\documentclass[reqno]{amsart} \usepackage{hyperref} \usepackage{cite} \AtBeginDocument{{\noindent\small \emph{Electronic Journal of Differential Equations}, Vol. 2015 (2015), No. 86, 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} \thanks{\copyright 2015 Texas State University - San Marcos.} \vspace{9mm}} \begin{document} \title[\hfilneg EJDE-2015/86\hfil Existence and multiplicity of solutions] {Existence and multiplicity of solutions for a prescribed mean-curvature problem with critical growth} \author[G. M. Figueiredo, M. T. O. Pimenta \hfil EJDE-2015/86\hfilneg] {Giovany M. Figueiredo, Marcos T. O. Pimenta} \address{Giovany M. Figueiredo \newline Universidade Federal do Par\'a, Faculdade de Matem\'atica, CEP: 66075-110 Bel\'em - Pa, Brazil} \email{giovany@ufpa.br} \address{Marcos T. O. Pimenta \newline Faculdade de Ci\^encias e Tecnlogia, UNESP - Univ Estadual Paulista, 19060-900, Presidente Prudente - SP, Brazil.\newline Phone (55) 18 - 3229-5625 FAX (55) 18 - 3221-8333} \email{pimenta@fct.unesp.br} \thanks{Submitted March 24, 2014. Published April 7, 2015.} \makeatletter \@namedef{subjclassname@2010}{\textup{2010} Mathematics Subject Classification} \makeatother \subjclass[2010]{35J93, 35J62, 35J20} \keywords{Prescribed mean-curvature problem; critical exponent; \hfill\break\indent variational methods} \begin{abstract} In this work we study an existence and multiplicity of solutions for the prescribed mean-curvature problem with critical growth, \begin{gather*} -\operatorname{div}\Bigl(\frac{\nabla u}{\sqrt{1+|\nabla u|^{2}}}\Bigl) = \lambda |u|^{q-2}u+ |u|^{2^*-2}u \quad \text{in }\Omega \\ u = 0 \quad \text{on } \partial \Omega, \end{gather*} where $\Omega$ is a bounded smooth domain of $\mathbb{R}^{N}$, $N\geq 3$ and $1 < q<2$. To employ variational arguments, we consider an auxiliary problem which is proved to have infinitely many solutions by genus theory. A clever estimate in the gradient of the solutions of the modified problem is necessary to recover solutions of the original problem. \end{abstract} \maketitle \numberwithin{equation}{section} \newtheorem{theorem}{Theorem}[section] \newtheorem{lemma}[theorem]{Lemma} \newtheorem{proposition}[theorem]{Proposition} \newtheorem{definition}[theorem]{Definition} \newtheorem{corollary}[theorem]{Corollary} \allowdisplaybreaks \section{Introduction} In this work we study the existence and multiplicity of solutions for quasilinear problems with nonlinearity of Br\'ezis-Nirenberg type (see \cite{BrezNiremb}) \begin{equation} \label{ePlambda} \begin{gathered} -\operatorname{div}\Bigl(\frac{\nabla u}{\sqrt{1+|\nabla u|^{2}}}\Bigl) = \lambda |u|^{q-2}u+ |u|^{2^*-2}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, $\lambda > 0$, $1 < q<2$ and $2^{*}=\frac{2N}{N-2}$. This problem has applications not just to describe a surface given by $u(x)$, whose mean curvature is described by the right hand side of \eqref{ePlambda}, but also in \emph{capillarity theory} where when the nonlinearity is replaced by $\kappa u$, the resultant equation describe the equilibrium of a liquid surface with constant surface tension in a uniform gravity field \cite[p. 262]{Trudinger}. Problems like \eqref{ePlambda} has been intensively studied over the previous decades. In \cite{Coffman}, the authors studied a related subcritical problem in which they obtained positive solutions. In \cite{Montenegro}, the authors proved the existence of infinitely many solutions for a subcritical version of \eqref{ePlambda}. In the recent work \cite{Bonheure1}, Bonheure, Derlet and Valeriola studied a purely subcritical version of \eqref{ePlambda}, where they proved the existence and multiplicity of nodal $H^1_0(\Omega)$ solutions, to sufficiently large values of $\lambda$. They overcame the difficulty in working in the $BV(\Omega)$ space, which is the natural functional space to treat \eqref{ePlambda}, by doing a truncation in the degenerate part of the mean-curvature operator in order to make possible construct a variational framework in the Sobolev space $H^1_0(\Omega)$. Nevertheless, this truncation requires sharp estimates on the gradient of the solutions, in order to prove that the solutions of the modified problem in fact are solutions of the original one. When $\Omega = \mathbb{R}^N$ and the nonlinearity is substituted by $u^q$; i.e., the Gidas-Spruck analogue for the mean-curvature operator, Ni and Serr\cite{NiSerrin1,NiSerrin2} proved that if $1 < q < \frac{N}{N-2}$ no positive solution exist, while for $q \geq 2^*-1$ there exist infinitely many solutions. In the range $\frac{N}{N-2} < q < 2^*-1$ some contributions has been given by Cl\'ement et al \cite{Clement} and by Del Pino and Guerra \cite{DelPino}, where in the latter the authors prove that many positive