\documentclass[twoside]{article} \usepackage{amsfonts,amsmath, amsthm} \pagestyle{myheadings} \markboth{\hfil Multiple positive solutions \hfil EJDE--2003/07} {EJDE--2003/07\hfil Kanishka Perera \hfil} \begin{document} \title{\vspace{-1in}\parbox{\linewidth}{\footnotesize\noindent {\sc Electronic Journal of Differential Equations}, Vol. {\bf 2003}(2003), No. 07, pp. 1--5. \newline ISSN: 1072-6691. URL: http://ejde.math.swt.edu or http://ejde.math.unt.edu \newline ftp ejde.math.swt.edu (login: ftp)} \vspace{\bigskipamount} \\ % Multiple positive solutions for a class of quasilinear elliptic boundary-value problems % \thanks{ {\em Mathematics Subject Classifications:} 35J20, 35J65. \hfil\break\indent {\em Key words:} $p$-linear $p$-Laplacian problems, positive solutions, non-existence, \hfil\break\indent multiplicity, variational methods. \hfil\break\indent \copyright 2003 Southwest Texas State University. \hfil\break\indent Submitted May 29, 2002. Published January 23, 2003.} } \date{} % \author{Kanishka Perera} \maketitle \begin{abstract} Using variational arguments we prove some nonexistence and multiplicity results for positive solutions of a class of elliptic boundary-value problems involving the $p$-Laplacian and a parameter. \end{abstract} \newcommand{\norm}[2]{\|#1\|_{#2}} \newtheorem{lemma}{Lemma}[section] \newtheorem{theorem}[lemma]{Theorem} \numberwithin{equation}{section} \section{Introduction} \label{S1} In a recent paper, Maya and Shivaji \cite{MaSh} studied the existence, multiplicity, and non-existence of positive classical solutions of the semilinear elliptic boundary-value problem $$\label{101} \begin{gathered} - \Delta u = \lambda f(u) \quad \text{in } \Omega,\\ u = 0 \quad \text{on } \partial \Omega \end{gathered}$$ where $\Omega$ is a smooth bounded domain in $\mathbb{R}^n$, $n \ge 1$, $\lambda > 0$ is a parameter, and $f$ is a $C^1$ function such that $$\label{102} f(0) = 0, \quad \lim_{t \to \infty} \frac{f(t)}{t} = 0.$$ Assuming \begin{enumerate} \item[(f$_1$)] $f'(0) < 0$, \item[(f$_2$)] $\exists \beta > 0$ such that $f(t) < 0$ for $0 < t < \beta$ and $f(t) > 0$ for $t > \beta$, \item[(f$_3$)] $f$ is eventually increasing, \end{enumerate} they showed using sub-super solutions arguments and recent results from semipositone problems that there are $\underline{\lambda}$ and $\overline{\lambda}$ such that \eqref{101} has no positive solution for $\lambda < \underline{\lambda}$ and at least two positive solutions for $\lambda \ge \overline{\lambda}$. In the present paper we consider the corresponding quasilinear problem $$\label{103} \begin{gathered} - \Delta_p u = \lambda f(x,u) \quad \text{in } \Omega,\\ u = 0 \quad \text{on } \partial \Omega \end{gathered}$$ where $\Delta_p u = \mathop{\rm div} \big(|\nabla u|^{p-2} \nabla u\big)$ is the $p$-Laplacian, $1 < p < \infty$, $\lambda > 0$, and $f$ is a Carath\'{e}odory