,p}(\Omega)$ there are $u_1 \in W_0^{1,p}(\Omega)$ and $\vartheta \in \mathbb{R}$ such that, up to a subsequence, $$ \|u_n\|^{p}_{1,p} \to \vartheta^{p},\quad u_n \rightharpoonup u_1 \quad \text{in } W^{1,p}_0(\Omega ). $$ Consequently, $u_1\geq 0$. Let us now show that $u_n\to u_1$ in $W^{1,p}_0(\Omega )$. From the continuity of $M$ and $\|u_n\|_{1,p}^{p}\to \vartheta^{p}$, we obtain $M(\|u_n\|_{1,p}^{p})\to M(\vartheta^{p})$ and because $M(\vartheta^{p})>0$, there is $K>0$ such that $$ M(\|u_n\|_{1,p}^{p})\geq K>0 \quad \text{for $n$ large enough}. $$ Using the well known Simon inequality (see \cite{peral}), we obtain \begin{align*} & K^{p-1}C_{p} \int_{\Omega}|\nabla u_n - \nabla u_1|^{p}\\ &\leq [M(\|u_n\|^p)]^{p-1}\int_{\Omega}\langle |\nabla u_n|^{p-2}\nabla u_n - |\nabla u_1|^{p-2}\nabla u_1, \nabla u_n - \nabla u_1\rangle \end{align*} and so \begin{align*} &K^{p-1}C_{p}\int_{\Omega}|\nabla u_n-\nabla u_1|^{p}\\ & \leq [M(\|u_n\|^{p}_{1,p})]^{p-1}\|u_n\|^{p}_{1,p} -[M(\|u_n\|^{p}_{1,p})]^{p-1}\int_{\Omega}|\nabla u_n|^{p-2}\nabla u_n\nabla u_1\\ &\quad - [M(\|u_n\|^{p}_{1,p})]^{p-1}\int_{\Omega}|\nabla u_1|^{p-2}\nabla u_1\nabla u_n + [M(\|u_n\|^{p}_{1,p})]^{p-1}\|u_1\|^{p}_{1,p}. \end{align*} Noticing that $$ [M(\|u_n\|^{p}_{1,p})]^{p-1}\|u_1\|^{p}_{1,p}-[M(\|u_n\|^{p}_{1,p})]^{p-1} \int_{\Omega}|\nabla u_1|^{p-2}\nabla u_1\nabla u_n=o_n(1), $$ we obtain \begin{align*} &K^{p-1}C_{p}\int_{\Omega}|\nabla u_n-\nabla u_1|^{p}\\ & \leq [M(\|u_n\|^{p}_{1,p})]^{p-1}\|u_n\|^{p}_{1,p} -[M(\|u_n\|^{p}_{1,p})]^{p-1}\int_{\Omega}|\nabla u_n|^{p-2}\nabla u_n\nabla u_1 + o_n(1). \end{align*} We point out that \begin{gather*} \int_{\Omega}h(x)u^{s+1}_n \to \int_{\Omega}h(x)u_1^{s+1}\\ \int_{\Omega}h(x)u^{s}_1u_1 \to \int_{\Omega}h(x)u^{s+1}_1\\ \begin{aligned} |\langle \rho_n,u_1\rangle -\langle\rho_n,u_n\rangle| & = |\langle \rho_n,u_n\rangle -\langle \rho_n,u_1\rangle|\\ & = |\langle \rho_n, u_n-u_1\rangle |\\ & \leq \|\rho_n\|_{\ast}|u_n-u_1|_{p} \to 0, \end{aligned} \end{gather*} From \eqref{eq2} and boundedness of $\{u_n\}$, it follows that $(\rho_n)$ is bounded in $(W^{1,p}_0(\Omega ))^{\ast}$, and since $u_n\to u_1$ in $L^{\alpha}(\Omega )$, $1\le \alpha 0 \end{equation} which implies that $u_1\not\equiv 0$. Since $\{u_n\}$ is (P.S) sequence, there are $\{\omega_n\} \subset \partial I_{\lambda,a}(u_n)$ and $\{\rho_n\}\subset \partial \Psi(u_n)$ verifying $\|\omega_n\|_{*} \to 0$ and \begin{equation} \label{convergencia} \langle \omega_n,\phi\rangle = [M(\|u_n\|^{p}_{1,p})]^{p-1}\int_{\Omega}|\nabla u_n|^{p-2}\nabla u_n\nabla \varphi -\int_{\Omega}h(x)u_n^{s}-\lambda\int_{\Omega}\rho_n\varphi \end{equation} where, by Lemma \ref{lem2}, \begin{equation}\label{eq5} \rho_n \in [\underline{f}(u_n(x)),\overline{f}(u_n(x))] \quad\text{a.e in } \Omega. \end{equation} The boundedness of $\{u_n\}$ combined with \eqref{eq5} implies in particular that $\{\rho_n\}$ is bounded in $L^{\frac{q+1}{q}}(\Omega)$. Thus, there is $\rho_0\in L^{\frac{q+1}{q}}(\Omega)$ such that, up to a subsequence $\rho_n \rightharpoonup \rho_0$ in $L^{\frac{q+1}{q}}(\Omega)$, or equivalently \begin{equation} \label{rho} \int_{\Omega}\rho_n\varphi \to \int_{\Omega}\rho_0\varphi, \quad \forall \varphi \in L^{q+1}(\Omega) \end{equation} By \cite[Lemma 3.3]{alves-nascimento}, $\rho_0(x) \in [\underline{f}(u_1(x)),\overline{f}(u_1(x))]$ in $\Omega$. Letting $n\to + \infty$ in \eqref{convergencia}, and using \eqref{rho}, we obtain the identity \[ [M(\|u_1\|^{p}_{1,p})]^{p-1}\int_{\Omega}|\nabla u_1|^{p-2}\nabla u_1\nabla \varphi -\int_{\Omega}h(x)u_1^{s}\varphi=\lambda\int_{\Omega}\rho_0\varphi. \] Showing that $u_1$ is a weak solution of the problem \begin{gather*} -[M(\|u_1\|^{p}_{1,p})]^{p-1}\Delta_{p}u_1-hu_1^{s} =\lambda \rho_0 \quad \text{in } \Omega \\ u_1 > 0 \quad \text{in } \Omega. \end{gather*} By elliptic regularity, once $\rho_0\in L^{\frac{q+1}{q}}(\Omega)$, it follows that $u_1\in W^{p,\frac{q+1}{q}}(\Omega)$ and $$ -[M(\|u_1\|^{p}_{1,p})]^{p-1}\Delta_{p}u_1(x)-h(x)u_1^{s}(x)=\lambda \rho_0(x) \quad\text{a.e. in } \Omega $$ which implies $$ -[M(\|u_1\|^{p}_{1,p})]^{p-1}\Delta_{p}u_1(x)-h(x)u_1^{s}(x)\in \lambda [\underline{f}(u_1(x)),\overline{f}(u_1(x))] \quad \text{a.e. in } \Omega . $$ This shows that $u_1$ is a solution of \eqref{P-lambda}. \noindent{\bf Proof of (i).} Let us show that $|\Gamma_a(u_1)|=0$, where $\Gamma_a(u_1)=\{x\in \Omega ;u_1(x)=a\}$. Let us suppose, by contradiction, that $|\Gamma_a(u_1)|>0$. From the Morrey-Stampacchia Theorem \cite{morrey}, $-\Delta_{p}u_1(x)=0$ a.e. in $\Gamma_a(u_1)$, and so \begin{equation}\label{eq6} -[M(\|u_1\|^{p}_{1,p})]^{p-1}\Delta_{p}u_1(x)=0 \quad \text{a.e. in } \Gamma_a(u_1). \end{equation} Since $u_1$ is a critical point, it follows that $$ -[M(\|u_1\|^{p}_{1,p})]^{p-1}\Delta_{p}u_1(x)- h(x)u^{s}_1(x)\in \lambda [\underline{f}(u_1(x)), \overline{f}(u_1(x))] \quad \text{a.e. in } \Omega . $$ From \eqref{eq6}, we obtain $$ - h(x)u^{s}_1(x)\in \lambda [\underline{f}(u_1(x)), \overline{f}(u_1(x))] \quad \text{a.e. in } \Omega . $$ As $0\leq H(u_1-a)(u^{+}_1)^{q}\leq (u^{+}_1)^{q}$, it follows from the definition of $\underline{f}(u_1(x)), \overline{f}(u_1(x))$ and from the fact that $u_1\geq 0$, that $$ 0\leq \underline{f}(u_1(x))\leq \overline{f}(u_1(x))\leq (u_1)^{q}. $$ Thus, $- h(x)u^{s}_1(x)\in [0, \lambda a^{q}]$ which is impossible. Hence $|\Gamma_a(u_1)|=0$. \subsection*{Second Solution (Ekeland Variational Principle)} By Lemma \ref{lem1}, we obtain $I_{\lambda , a}(u)\geq \eta$ for $0<\|u\|_{1,p}=\rho$ and so $I_{\lambda ,a}(u) $ is bounded from below on $\overline{B}_{r}$ and so there is $ \inf_{\overline{B}_{r}}I_{\lambda ,a}(u)$. \begin{claim} There is $a^{\ast}>0$ such that for $a\in (0, a^{\ast})$ we have $\inf _{B_{r}(0)}I_{\lambda ,a}(u)<0$. \end{claim} Indeed, let us define the auxiliary function $$ \varphi_{\tau}(x)=\begin{cases} \tau a & |x|\leq \frac{1}{2},\\ 2\tau a(1-|x|) & \frac{1}{2}\leq |x|\leq 1,\\ 0 & |x|\geq 1. \end{cases} $$ where $\tau >(q+2)^{1/(q+1)}$. We point out that $\varphi_{\tau}\in W^{1,p}_0(\Omega )$ and $$ \|\varphi_{\tau}\|^{p}_{1,p}=\int_{\Omega}|\nabla \varphi_{\tau}|^{p}=\int_{\{|x|\leq 1\}}|\nabla \varphi_{\tau}|^{p}=(2\tau a)^{p}\int_{\{|x|\leq 1\}}|\nabla |x|)|^{p}. $$ Using the change of variables $x=\omega_{\tau}$, with $\omega \in S^{N-1}$ implies $dx=\tau^{N-1}ds(\omega )d\tau$. Then we obtain $|x|=\tau$. Hence, $\frac{\partial \tau}{\partial x_i}=\frac{x_i}{\tau}$ which implies $|\nabla r|=1$. In this way, $$ \|\varphi_{\tau}\|^{p}_{1,p} \leq (2\tau a)^{p}\alpha_{N}, $$ where $\alpha_{N}$ is the volume of the unit ball. If $a<\frac{\sqrt[p]{r}}{2\tau \sqrt[p]{\alpha_{N}}}=:a_1$, where $r$ is given by the geometry of the Mountain Pass Theorem, $\varphi_{\tau}\in B_{r}$. On the other hand, \begin{align*} \int_{\Omega}\int^{\varphi_{r}}_0H(t-a)(t^{+})^{q}\,dt\,dx &\geq \int_{\{|x|\leq 1/2\}}\int^{\varphi_{r}}_0H(t-a)(t^{+})^{q}\,dt\,dx\\ & =\frac{a^{q+1}(\tau^{q+1}-1)}{q+1}\int_{\{|x|\leq 1/2 \}}dx; \end{align*} that is, $$ \int_{\Omega}\int^{\varphi}_0H(t-a)(t^{+})^{q}\,dt\,dx\geq \frac{a^{q+1}(\tau^{q+1}-1)}{q+1}C_1 $$ and so $$ I_{\lambda ,a}(\varphi_{\tau})=\frac{1}{p}\widehat{M} (\|\varphi_{\tau}\|^{p}_{1,p})-\lambda\int_{\Omega} \int^{\varphi_{\tau}}_0H(t-a)(t^{+})^{q}\,dt\,dx -\frac{t^{s+1}}{s+1}\int_{\Omega}h(x)(\varphi^{+}_{\tau})^{q+1}. $$ As $\frac{t^{s+1}}{s+1}\int_{\Omega}h(x)(\varphi^{+}_{\tau})^{q+1}\geq 0$, it follows that $$ I_{\lambda ,a}(\varphi_{\tau})\geq \frac{1}{p}\widehat{M}(\|\varphi_{\tau}\|^{p}_{1,p})-\frac{\lambda C_1a^{q+1}(\tau^{q+1}-1)}{q+1}. $$ Because $M$ is a continuous function on $[0,\|\varphi_{\tau}\|^{p}_{1,p}]$, \begin{align*} I_{\lambda ,a}(\varphi_{\tau}) & \leq \frac{C_2}{p}\|\varphi_{\tau}\|^{p}_{1,p}-\frac{\lambda C_1a^{q+1}(\tau^{q+1}-1)}{q+1}\\ & \leq \frac{C_2}{p}2^{p}a^{p}\tau^{p}\alpha_{N}-\frac{\lambda C_1a^{q+1}(\tau^{q+1}-1)}{q+1}\\ & =a^{q+1}\Big(\frac{C_22^{p}a^{p-(q+1)}\tau^{p}\alpha_{N}}{p} -\frac{\lambda C_1(\tau^{q+1}-1)}{q+1}\Big). \end{align*} We now point out that $$ \frac{C_22^{p}a^{p-(q+1)}\tau^{p}\alpha_{N}}{p}-\frac{\lambda C_1(\tau^{q+1}-1)}{q+1}\leq 0 \Leftrightarrow a^{p-(q+1)}\leq \frac{\lambda C_1(\tau^{q+1}-1)p}{C_22^{p}(q+1)\alpha_{N}\tau^{p}}. $$ Setting $$ a_2=\frac{1}{2} \frac{\lambda C_1(\tau^{q+1}-1)p}{C_22^{p}(q+1)\alpha_{N}\tau^{p}} $$ and taking $a^{\ast}=a^{\ast}(\lambda )=\min \{a_1,a_2\}$ it follows that $\inf_{B_{r}}I_{\lambda ,a}<0$ for $a\in (0,a^{\ast})$, which proves the claim. \medskip By the Ekeland Variational Principle, there exists $u_{\epsilon}\in B_{r}$ such that \begin{gather}\label{inf} I_{\lambda ,a}(u_{\epsilon})<\inf_{B_{r}}I_{\lambda ,a}+\epsilon,\\ \label{norma} I_{\lambda ,a}(u_{\epsilon})0$ in such a way that \[ 0< \epsilon < \inf_{\partial B_{r}}I_{\lambda ,a}-\inf_{B_{r}}I_{\lambda ,a} \] and so $u_{\epsilon} \in B_{r}$. Let $\gamma >0$ be small enough and $v\in W^{1,p}_0(\Omega )$ with $\|v\|_{1,p}<1$ so that $u_{\gamma}=u_{\epsilon}+\gamma v\in B_{r}$. From \eqref{norma} we have $$ I_{\lambda ,a}(u_{\epsilon}) < I_{\lambda ,a}(u_{\epsilon}+\gamma v)+\epsilon \gamma \|v\|_{1,p} $$ which implies $$ I_{\lambda ,a}(u_{\epsilon}+\gamma v)-I_{\lambda ,a}(u_{\epsilon})+\epsilon \gamma \|v\|_{1,p}\geq 0. $$ Consequently, $$ -\epsilon \|v\|_{1,p}\leq \frac{I_{\lambda ,a}(u_{\epsilon}+\gamma v)-I_{\lambda ,a}(u_{\epsilon})}{\gamma} $$ and so $$ -\epsilon \|v\|_{1,p}\leq \limsup_{\gamma \to 0}\frac{I_{\lambda ,a}(u_{\epsilon}+\gamma v)-I_{\lambda ,a}(u_{\epsilon})}{\gamma}\leq I^{0}_{\lambda ,a}(u_{\epsilon };v). $$ Now, since $$ I_{\lambda, a}^{0}(u ; v) = \max_{\mu \in \partial I_{\lambda , a}(u_{\epsilon})}\langle\mu , v\rangle , \quad \text{for all } u,v \in W_0^{1,p}(\Omega), $$ it follows that $$ -\epsilon\|v\|_{1,p} \leq I_{\lambda, a}^{0}(u_{\epsilon} ; v) = \max_{\omega \in \partial I_{\lambda ,a}(u_{\epsilon})}\langle \omega , v\rangle. $$ Interchanging $v$ and $-v$ we obtain $$ -\epsilon\|v\|_{1,p} \leq \max_{\omega \in \partial I_{\lambda , a}(u_{\epsilon})}\langle \omega , -v\rangle = -\min_{\omega \in \partial I_{\lambda , a}(u_{\epsilon})}\langle\omega, v\rangle. $$ Therefore, $$ \min_{\omega \in \partial I_{\lambda , a}(u_{\epsilon})}\langle\omega, v\rangle \leq \epsilon\|v\|_{1,p}, \quad \text{for all } v \in W_0^{1,p}(\Omega), $$ concluding that $$ \sup_{\|v\|_{1,p} < 1}\min_{\omega \in \partial I_{\lambda , a}(u_{\epsilon})}\langle\omega, v\rangle \leq \epsilon. $$ By Fan's Min-max theorem, we obtain $$ \min_{\omega \in \partial I_{\lambda , a}(u_{\epsilon})}\sup_{\|v\|_{1,p} < 1}\langle\omega, v\rangle \le \epsilon. $$ Which along with \eqref{inf} yields the existence of $u_n \in B_{r}$ such that $$ I_{\lambda , a}(u_n) \to \widetilde{c}, \quad m(u_n)=\min_{\omega \in \partial I_{\lambda , a}(u_n)}\|\omega\|_{*} \to 0; $$ that is, $(u_n)$ is a Palais-Smale sequence at the level $\widetilde{c}$. By lemma \ref{lem3}, there exists $u_2 \in W_0^{1,p}(\Omega)$, where, passing to a subsequence if necessary, we obtain \begin{gather} u_n \to u_2 \quad \text{in } W_0^{1,p}(\Omega),\\ \label{inf2} I_{\lambda , a}(u_2) = \widetilde{c}= \inf_{B_{r}(0)}I_{\lambda , a} < 0. \end{gather} Thus, $u_2$ is a local minimum point and consequently is a critical point of $I_{\lambda , a}$. Hence, following the same arguments made before, we have that $u_2$ is also solution of the problem \eqref{P-lambda} and using the same arguments used in (i) with $u_1$ solution, we obtain also $u_2$ satisfy (i). \noindent\textbf{Proof of (ii):} By \eqref{ptcritico} and \eqref{inf2}, it follows that $I_{\lambda , a}(u_2) < 0 < I_{\lambda , a}(u_1)$. \noindent\textbf{Proof of (iii):} We will show now that $|\{x \in \Omega; u_i(x) > a \}| > 0$, $i=1,2$. We begin with the solution $u_1$ obtained via the Mountain Pass Theorem. Suppose, by contradiction, that $u_1(x) \leq a$ a.e, in $\Omega$. So $$ \lambda\int_{\Omega}\int_0^{u_1}H(t-a)(t^{+})^{q}=0. $$ By the above equality and since $u_1$ is critical point of $I_{\lambda,a}$, we obtain $$ [M(\|u_1\|_{1,p}^p)]^{p-1}\|u_1\|_{1,p}^{p} = \int_{\Omega}hu_1^{s+1}. $$ Note that $$ \int_{\Omega}hu_1^{s+1} \leq |h|_{\infty}\int_{\Omega}u_1^{p}u_1^{s+1 - p} $$ and thus \[ [M(\|u_1\|_{1,p}^p)]^{p-1}\|u_1\|_{1,p}^{p} \leq |h|_{\infty}\int_{\Omega}u_1^{p}u_1^{s+1 - p} \leq