solutions do exist if $q < 2^*-1$ is sufficiently close to $2^*-1$. Still in the case $\Omega = \mathbb{R}^N$ but with nonlinearity given by $\lambda u + u^p$, Peletier and Serr\cite{Peletier} succeed in proving the existence of positive radial solutions when $\lambda < 0$ is small enough and $p$ is subcritical. In the case $\lambda > 0$, they stated there is no regular solution to that problem no matter how much small or large $p$ is. In this work, because of the boundedness of $\Omega$, we prove a result in a strike opposition of that \cite{Peletier}, in which we obtain the existence of infinitely many regular solutions of \eqref{ePlambda}, for small enough $\lambda > 0$. More specifically, we prove the following result. \begin{theorem}\label{Thm1} If $1 0$ such that if $0 < \lambda < \lambda^*$, $\eqref{ePlambda}$ has infinitely many solutions. Moreover, if $u_{\lambda}$ is a solution of $\eqref{ePlambda}$, then $u_{\lambda} \in H^1_0(\Omega) \cap C^{1,\alpha}(\overline{\Omega})$ with $\alpha \in (0,1)$, and $$ \lim_{\lambda\to 0}\|u_{\lambda}\| = \lim_{\lambda\to 0}\|u_{\lambda}\|_{\infty} = \lim_{\lambda\to 0}\|\nabla u_{\lambda}\|_{\infty}=0, $$ where $\|\cdot\|$ is the Sobolev norm in $H^1_0(\Omega)$. \end{theorem} Our approach follows the main ideas of Bonheure et al \cite{Bonheure1}, to make possible consider a related modified problem in $H^1_0(\Omega)$. Afterwards, to get solutions of the modified problem we apply Krasnoselskii genus theory in the same way that Azorero and Alonso \cite{GP}. Finally, we use the Moser iteration technique and a regularity result by Lewy and Stampacchia \cite{Stampacchia} to get decay in $\lambda$ of the gradient of the solutions, which will imply that the solutions of the modified problem in fact are solutions of the original one. It is worth pointing out that in fact, our result with minor modifications could be used to prove the existence of infinitely many solutions of a supercritical problem like \begin{equation} \begin{gathered} -\operatorname{div}\Bigl(\frac{\nabla u}{\sqrt{1+|\nabla u|^{2}}}\Bigl) = \lambda |u|^{q-2}u+ |u|^{s-2}u \quad \text{in } \Omega \\ u = 0 \quad \text{on } \partial \Omega, \end{gathered} \label{ebPlambda} \end{equation} where $s > 2^*$. If this were the case, its enough to proceed as in \cite{Figueiredo,Yang, Rabinowitz0}, by truncating the nonlinearity, substituting it by $$ f_K(t)=\begin{cases} \lambda |t|^{q-2} t + |t|^{s-2}t & \text{if } |t| \leq K\\ \lambda |t|^{q-2} t + K^{s-2^*}|t|^{2^*-2}t & \text{if } |t| > K \end{cases} $$ where $K > 0$. Since $f_K$ is an odd continuous function and $f_K(t)| \leq \lambda |t|^{q-1} + K^{s-2^*}|t|^{2^*-1}$ for all $t \in \mathbb{R}$, just few modifications in some of our technical results allow one to obtain infinitely many solutions of the truncated problem for $\lambda$ small enough. Hence, if $u_\lambda$ is one of such solutions, as $\lim_{\lambda \to 0}\|\nabla u_\lambda\|_\infty = 0$, for $\lambda$ small enough $\|\nabla u_\lambda\|_\infty < K$ and then $u_\lambda$ would be a solution of $(\bar{P}_\lambda)$. This article is organized as follows. In the second section we present the auxiliary problem and the variational framework. In the third one we make a brief review of Genus theory. In the fourth we prove some technical results which imply on the existence of infinitely many solutions of the auxiliary problem. The last one is dedicated to present the proof of the main result, which consists in estimates in $L^\infty(\Omega)$ norm of the gradient of solutions. \section{Auxiliary problem and variational framework} Let us consider $r \geq 0$, $\delta >0$ and a function $ \eta \in C^{1}([r,r+\delta])$ such that \begin{gather*} \eta (r)=\frac{1}{\sqrt{1+r}}, \quad \eta (r+\delta)=\frac{1}{\sqrt{1+r+\delta}}, \\ \eta' (r)=-\frac{1}{2\sqrt{(1+r)^{3}}}, \quad \eta' (r+ \delta)=0. \end{gather*} Now we define $$ a(t):= \begin{cases} \frac{1}{\sqrt{1+t}}, &\text{if } 0 \leq t \leq r, \\ \eta(t), & \text{if } r\leq t \leq r + \delta, \\ K_0=\frac{1}{\sqrt{1+r+\delta }}, &\text{if } t \geq r+\delta. \end{cases} $$ Note that $ a \in C^{1}([0, \infty))$ is decreasing and $K_0 \leq a(t)\leq 1$ for $t \in [0, \infty)$. Let us fix $r>0$ such that \begin{equation}\label{pradacaerto1} \frac{2}{2^{*}}< K_0< 1. \end{equation} The proof of the Theorem \ref{Thm1} is based on a careful study of solutions of the auxiliary problem \begin{equation} \begin{gathered} -\operatorname{div}(a(|\nabla u|^2)\nabla u) = \lambda |u|^{q-2}u+ |u|^{2^*-2}u \quad \text{in } \Omega \\ u = 0 \quad \text{on } \partial \Omega, \end{gathered} \label{eTlambda} \end{equation} We say that $u \in H^{1}_0(\Omega)$ is a weak solution \eqref{eTlambda} if it satisfies \[ \int_{\Omega}a(|\nabla u|^{2})\nabla u \nabla \phi \,dx = \lambda\int_{\Omega}|u|^{q-2}u\phi \,dx +\int_{\Omega}|u|^{2^{*}-2}u\phi \,dx, \] for all $\phi \in H^{1}_0(\Omega)$. Let us consider $H^1_0(\Omega)$ with its usual norm $\|u\|= \big(\int_\Omega|\nabla u|^2\big)^{1/2}$ and define the $C^1$-functional $I_{\lambda}: H^{1}_0(\Omega)\to \mathbb{R}$ by $$ I_{\lambda}(u) = \frac{1}{2}\int_{\Omega}A(|\nabla u|^{2}) \,dx - \frac{\lambda}{q}\int_{\Omega}|u|^{q} \,dx -\frac{1}{2^{*}}\int_{\Omega}|u|^{2^{*}} \,dx , $$ where $A(t)=\int^{t}_0a(s) \,ds$. Note that \[ I_{\lambda}'(u)\phi = \int_{\Omega}a(|\nabla u|^{2}) \nabla u \nabla \phi \,dx - \lambda\int_{\Omega}|u|^{q-2}u\phi \,dx-\int_{\Omega}|u|^{2^{*}-2}u\phi \,dx, \] for all $\phi \in H^{1}_0(\Omega)$ and then, critical points of $I_{\lambda}$ are weak solutions of \eqref{eTlambda}. To use variational methods, we first derive some results related to the Palais-Smale compactness condition. We say that a sequence $(u_n)\subset H^{1}_0(\Omega)$ is a $(PS)_{c_\lambda}$ sequence for $I_{\lambda}$ if \begin{equation}\label{***} I_{\lambda}(u_n)\to c_{\lambda} \ \text{and} \ \|I_{\lambda}'(u_n)\|_{H^{-1}(\Omega)}\to 0, \quad \text{as } n\to \infty, \end{equation} where \begin{gather*} c_{\lambda} = \inf_{\pi \in \Gamma} \max_{t \in [0,1]} I_{\lambda}(\pi(t))>0, \\ \Gamma := \{ \pi\in C([0,1],H^{1}_0(\Omega)) : \pi(0)=0, ~I_{\lambda}(\pi(1)) < 0\}. \end{gather*} If \eqref{***} implies the existence of a subsequence $(u_{n_j}) \subset (u_n)$ which converges in $H^1_0(\Omega)$, we say that $I_{\lambda}$ satisfies the Palais-Smale condition on the level $c_\lambda$. \section{Genus theory} We start by considering some basic facts on the Krasnoselskii genus theory that we will use in the proof of Theorem \ref{Thm1}. Let $E$ be a real Banach space. Let us denote by $\mathfrak{A}$ the class of all closed subsets $A\subset E\setminus \{0\}$ that are symmetric with respect to the origin, that is, $u\in A$ implies $-u\in A$. \begin{definition} \label{def3.1} \rm Let $A\in \mathfrak{A}$. The Krasnoselskii genus $\gamma(A)$ of $A$ is defined as being the least positive integer $k$ such that there is an odd mapping $\phi \in C(A,\mathbb{R}^{k})$ such that $\phi(x)\neq 0$ for all $x\in A$. When such number does not exist we set $\gamma(A)=\infty$. Furthermore, by definition, $\gamma(\emptyset)=0$. \end{definition} In the sequel we establish only the properties of the genus that will be used through this work. More informations on this subject may be found \cite{kras}. \begin{theorem} Let $E={\mathbb{R}}^{N}$ and $\partial\Omega$ be the boundary of an open, symmetric and bounded subset $\Omega \subset {\mathbb{R}}^{N}$ such that $0 \in \Omega$. Then $\gamma(\partial\Omega)=N$. \end{theorem} \begin{corollary}\label{esfera} $\gamma(S^{N-1})=N$. \end{corollary} \begin{proposition}\label{paracompletar} If $K \in \mathfrak{A}$, $0 \notin K$ and $\gamma(K) \geq 2$, then $K$ has infinitely many points. \end{proposition} \section{Technical results} The genus theory requires that the functional $I_{\lambda}$ is bounded from below. Since this is not the case, it is necessary to work with a related functional, which will be done employing some ideas contained \cite{GP}. In light of Proposition \ref{paracompletar}, it seems to be useful proving that the set of critical points of the related functional has genus greater than 2, to obtain infinitely many solutions of \eqref{eTlambda}. Let us present the way in which we truncate the function $I_\lambda$ . From \eqref{pradacaerto1} and Sobolev's embedding, we obtain \[ I_{\lambda}(u) \geq \frac{K_0}{2}\|u\|^{2} -\frac{\lambda}{q S_{q}^{q/2}}\|u\|^{q} - \frac{1}{2^{*}S^{2^{*}/2}}\|u\|^{2^{*}}=g(\|u\|^{2}), \] $S$ and $S_{q}$ are, respectively, the best constants of the Sobolev's embeddings $H^1_0(\Omega)\hookrightarrow L^{2^*}(\Omega)$ and $H^1_0(\Omega) \hookrightarrow L^{q}(\Omega)$ and \begin{equation}\label{gabriel1} g(t)=\frac{K_0}{2} t -\frac{\lambda}{q S_{q}^{q/2}}t^{q/2} - \frac{1}{2^{*}S^{2^{*}/2}}t^{2^{*}/2}. \end{equation} Hence, there exists $\tau_1>0$ such that, if $\lambda \in (0, \tau_1)$, $g$ attains its positive maximum. Let $R_0 < R_1$ the roots of $g$. We have that $R_0=R_0(\tau_1)$ and the following result holds. \begin{lemma}\label{comportamentoassimtotico100} \begin{equation}\label{comportamentoassimtotico1} R_0(\tau_1)\to 0 \quad \text{as } \lambda\to 0. \end{equation} \end{lemma} \begin{proof} From $g(R_0(\tau_1))=0$ and $g'(R_0(\tau_1))>0$, we have \begin{gather}\label{lambda1} \frac{K_0}{2}R_0(\tau_1)=\frac{\lambda}{q S_{q}^{q/2}}R_0(\tau_1)^{q/2}+ \frac{1}{2^{*}S^{2^{*}/2}}R_0(\tau_1)^{2^{*}/2},\\ \label{lambda2} \frac{K_0}{2}>\frac{\lambda}{2q S_{q}^{q/2}}R_0(\tau_1)^{(q-2)/2}+ \frac{1}{2S^{2^{*}/2}}R_0(\tau_1)^{(2^{*}-2)/2}, \end{gather} for all $\lambda\in (0,\tau_1)$. From \eqref{lambda1}, we conclude that $R_0(\tau_1)$ is bounded. Suppose that $R_0(\tau_1)\to R_0>0$ as $\lambda\to 0$. Then \begin{gather} \frac{K_0}{2}= \frac{1}{2^{*}S^{2^{*}/2}}R_0(\tau_1)^{(2^{*}-2)/2}, \\ \label{lambda2b}\\ \frac{K_0}{2}\geq \frac{1}{2S^{2^{*}/2}}R_0(\tau_1)^{(2^{*}-2)/2}, \end{gather} which is a contradiction, because $2^{*}>2$. Therefore $R_0=0$. \end{proof} We consider $\tau_1$ such that $R_0\leq r$ and we modify the functional $I_{\lambda}$ in the following way. Take $\phi \in C^{\infty}([0,+\infty))$, $0\leq \phi\leq 1$ such that $\phi(t)=1$ if $t\in [0,R_0]$ and $\phi(t)=0$ if $t\in [R_1, +\infty)$. Now, we consider the truncated functional $$ J_{\lambda}(u) = \frac{1}{2}\int_{\Omega}A(|\nabla u|^{2}) \,dx - \frac{\lambda}{q}\int_{\Omega}|u|^{q}\,dx - \phi(\|u\|^{2})\frac{1}{2^{*}}\int_{\Omega}|u|^{2^{*}}\,dx. $$ Note that $J_{\lambda} \in C^{1}(H^{1}_0(\Omega),\mathbb{R})$ and, as in \eqref{gabriel1}, $J_{\lambda}(u)\geq \overline{g}(\|u\|^{2})$, where $$ \overline{g}(t)= \frac{K_0}{2} t -\frac{\lambda}{q S_{q}^{q/2}}t^{q/2} - \phi(t)\frac{1}{2^{*}S^{2^{*}/2}}t^{2^{*}/2}. $$ Let us remark that if $\|u\|^{2}\leq R_0$, then $J_{\lambda}(u)=I_{\lambda}(u)$ and if $\|u\|^{2}\geq R_1$, then \[ J_{\lambda}(u)= \frac{1}{2}\int_{\Omega}A(|\nabla u|^{2}) \,dx - \frac{\lambda}{q}\int_{\Omega}|u|^{q} \,dx, \] which implies that $J_{\lambda}$ is coercive and hence bounded from below. Now we show that $J_{\lambda}$ satisfy the local Palais-Smale condition. For this, we need the following technical result, which is analogous of \cite[Lemma 4.2]{GP}. \begin{lemma}\label{nivelbaixo} Let $(u_n) \subset H^{1}_0(\Omega)$ be a bounded sequence such that $$ I_{\lambda}(u_n)\to c_{\lambda} \quad \text{and} \quad I_{\lambda}'(u_n)\to 0. $$ If \begin{align*} c_{\lambda}&< (\frac{K_0}{2}- \frac{1}{2^{*}})K_0^{(N-2)/2}S^{N/2} - \lambda(\frac{1}{q}-\frac{1}{2^{*}})|\Omega|^{\frac{(2^{*}-q)}{2}}\\ &\quad\times \Bigl[\frac{q}{2^{*}}\lambda\bigl(\frac{1}{q}-\frac{1}{2^{*}}\bigl)|\Omega|^{(2^{*}-q)/2^{*}} \Bigl(\bigl(\frac{K_0}{2}- \frac{1}{2^{*}})\frac{1}{S^{2^{*}/2}}\Bigl)^{-1}\Bigl]^{\frac{q}{(2^{*}-q)}} \end{align*} hold, then, up to a subsequence, $(u_n)$ is strongly convergent in $H^{1}_0(\Omega)$. \end{lemma} \begin{proof} Taking a subsequence, we may suppose that \[ |\nabla u_n|^2 \rightharpoonup |\nabla u|^2 + \sigma \ \ \text{ and } \ \ |u_n|^{2^{*}} \rightharpoonup |u|^{2^{*}} + \nu \] in the weak* sense of measures. Using the concentration compactness-principle due to Lions \cite[Lemma 2.1]{Lio2}, we obtain an at most countable index set $\Lambda$, sequences $(x_i) \subset \Omega$, $(\mu_i), (\sigma_i), (\nu_i), \subset [0,\infty)$, such that \begin{equation} \nu = \sum_{i \in \Lambda}\nu_i\delta_{x_i}, \quad \sigma\geq \sum_{i \in \Lambda}\sigma_i\delta_{x_i}, \quad S \nu_i^{2/2^{*}}\leq \sigma_i, \label{lema_infinito_eq11} \end{equation} for all $i \in\Lambda$, where $\delta_{x_i}$ is the Dirac mass at $x_i \in \Omega$. Now we claim that $\Lambda=\emptyset$. Arguing by contradiction, assume that $\Lambda\neq\emptyset$ and fix $i \in \Lambda$. Consider $\psi \in C_0^{\infty}(\Omega,[0,1])$ such that $\psi \equiv 1$ on $B_1(0)$, $\psi \equiv 0$ on $\Omega \setminus B_2(0)$ and $|\nabla \psi|_{\infty} \leq 2$. Defining $\psi_{\varrho}(x) := \psi((x-x_i)/\varrho)$ where $\varrho>0$, we have that $(\psi_{\varrho}u_n)$ is bounded. Thus $I_{\lambda}'(u_n)(\psi_{\varrho}u_n) \to 0$; that is, \begin{align*} &\int_{\Omega}a(|\nabla u_n|^{2})u_n \nabla u_n \nabla \psi_{\varrho} \,dx + \int_{\Omega} a(|\nabla u_n|^{2})\psi_{\varrho}|\nabla u_n|^2 \,dx\\ &= \lambda\int_{\Omega}|u_n|^{q}\psi_{\varrho} \,dx+ \int_{\Omega} \psi_{\varrho}|u_n|^{2^{*}} \,dx + o_n(1). \end{align*} Since $\operatorname{supp}(\psi_{\varrho}) \subset B_{2\varrho}(x_i)$, we obtain $$ \bigl|\int_{\Omega}u_n\nabla u_n \nabla \psi_{\varrho} \,dx \bigl|\leq \int_{B_{2\rho}(x_i)} |\nabla u_n||u_n \nabla \psi_{\varrho}| \,dx. $$ By H\"older inequality and the fact that the sequence $(u_n)$ is bounded in $H^{1}_0(\Omega)$ we have $$ \big|\int_{\Omega}u_n\nabla u_n \nabla \psi_{\varrho} \,dx \big| \leq C \Big( \int_{B_{2\varrho}(x_i)}|u_n \nabla \psi_{\varrho}|^{2} \,dx\Big)^{1/2}. $$ By the Dominated Convergence Theorem $\int_{B_{2\varrho}(x_i)} |u_n\nabla \psi_{\varrho}|^{2} \,dx \to 0$ as $n \to +\infty$ and $\varrho \to 0$. Thus, we obtain \begin{equation*} \lim_{\varrho\to 0}\Big[\lim_{n\to \infty}\int_{\Omega} u_n\nabla u_n \nabla \psi_{\varrho} \,dx \Big]=0. \end{equation*} Since $00$, at the point $$ \alpha_0=\Bigl[\frac{q}{2^{*}}\lambda\bigl(\frac{1}{q}-\frac{1}{2^{*}}\bigl)|\Omega|^{(2^{*}-q)/2^{*}} \Bigl(\bigl(\frac{K_0}{2}- \frac{1}{2^{*}})\frac{1}{S^{2^{*}/2}}\Bigl)^{-1}\Bigl]^{\frac{1}{(2^{*}-q)}}. $$ Then \[ c_{\lambda} \geq (\frac{K_0}{2}- \frac{1}{2^{*}})K_0^{(N-2)/2}S^{N/2}+ (\frac{K_0}{2}- \frac{1}{2^{*}})\frac{1}{S^{2^{*}/2}}\alpha^{2^{*}}_0- \lambda (\frac{1}{q}-\frac{1}{2^{*}})|\Omega|^{\frac{(2^{*}-q)}{2}}\alpha_0^{q}. \] So \[ c_{\lambda} \geq (\frac{K_0}{2}- \frac{1}{2^{*}})K_0^{(N-2)/2}S^{N/2} - \lambda (\frac{1}{q}-\frac{1}{2^{*}})|\Omega|^{\frac{(2^{*}-q)}{2}}\alpha_0^{q}. \] Thus, we conclude that \begin{align*} c_{\lambda} &\geq (\frac{K_0}{2}- \frac{1}{2^{*}})K_0^{(N-2)/2}S^{N/2} - \lambda (\frac{1}{q}-\frac{1}{2^{*}})|\Omega|^{\frac{(2^{*}-q)}{2}}\\ &\quad\times \Bigl[\frac{q}{2^{*}}\lambda\bigl(\frac{1}{q}-\frac{1}{2^{*}}\bigl)|\Omega|^{(2^{*}-q)/2^{*}} \Bigl(\bigl(\frac{K_0}{2}- \frac{1}{2^{*}})\frac{1}{S^{2^{*}/2}}\Bigl)^{-1}\Bigl]^{\frac{q}{(2^{*}-q)}}, \end{align*} which is a contradiction. Thus $\Lambda$ is empty and it follows that $u_n \to u$ in $L^{2^{*}}(\Omega)$. Thus, up to a subsequence, $$ \|u_n-u\|^{2} \leq \frac{1}{K_0}\int_{\Omega}a(|\nabla u_n|^{2})|\nabla u_n - \nabla u|^{2} = I_{\lambda}(u_n)u_n- I_{\lambda}(u_n)u+ o_n(1)=o_n(1). $$ \end{proof} By Lemma \ref{nivelbaixo} we conclude that, there exists $\tau_{2}>0$ such that, for all $\lambda \in (0,\tau_{2})$ we obtain \begin{align*} &(\frac{K_0}{2}-\frac{1}{2^{*}})K_0^{(N-2)/2}S^{N/2} - \lambda(\frac{1}{q}-\frac{1}{2^{*}})|\Omega|^{\frac{(2^{*}-q)}{2}}\\ &\times \Bigl[\frac{q}{2^{*}}\lambda \bigl(\frac{1}{q}-\frac{1}{2^{*}}\bigl)|\Omega|^{(2^{*}-q)/2^{*}} \Bigl(\bigl(\frac{K_0}{2}- \frac{1}{2^{*}})\frac{1}{S^{2^{*}/2}}\Bigl)^{-1}\Bigl]^{\frac{q}{(2^{*}-q)}} >0 \end{align*} and, hence, if $(u_n)$ is a bounded sequence such that $I_{\lambda}(u_n)\to c$, $I_{\lambda}'(u_n)\to 0$ with $c<0$, then $(u_n)$ has a convergent subsequence. \begin{lemma}\label{faltava} If $J_{\lambda}(u) <0$, then $\|u\|^{2}< R_0\leq r$ and $J_{\lambda}(v)=I_{\lambda}(v)$, for all $v$ in a small neighborhood of $u$. Moreover, $J_{\lambda}$ satisfies a local Palais-Smale condition for $c <0$. \end{lemma} \begin{proof} Since $\overline{g}(\|u\|^{2})\leq J_{\lambda}(u) <0$, then $\|u\|^{2} < R_0\leq r$. By the choice of $\tau_1$ in \eqref{comportamentoassimtotico1} we have that $J_{\lambda}(u)=I_{\lambda}(u)$. Moreover, since $J_{\lambda}$ is continuous, we conclude that $J_{\lambda}(v)=I_{\lambda}(v)$, for all $v \in B_{R_0/2}(0)$. Besides, if $(u_n)$ is a sequence such that $J_{\lambda}(u_n)\to c<0$ and $J_{\lambda}'(u_n)\to 0$ as $n\to \infty$, then for $n$ sufficiently large $I_{\lambda}(u_n)=J_{\lambda}(u_n)\to c<0$ and $I_{\lambda}'(u_n)=J_{\lambda}'(u_n)\to 0$ as $n \to \infty$. Since $J_{\lambda}$ is coercive, we obtain that $(u_n)$ is bounded in $H^{1}_0(\Omega)$. From Lemma \ref{nivelbaixo}, for all $\lambda \in (0,\tau_{2})$, we obtain \begin{align*} c<0 &< (\frac{K_0}{2}- \frac{1}{2^{*}})K_0^{(N-2)/2}S^{N/2} - \lambda (\frac{1}{q}-\frac{1}{2^{*}})|\Omega|^{\frac{(2^{*}-q)}{2}}\\ &\quad\times \Bigl[\frac{q}{2^{*}}\lambda\bigl(\frac{1}{q} -\frac{1}{2^{*}}\bigl)|\Omega|^{(2^{*}-q)/2^{*}} \Bigl(\bigl(\frac{K_0}{2}- \frac{1}{2^{*}})\frac{1}{S^{2^{*}/2}}\Bigl)^{-1}\Bigl]^{\frac{q}{(2^{*}-q)}} \end{align*} and hence, up to a subsequence $(u_n)$ is strongly convergent in $H^{1}_0(\Omega)$. \end{proof} Now, we construct an appropriate minimax sequence of negative critical values. \begin{lemma}\label{minimax} Given $k \in \mathbb{N}$, there exists $\epsilon = \epsilon(k)>0$ such that $$ \gamma(J_{\lambda}^{-\epsilon}) \geq k, $$ where $J_{\lambda}^{-\epsilon}=\{u \in H^{1}_0(\Omega): J_{\lambda}(u) \leq -\epsilon\}$. \end{lemma} \begin{proof} Consider $k \in \mathbb{N}$ and let $X_k$ be a k-dimensional subspace of $H^{1}_0(\Omega)$. Since in $X_k$ all norms are equivalent, there exists $C(k)>0$ such that $$ -C(k)\|u\|^{q}\geq - \int_{\Omega}|u|^{q} \,dx, $$ for all $u \in X_k$. We now use the inequality above to conclude that \[ J_{\lambda}(u)\leq \frac{1}{2}\| u\|^{2}-\frac{C(k)}{q}\|u\|^{q}= \| u \|^{q}\Big(\frac{1}{2}\| u\|^{2-q}-\frac{C(k)}{q}\Big). \] Considering $R>0$ sufficiently small, there exists $\epsilon=\epsilon(R)>0$ such that $$ J_{\lambda}(u)<-\epsilon < 0, $$ for all $u\in {\mathcal{S}_R}=\{u\in X_k; \| u \|=R \}$. Since $X_k$ and $\mathbb{R}^k$ are isomorphic and $\mathcal{S}_R$ and $S^{k-1}$ are homeomorphic, we conclude from Corollary \ref{esfera} that $\gamma(\mathcal{S}_R)=\gamma(S^{k-1})=k$. Moreover, once that ${\mathcal{S}_R} \subset J_{\lambda}^{-\epsilon}$ and $J_{\lambda}^{-\epsilon}$ is symmetric and closed, we have $$ k= \gamma ({\mathcal{S}_R})\leq \gamma( J_{\lambda}^{-\epsilon}). $$ \end{proof} We define now, for each $k \in \mathbb{N}$, the sets \begin{gather*} \Gamma_k=\{C \subset H: C \text{ is closed }, C=-C \text{ and } \gamma(C) \geq k\}, \\ K_c=\{u \in H: J_{\lambda}'(u)=0 \text{ and } J_{\lambda}(u)=c\} \end{gather*} and the number $$ c_k=\inf_{C\in \Gamma_k}\sup_{u \in C}J_{\lambda}(u). $$ \begin{lemma}\label{minimax1} Given $k \in \mathbb{N}$, the number $c_k$ is negative. \end{lemma} \begin{proof} From Lemma \ref{minimax}, for each $k\in \mathbb{N}$ there exists $\epsilon >0$ such that $\gamma(J_{\lambda}^{-\epsilon}) \geq k$. Moreover, $ 0 \notin J_{\lambda}^{-\epsilon}$ and $J_{\lambda}^{-\epsilon}\in \Gamma_k$. On the other hand $$ \sup_{u\in J_{\lambda}^{-\epsilon}}J_{\lambda}(u)\leq -\epsilon. $$ Hence, $$ -\infty < c_k=\inf_{C\in \Gamma_k}\sup_{u \in C}J_{\lambda}(u) \leq \sup_{u\in J_{\lambda}^{-\epsilon}}J_{\lambda}(u) \leq -\epsilon <0. $$ \end{proof} The next Lemma allows us to prove the existence of critical points of $J_{\lambda}$. \begin{lemma}\label{minimax2} If $c=c_k=c_{k+1}=\dots =c_{k+r}$ for some $r \in \mathbb{N}$, then there exists $\lambda^{*}>0$ such that $$ \gamma(K_c)\geq r+1, $$ for $\lambda \in ( 0, \lambda^{*})$. \end{lemma} \begin{proof} Since $c=c_k=c_{k+1}=\dots =c_{k+r} <0$, for $\lambda^{*}=\min\{\tau_1,\tau_{2}\}$ and for all $\lambda \in (0,\lambda^{*})$, from Lemma \ref{nivelbaixo} and Lemma \ref{minimax1}, we obtain that $K_c$ is compact. Moreover, $K_c= - K_c$. If $\gamma(K_c)\leq r$, there exists a closed and symmetric set $U$ with $ K_c\subset U$ such that $\gamma(U)= \gamma(K_c) \leq r$. Note that we can choose $U\subset J_{\lambda}^{0}$ because $c<0$. By the deformation lemma \cite{benci} we have an odd homeomorphism $ \eta: H\to H$ such that $\eta(J_{\lambda}^{c+\delta}-U)\subset J_{\lambda}^{c-\delta}$ for some $\delta > 0$ with $0<\delta < -c$. Thus, $J_{\lambda}^{c+\delta}\subset J_{\lambda}^{0}$ and by definition of $c=c_{k+r}$, there exists $A \in \Gamma_{k+r}$ such that $\sup_{u \in A} < c+\delta$; that is, $A \subset J_{\lambda}^{c+\delta}$ and \begin{equation}\label{estrela1} \eta(A-U) \subset \eta ( J_{\lambda}^{c+\delta}-U)\subset J_{\lambda}^{c-\delta}. \end{equation} But $\gamma(\overline{A-U})\geq \gamma(A)-\gamma(U) \geq k$ and $\gamma(\eta(\overline{A-U}))\geq \gamma(\overline{A-U})\geq k$. Then $\eta(\overline{A-U}) \in \Gamma_k$ which contradicts \eqref{estrela1}. \end{proof} \section{Proof of Theorem \ref{Thm1}} If $-\infty< c_1 < c_{2} < \dots < c_k< \dots <0$ with $c_i\neq c_j$, once each $c_k$ is a critical value of $J_{\lambda}$, we obtain infinitely many critical points of $J_{\lambda}$ and then, \eqref{eTlambda} has infinitely many solutions. On the other hand, if $c_k=c_{k+r}$ for some $k$ and $r$, then $c=c_k=c_{k+1}=\dots =c_{k+r}$ and from Lemma \ref{minimax2}, there exists $\lambda^{*}>0$ such that $$ \gamma(K_c)\geq r+1 \geq 2 $$ for all $\lambda \in (0,\lambda^{*})$. From Proposition \ref{paracompletar} $K_c$ has infinitely many points; that is, \eqref{eTlambda} has infinitely many solutions. Let $\lambda^{*}$ be as in Lemma \ref{minimax2} and, for $\lambda \in (0,\lambda^*)$, let $u_{\lambda}$ be a solution of \eqref{eTlambda}. Thus $J_{\lambda}(u_{\lambda})=I_{\lambda}(u_{\lambda}) <0$. Hence, \[ \|u_{\lambda}\|^{2} \leq R_0, \] which together with \eqref{comportamentoassimtotico1} implies \begin{equation}\label{C1} \lim_{\lambda \to 0}\|u_{\lambda}\|=0. \end{equation} Now we use the Moser iteration technique in order to prove that there exists a positive constant $C$, independent on $\lambda$ such that \begin{equation}\label{INFTY} \|u_{\lambda}\|_{\infty} \leq C \|u_{\lambda}\|. \end{equation} Using \eqref{INFTY} we can conclude that \begin{equation} \lim_{\lambda\to 0}\|u_{\lambda}\|_{\infty}=0. \label{Linfinity} \end{equation} To save notation, from now on we denote $u_{\lambda}$ by $u$. In what follows, we fix $R>R_1>0$, $R>1$ and take a cut-off function $\eta_R \in C^{\infty}_0(\Omega)$ such that $0 \leq \eta_R \leq 1$, $\eta_R \equiv 0$ in $B_R^{c}$, $\eta_R \equiv 1$ in $B_{R_1}$ and $|\nabla \eta_R|\leq C/R$, where $B_R\subset \Omega$ and $C>0$ is a constant. Let $h(t)= \lambda t^{q-1}+ t^{2^{*}-1}$. Thus \begin{gather*} |h(t)|\to 0 \quad \text{as } t\to 0,\\ \frac{|h(t)|}{t^{2^{*}-1}} \to 1 \quad \text{as } t\to \infty. \end{gather*} Thus, for all $\delta >0$ there is $C_{\delta}(\lambda)>0$ such that \begin{equation}\label{1moser} h(t)\leq \delta + C_{\delta}(\lambda) t^{2^{*}-1}. \end{equation} Moreover, for $\lambda \in [0,\lambda_0]$, $C_\delta(\lambda)$ can be chosen uniformly in $\lambda$ in such a way that \eqref{1moser} holds independently of $\lambda$. For each $L> 0$, define \begin{gather*} u_L(x) = \begin{cases} u(x), & \text{if } u(x) \leq L \\ L, & \text{if } u(x) \geq L, \end{cases} \\ z_L = \eta_R^{2}u_L^{2(\sigma - 1)}u \quad\text{and} \quad w_L = \eta_Ru u_L^{\sigma - 1} \end{gather*} with $\sigma > 1$ to be determined later. In the course of this proof, $C_1$, $C_2$\dots , denote constants independent of $\lambda$. Taking $z_L$ as a test function we obtain $I'_{\lambda}(u)z_L=0$. More specifically, $$ \int_{\Omega}a(|\nabla u|^{2})\nabla u \nabla z_L = \lambda \int_{\Omega}u^{q-1}z_L+\int_{\Omega}u^{2^{*}-1}z_L. $$ Hence $$ K_0\int_{\Omega}\nabla u \nabla z_L \leq \int_{\Omega}h(u)z_L. $$ By \eqref{1moser} we obtain $$ \int_{\Omega}\nabla u \nabla z_L \leq \delta K^{-1}_0\int_\Omega z_L + K^{-1}_0C_{\delta}\int_{\Omega}u^{2^{*}-1}z_L. $$ Let us fix $\delta > 0$ small enough in such a way that $$ \int_{\Omega}\nabla u \nabla z_L \leq C\int_{\Omega}u^{2^{*}-1}z_L. $$ Using $z_L$ we obtain \begin{align*} \int_{\Omega}\eta_R^{2}u_L^{2(\sigma -1)}|\nabla u|^{2} \,dx &\leq - \int_{\Omega}\eta_Ruu_L^{2(\sigma-1)}\nabla \eta_R\nabla u \,dx\\ &\quad - 2(\sigma-1)\int_{\Omega}u_L^{(2\sigma-3)}u\nabla u \nabla u_L + \int_{\Omega}\eta_R^{2}u^{2^{*}}u_L^{2(\sigma-1)}\,dx, \end{align*} and the definition of $u_L$ implies $$ -2(\sigma-1)\int_{\Omega}u_L^{(2\sigma-3)}u\nabla u \nabla u_L\leq 0. $$ Thus \[ \int_{\Omega}\eta_R^{2}u_L^{2(\sigma -1)}|\nabla u|^{2} \,dx\leq + \int_{\Omega}\eta_Ruu_L^{2(\sigma-1)}|\nabla \eta_R| |\nabla u| \,dx + \int_{\Omega}\eta_R^{2}u^{2^{*}}u_L^{2(\sigma-1)}\,dx. \] Taking $z_L$ as a test function and using \eqref{1moser}, we obtain \[ \int_{\Omega}\eta_R^{2}u_L^{2(\sigma -1)}|\nabla u|^{2} \,dx\leq C_1 \int_{\Omega}\eta_Ruu_L^{2(\sigma-1)}|\nabla \eta_R||\nabla u| \,dx + C_1 \int_{\Omega}\eta_R^{2}u^{2^{*}}u_L^{2(\sigma-1)}\,dx. \] Fixing $\widetilde{\tau}>0$ and using Young's inequality, we obtain \begin{align*} \int_{\Omega}\eta_R^{2}u_L^{2(\sigma -1)}|\nabla u|^{2} \,dx &\leq C_1 \int_{\Omega}\Big(\widetilde{\tau}\eta_R^{2}|\nabla u|^{2}+ C_{\widetilde{\tau}}u^{2}|\nabla \eta_R|^{2} \Bigl)u_L^{2(\sigma-1)} \,dx \\ &\quad + C_1 \int_{\Omega}\eta_R^{2}u^{2^{*}}u_L^{2(\sigma-1)}\,dx. \end{align*} Choosing $\widetilde{\tau} \leq 1/4$, it follows that \begin{equation}\label{2moser} \int_{\Omega}\eta_R^{2}u_L^{2(\sigma -1)}|\nabla u|^{2} \,dx\leq C_{2}\Big(\int_{\Omega}u^{2}u_L^{2(\sigma-1)}|\nabla \eta_R|^{2} \,dx + \int_{\Omega}\eta_R^{2}u^{2^{*}}u_L^{2(\sigma-1)}\,dx\Big). \end{equation} On the other hand, we obtain \begin{align*} S \|w_L\|^{2}_{L^{2^{*}}(\Omega)} & \leq \int_{\Omega} |\nabla (\eta_Ru u_L^{\sigma-1})|^2 \\ & \leq \int_{\Omega} |u|^2 u_L^{2(\sigma-1)}|\nabla \eta_R|^2 + \int_{\Omega} \eta_R^2\left|\nabla \left(u u_L^{\sigma-1}\right)\right|^2. \end{align*} But \begin{align*} \int_{\Omega} \eta_R^2\left|\nabla \left(u u_L^{\sigma-1}\right)\right|^2 &= \int_{\{|u| \leq L\}}\eta_R^2\left|\nabla \left(u u_L^{\sigma-1}\right)\right|^2 + \int_{\{|u|> L\}} \eta_R^2\left|\nabla \left(u u_L^{\sigma-1}\right)\right|^2 \\ &= \int_{\{|u| \leq L\}} \eta_R^2\left|\nabla u^{\sigma}\right|^2 + \int_{\{|u| > L\}} \eta_R^2 L^{2(\sigma-1)}\left|\nabla u\right|^2 \\ & \leq \sigma^2 \int_{\Omega}\eta_R^2 u_L^{2(\sigma-1)}|\nabla u|^2, \end{align*} and therefore $$ \|w_L\|^{2}_{L^{2^{*}}(\Omega)} \leq C_3\sigma^{2} \Big( \int_{\Omega} |u|^{2}u_L^{2(\sigma - 1)}|\nabla \eta_R|^{2} + \int_{\Omega} \eta_R^{2} u_L^{2(\sigma - 1)}|\nabla u|^{2}\Big). $$ From this and \eqref{2moser}, \begin{equation}\label{4moser} \|w_L\|^{2}_{L^{2^{*}}(\Omega)} \leq C_4\sigma^{2} \Big( \int_{\Omega} |u|^{2}u_L^{2(\sigma - 1)}|\nabla \eta_R|^{2} + \int_{\Omega} \eta_R^{2}|u|^{2^{*}}u_L^{2(\sigma - 1)}\Big), \end{equation} for all $\sigma>1$. The above expression, the properties of $\eta_R$ and $u_L \leq u$, imply \begin{equation} \label{5moser} \|w_L\|^{2}_{L^{2^{*}}(\Omega)} \leq C_4\sigma^{2} \int_{B_R} \left( |u|^{2\sigma}|\nabla \eta_R|^{2} + |u|^{2^{*}-2}|u|^{2\sigma}\right). \end{equation} If we set \begin{equation} \label{def_t} t:= \frac{2^*2^*}{2(2^*-2)}>1,\quad \alpha := \frac{2t}{t-1}< 2^*, \end{equation} we can apply H\"{o}lder's inequality with exponents $t/(t-1)$ and $t$ in \eqref{5moser} to get \begin{equation} \label{6moser} \begin{aligned} \|w_L\|^{2}_{L^{2^{*}}(\Omega)} &\leq C_4\sigma^{2} \|u\|_{L^{\sigma\alpha}(B_R)}^{2\sigma} \Big( \int_{B_R}|\nabla \eta_R|^{2t}\Big)^{1/t} \\ &\quad +C_4\sigma^{2} \|u\|_{L^{\sigma\alpha}(B_R)}^{2\sigma} \Big( \int_{B_R}|u|^{2^*(2^*/2)}\Big)^{1/t}. \end{aligned} \end{equation} Since $\eta_R$ is constant on $B_{R_1} \cup B_R^c$ and $|\nabla \eta_R| \leq C/R$, we conclude that \begin{equation} \label{61moser} \int_{B_R}|\nabla \eta_R|^{2t} = \int_{B_R\backslash B_{R_1}}|\nabla \eta_R|^{2t} \leq \frac{C_5}{R^{2t-N}} \leq C_5. \end{equation} We have used $R>1$ and $2t=\frac{2^{*}}{2}N > N$ in the last inequality. \smallskip \noindent \textbf{Claim.