function on $\Omega \times [0,\infty)$ satisfying $$\label{104} f(x,0) = 0, \quad |f(x,t)| \le C t^{p-1}$$ for some constant $C > 0$. Note that when $p = 2$ and $f$ is $C^1$ and satisfies \eqref{102}, the existence of the limits $\lim_{t\to 0} f(t)/t = f'(0)$ and $\lim_{t \to \infty} f(t)/t$ imply \eqref{104}. Using variational methods, we shall prove the following theorems. \begin{theorem} \label{T101} There is a $\underline{\lambda}$ such that \eqref{103} has no positive solution for $\lambda < \underline{\lambda}$. \end{theorem} \begin{theorem} \label{T102} Set $F(x,t) = \int_0^t f(x,s) ds$, and assume \begin{enumerate} \item[\em (F$_1$)] $\exists \delta > 0$ such that $F(x,t) \le 0$ for $0 \le t \le \delta$, \item[\em (F$_2$)] $\exists t_0 > 0$ such that $F(x,t_0) > 0$, \item[\em (F$_3$)] $\displaystyle \varlimsup_{t \to \infty} \frac{F(x,t)}{t^p} \le 0$ uniformly in $x$. \end{enumerate} Then there is a $\overline{\lambda}$ such that \eqref{103} has at least two positive solutions $u_1 > u_2$ for $\lambda \ge \overline{\lambda}$. \end{theorem} Note that we have substantially relaxed the assumptions in \cite{MaSh} and therefore our results seem to be new even in the semilinear case $p = 2$. More specifically, we have let $f$ depend on $x$ and dropped the assumption of differentiability in $t$, and replaced (f$_1$), (f$_2$), and (f$_3$) with the much weaker assumptions (F$_1$) and (F$_2$) on the primitive $F$. We emphasize that (F$_1$) follows from (f$_1$), while (f$_2$) and (f$_3$) together imply (F$_2$), and that we make no monotonicity assumptions. The limit in (F$_3$) equals $0$ in the $p$-sublinear case $$\label{105} \lim_{t \to \infty} \frac{f(x,t)}{t^{p-1}} = 0 \text{ uniformly in } x,$$ in particular, in the special case considered in \cite{MaSh}. \section{Proofs of Theorems \ref{T101} and \ref{T102}} \label{S2} Recall that the first Dirichlet eigenvalue of $- \Delta_p$ is positive and is given by $$\label{201} \lambda_1 = \min_{u \ne 0} \frac{ \int_\Omega |\nabla u|^p}{ \int_\Omega |u|^p}$$ (see Lindqvist \cite{Lin1}). If \eqref{103} has a positive solution $u$, multiplying \eqref{103} by $u$, integrating by parts, and using \eqref{104} gives $$\label{202} \int_\Omega |\nabla u|^p = \lambda \int_\Omega f(x,u) u \le C \lambda \int_\Omega u^p,$$ and hence $\lambda \ge \lambda_1/C$ by \eqref{201}, proving Theorem \ref{T101}. We will prove Theorem \ref{T102} using critical point theory. Set $f(x,t) = 0$ for $t < 0$, and consider the $C^1$ functional $$\label{203} \Phi_\lambda(u) = \int_\Omega |\nabla u|^p - \lambda p F(x,u), \quad u \in W^{1, p}_0(\Omega).