C|h|_{\infty}a^{s+1 - p}\|u_1\|_{1,p}^p \] which implies $$ [M(\|u_1\|_{1,p}^p)]^{p-1} \leq C|h|_{\infty}a^{s+1 - p}. $$ Note that, there exists $C>0$ such that $\|u_1\| \geq C$. Hence, using (H4); that is, $M(t) > 0$ for all $ t \geq 0$, there exists $\widetilde{C} > 0$ such that $$ 0 < \widetilde{C} \leq [M(\|u_1\|_{1,p}^p)]^{p-1} \leq C|h|_{\infty}a^{s+1 - p}, $$ for all $a >0$, which is impossible. We now consider the solution $u_2$ obtained via Ekeland Variational Principle. Suppose, by contradiction, that $u_2(x) \leq a$ a.e. in $\Omega$. Thus $$ \lambda\int_{\Omega}\int_0^{u_2}H(t-a)(t^{+})q=0. $$ By the above equality and since $u_2$ is a critical point of $I_{\lambda,a}$, we obtain $$ [M(\|u_2\|_{1,p}^p)]^{p-1}\|u_2\|_{1,p}^{p} = \int_{\Omega}hu_2^{s+1}. $$ We will consider two cases: \noindent\textbf{Case (1):} If $0< \|u_2\|_{1,p}^{p} \leq t_1$, from $(H_1)$ we have \begin{eqnarray}\label{H1} M(\|u_2\|_{1,p}^{p}) \geq m_1 > 0. \end{eqnarray} Hence, \[ [M(\|u_2\|_{1,p}^p)]^{p-1}\|u_2\|_{1,p}^{p} \leq |h|_{\infty}\int_{\Omega}u_2^{p}u_2^{s+1 - p} \leq C|h|_{\infty}a^{s+1 - p}\|u_2\|_{1,p}^p; \] that is, $$ [M(\|u_2\|_{1,p}^p)]^{p-1} \leq C|h|_{\infty}a^{s+1 - p}. $$ So $[M(\|u_2\|_{1,p}^p)] \to 0$ as $a\to 0$ which cannot happen from \eqref{H1}. \noindent\textbf{Case (2):} If $\|u_2\|^p_{1,p} \geq t_1$, then $\widehat{M}(\|u_2\|_{1,p}^p) \geq \widehat{M}(t_1)>0$, because $\widehat{M}$ is increasing. Moreover, $$ 0 \leq [M(\|u_2\|_{1,p}^p)]^{p-1}\|u_2\|_{1,p}^{p}= \int_{\Omega}hu_2^{s+1} \leq |h|_{\infty}a^{s+1}. $$ Thus $[M(\|u_2\|_{1,p}^p)]^{p-1}\|u_2\|_{1,p}^{p} \to 0$ as $a \to 0$. Since \[ I_{\lambda,a}(u_2) =\widehat{M}(\|u_2\|_{1,p}^p)-\frac{1}{s+1}\int_{\Omega}hu_2^{s+1} \geq \widehat{M}(t_1) - [M(\|u_2\|_{1,p}^p)]^{p-1}\|u_2\|_{1,p}^{p}, \] we obtain $$ 0 < \widehat{M}(t_1)\leq I_{\lambda,a}(u_2) + [M(\|u_2\|_{1,p}^p)]^{p-1}\|u_2\|_{1,p}^{p}. $$ Because $I_{\lambda,a}(u_2) < 0$, we obtain $$ 0 < \widehat{M}(t_1)\leq [M(\|u_2\|_{1,p}^p)]^{p-1}\|u_2\|_{1,p}^{p}. $$ Hence, as $a\to 0$, we have $\widehat{M}(t_1)=0$ for $t_1>0$, which is an absurd. With this, we conclude the proof of the theorem. \begin{thebibliography}{00} \bibitem{alves-bertone} C .O. Alves, A. M. Bertone; \emph{A discontinuous problem involving the p-Laplacian}, Electron. J. Differential Equations, vol 2003, no. 42 (2003), 1-10. \bibitem{alves-bertone-valdo} C. O. Alves, A .M. Bertone, J. V. 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