} There exist a constants $K >0$ independent on $\lambda$ such that, $$ \int_{\Omega}|u|^{2^*(2^*/2)}\leq K. $$ Assuming the claim is true, we can use \eqref{6moser} and \eqref{61moser} to conclude that \[ \|w_L\|^{2}_{L^{2^{*}}(\Omega)} \leq C_6\sigma^{2}\|u\|^{2\sigma}_{L^{\sigma\alpha}(B_R)}. \] Since \begin{align*} \|u_L\|^{2\sigma}_{L^{\sigma 2^{*}}(B_R)} &= \Big(\int_{B_R}u_L^{\sigma 2^{*}}\Big)^{2/2^*} \\ &\leq \Big( \int_{\Omega} \eta_R^{2^*} |u|^{2^{*}}u_L^{2^{*}(\sigma -1) }\Big)^{2/2^*} \\ &= \|w_L\|^{2}_{L^{2^{*}}(\Omega)} \leq C_6\sigma^{2}\|u\|^{2\sigma}_{L^{\sigma\alpha}(\Omega)}, \end{align*} we can apply Fatou's lemma in the variable $L$ to obtain \[ \|u\|_{L^{\sigma 2^{*}}(B_R)}\leq C_7^{1/\sigma}\sigma^{1/\sigma}\|u\|_{L^{\sigma\alpha}(\Omega)}, \] whenever $|u|^{\sigma\alpha} \in L^1(B_R)$. Here, $C_7$ is a positive constant independent on $R$. Iterating this process, for each $k\in\mathbb{N}$, it follows that $$ \|u\|_{L^{\sigma^k 2^{*}}(B_R)} \leq C_7^{\sum_{i=1}^k \sigma^{-i}} \sigma^{\sum_{i=1}^m i \sigma^{-i}} \|u\|_{L^{2^*}(\Omega)}. $$ Since $\Omega$ can be covered by a finite number of balls $B_R^j$, we have that $$ \|u\|_{L^{\sigma^k 2^{*}}(\Omega)}\leq \sum_j^{finite} \|u\|_{L^{\sigma^k 2^{*}}(B_R^j)} \leq \sum_j^{finite} C_7^{\sum_{i=1}^k \sigma^{-i}} \sigma^{\sum_{i=1}^m i \sigma^{-i}} \|u\|_{L^{2^*}(\Omega)} . $$ Since $\sigma>1$, we let $k \to \infty$ to obtain $$ \|u\|_{L^{\infty}(\Omega)} \leq K_{2}\|u\|, $$ for some $K_{2}>0$ independent on $\lambda$. It remains to prove the claim. From \eqref{4moser} \begin{equation}\label{7moser} \|w_L\|^{2}_{L^{2^{*}}(\Omega)} \leq C_{9}\sigma^{2} \Big( \int_{\Omega} |u|^{2}u_L^{2(\sigma - 1)}|\nabla \eta_R|^{2} + \int_{\Omega} \eta_R^{2}|u|^{2^{*}}u_L^{2(\sigma - 1)}\Big), \end{equation} We set $\sigma:= 2^*/2$ in \eqref{4moser} to obtain $$ \|w_L\|^{2}_{L^{2^{*}}(\Omega)} \leq C_{10}\Big( \int_{\Omega} |u|^{2}u_L^{(2^{*}-2)}|\nabla \eta_R|^{2} + \int_{B_R} \eta_R^{2}|u|^{2}u_L^{(2^{*}-2)}|u|^{(2^{*}-2)}\Big). $$ By H\"{o}lder's inequality with exponents $2^*/2$ and $2^*/(2^*-2)$ we obtain \begin{align*} \|w_L\|^{2}_{L^{2^{*}}(\Omega)} &\leq C_{10} \int_{\Omega} |u|^{2}u_L^{(2^{*}-2)}|\nabla \eta_R|^{2}\\ &\quad + C_{10} \Big(\int_{B_R}\big(\eta_R|u| u_L^{(2^*-2)/2}\big)^{2^{*}}\Big)^{2/2^*} \|u\|_{L^{2^*}(\Omega)}^{2^*-2}. \end{align*} From \eqref{C1} and recalling that $\eta_R u u_L^{(2^*-2)/2}=w_L$, $u_L \leq u$ and $\nabla \eta_R$ is bounded, we obtain \[ \|w_L\|^{2}_{L^{2^{*}}(\Omega)} \leq C_{11} \int_{\Omega} |u|^{2}u_L^{(2^{*}-2)}|\nabla \eta_R|^{2} \leq C_{11} \int_{\Omega} |u|^{2^*} \leq C_{12}. \] The definition of $\eta_R$ and $w_L$ and the above inequality imply $$ \Big( \int_{B_R} |u|^{2^*}u_L^{2^*(2^*-2)/2}\Big)^{2/2^*} \leq |w_L|^{2}_{L^{2^{*}}(\Omega)} \leq C_{12}. $$ Using Fatou's lemma in the variable $L$, we have \[ \int_{B_R}|u|^{2^*(2^*/2)}\leq K:=C_{12}^{2^*/2}. \] Since $\Omega$ can be covered by a finite number of balls $B^j_R$, we have $$ \int_{\Omega}|u|^{2^*(2^*/2)}\leq \sum_j^n \int_{B_R}|u|^{2^*(2^*/2)}\leq K_3, $$ for some $K_3>0$. To estimate $\|\nabla u_\lambda\|_ \infty$, we use the following result by Stampacchia \cite{Stampacchia}. \begin{lemma} Let $A(\eta)$ a given $C^1$ vector field in $\mathbb{R}^N$, and $f(x,s)$ a bounded Carath\'eodory function in $\Omega \times \mathbb{R}$. Let $u \in H^1_0(\Omega)$ be a solution of $$ \int_\Omega \left(A(|\nabla u|)\nabla \varphi + f(x,u)\varphi\right) = 0, $$ for all $\varphi \in H^1_0(\Omega)$. Assume that there exist $0 < \nu < M$ such that \begin{equation} \nu |\xi|^2 \leq \frac{\partial A_i}{\partial \eta_j}(\nabla u)\xi_i\xi_j \quad \text{and} \quad \big|\frac{\partial A_i}{\partial \eta_j}(\nabla u)\big| \leq M, \label{elipticidade} \end{equation} for $i,j = 1, \dots , N$ and $\xi \in \mathbb{R}^N$. Then $u \in W^{2,p}(\Omega) \cap C^{1,\alpha}(\overline{\Omega})$, for all $\alpha \in (0,1)$ and $p > 1$. Moreover \begin{equation} \|u\|_{1,\alpha} \leq C(\nu, M, \Omega) \|f(\cdot,u)\|_\infty. \end{equation} \end{lemma} \begin{proof} By the definition of $a$, for $r$ small enough \eqref{elipticidade} hold. This, together with the fact that $\|u_\lambda\|_\infty$ is bounded allow us to apply the last result. Then \eqref{Linfinity} implies \begin{equation} \|u\|_{1,\alpha} \leq \lambda \|u\|_\infty^{q-1} + \|u\|_\infty^{2^*-1} = o(\lambda), \label{limitegradiente} \end{equation} as $\lambda \to 0$. Then, there exists $\lambda^* > 0$ such that $\lambda \in (0,\lambda^*)$ implies $\|\nabla u\|_\infty \leq r$ and hence, $u_\lambda$ is a solution of \eqref{ePlambda}. \end{proof} \subsection*{Acknowledgments} G. M. Figueiredo is supported by PROCAD/CASADINHO: 552101/2011-7, CNPq/PQ 301242/2011-9 and CNPQ/CSF 200237/2012-8. M.T.O. Pimenta is supported by FAPESP: 2014/16136-1 and CNPq 442520/2014-0. \begin{thebibliography}{00} \bibitem{GP} J. G. Azorero, I. P. 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