$$ If $u$ is a critical point of $\Phi_\lambda$, denoting by $u^-$ the negative part of $u$, $$\label{204} 0 = ({\Phi_\lambda'(u)},{u^-}) = \int_\Omega |\nabla u|^{p-2} \nabla u \cdot \nabla u^- - \lambda f(x,u) u^- = \norm{u^-}{}^p$$ shows that $u \ge 0$. Furthermore, $u \in L^\infty(\Omega) \cap C^1(\Omega)$ by Anane \cite{An} and di Benedetto \cite{diBe}, so it follows from the Harnack inequality (Theorem 1.1 of Trudinger \cite{Tr}) that either $u > 0$ or $u \equiv 0$. Thus, nontrivial critical points of $\Phi_\lambda$ are positive solutions of \eqref{103}. By (F$_3$) and \eqref{104}, there is a constant $C_\lambda > 0$ such that $$\label{205} \lambda p F(x,t) \le \frac{\lambda_1}{2} |t|^p + C_\lambda$$ and hence $$\label{206} \Phi_\lambda(u) \ge \int_\Omega |\nabla u|^p - \frac{\lambda_1}{2} |u|^p - C_\lambda \ge \frac12 \norm{u}{}^p - C_\lambda \mu(\Omega)$$ where $\mu$ denotes the Lebesgue measure in $\mathbb{R}^n$, so $\Phi_\lambda$ is bounded from below and coercive. This yields a global minimizer $u_1$ since $\Phi_\lambda$ is weakly lower semicontinuous. \begin{lemma} \label{T201} There is a $\overline{\lambda}$ such that $\inf \Phi_\lambda < 0$, and hence $u_1 \ne 0$, for $\lambda \ge \overline{\lambda}$. \end{lemma} \begin{proof} Taking a sufficiently large compact subset $\Omega'$ of $\Omega$ and a function $u_0 \in W^{1, p}_0(\Omega)$ such that $u_0(x) = t_0$ on $\Omega'$ and $0 \le u_0(x) \le t_0$ on $\Omega \setminus \Omega'$, where $t_0$ is as in (F$_2$), we have $$\label{207} \int_\Omega F(x,u_0) \ge \int_{\Omega'} F(x,t_0) - C t_0^p \mu(\Omega \setminus \Omega') > 0$$ and hence $\Phi_\lambda(u_0) < 0$ for $\lambda$ large enough. \end{proof} Now fix $\lambda \ge \overline{\lambda}$, let $$\label{208} \widetilde{f}(x,t) = \begin{cases} f(x,t), & t \le u_1(x),\\[5pt] f(x,u_1(x)), & t > u_1(x), \end{cases} \quad\text{and}\quad \widetilde{F}(x,t) = \int_0^t \widetilde{f}(x,s) ds.$$ Then consider $$\label{209} \widetilde{\Phi}_\lambda(u) = \int_\Omega |\nabla u|^p - \lambda p \widetilde{F}(x,u).$$ If $u$ is a critical point of $\widetilde{\Phi}_\lambda$, then $u \ge 0$ as before, and \label{210} \begin{aligned} 0 = & \big({\widetilde{\Phi}_\lambda'(u) - \Phi_\lambda'(u_1)},{(u- u_1)^+}\big) \\ = & \int_\Omega \left(|\nabla u|^{p-2} \nabla u - |\nabla u_1|^{p-2} \nabla u_1\right) \cdot \nabla (u - u_1)^+\\ &- \lambda \big(\widetilde{f}(x,u) - f(x,u_1)\big) (u - u_1)^+ \\ = & \int_{u > u_1} \left(|\nabla u|^{p-2} \nabla u - |\nabla u_1|^{p-2} \nabla u_1\right) \cdot (\nabla u - \nabla u_1)\\ \ge & \int_{u > u_1} \left(|\nabla u|^{p-1} - |\nabla u_1|^{p-1}\right)(|\nabla u| - |\nabla u_1|) \ge 0 \end{aligned} implies that $u \le u_1$, so $u$ is a solution of \eqref{103} in the order interval $[0,u_1]$. We will obtain a critical point $u_2$ with $\widetilde{\Phi}_\lambda(u_2) > 0$ via the mountain-pass lemma, which would complete the proof since $\widetilde{\Phi}_\lambda(0) = 0 > \widetilde{\Phi}_\lambda(u_1)$. \begin{lemma} \label{T202} The origin is a strict local minimizer of $\widetilde{\Phi}_\lambda$. \end{lemma} \begin{proof} Setting $\Omega_u = \big\{x \in \Omega : u(x) > \min \big\{u_1(x), \delta\big\}\big\}$, by \eqref{208} and (F$_1$), $\widetilde{F}(x,u(x)) \le 0$ on $\Omega \setminus \Omega_u$, so $$\label{211} \widetilde{\Phi}_\lambda(u) \ge \norm{u}{}^p - \lambda p \int_{\Omega_u} \widetilde{F}(x,u).$$ By \eqref{104}, H\"{o}lder's inequality, and Sobolev imbedding, $$\label{212} \int_{\Omega_u} \widetilde{F}(x,u) \le C \int_{\Omega_u} u^p \le C \mu(\Omega_u)^{1 - \frac{p}{q}} \norm{u}{}^p$$ where $q = np/(n - p)$ if $p < n$ and $q > p$ if $p \ge n$, so it suffices to show that $\mu(\Omega_u) \to 0$ as $\norm{u}{} \to 0$. Given $\varepsilon > 0$, take a compact subset $\Omega_\varepsilon$ of $\Omega$ such that $\mu(\Omega \setminus \Omega_\varepsilon) < \varepsilon$ and let $\Omega_{u, \varepsilon} = \Omega_u \cap \Omega_\varepsilon$. Then $$\label{213} \norm{u}{p}^p \ge \int_{\Omega_{u, \varepsilon}} u^p \ge c^p \mu(\Omega_{u, \varepsilon})$$ where $c = \min \big\{\min u_1(\Omega_\varepsilon), \delta\big\} > 0$, so $\mu(\Omega_{u, \varepsilon}) \to 0$. But, since $\Omega_u \subset \Omega_{u, \varepsilon} \cup (\Omega \setminus \Omega_\varepsilon)$, $$\label{214} \mu(\Omega_u) < \mu(\Omega_{u, \varepsilon}) + \varepsilon,$$ and $\varepsilon$ is arbitrary. \end{proof} An argument similar to the one we used for $\Phi_\lambda$ shows that $\widetilde{\Phi}_\lambda$ is also coercive, so every Palais-Smale sequence of $\widetilde{\Phi}_\lambda$ is bounded and hence contains a convergent subsequence as usual. Now the mountain-pass lemma gives a critical point $u_2$ of $\widetilde{\Phi}_\lambda$ at the level $$\label{215} c := \inf_{\gamma \in \Gamma} \max_{u \in \gamma([0,1])} \widetilde{\Phi}_\lambda(u) > 0$$ where $\Gamma = \big\{\gamma \in C([0,1], W^{1, p}_0(\Omega)) : \gamma(0) = 0, \gamma(1) = u_1\big\}$ is the class of paths joining the origin to $u_1$ (see, e.g., Rabinowitz \cite{Ra}). \begin{thebibliography}{0} \frenchspacing \bibitem{An} A. Anane. \newblock {\em {E}tude des valeurs propres et de la r\'esonnance pour l'op\'erateur $p$-{L}aplacien}. \newblock PhD thesis, {U}niversit\'e {L}ibre de {B}ruxelles, 1987. \newblock {\em C. R. Acad. Sci. Paris S\'er. I Math.}, 305(16):725--728, 1987. \bibitem{diBe} E. di Benedetto. \newblock ${C}\sp{1+\alpha }$ local regularity of weak solutions of degenerate elliptic equations. \newblock {\em Nonlinear Anal.}, 7(8):827--850, 1983. \bibitem{Lin1} P. Lindqvist. \newblock On the equation ${\rm div}(\vert \nabla u\vert \sp {p-2}\nabla u)+\lambda\vert u\vert \sp {p-2}u=0$. \newblock {\em Proc. Amer. Math. Soc.}, 109(1):157--164, 1990. \newblock Addendum: {\em Proc. Amer. Math. Soc.}, 116(2):583--584, 1992. \bibitem{MaSh} C. Maya and R. Shivaji. \newblock Multiple positive solutions for a class of semilinear elliptic boundary value problems. \newblock {\em Nonlinear Anal.}, 38(4, Ser. A: Theory Methods):497--504, 1999. \bibitem{Ra} P. H. Rabinowitz. \newblock {\em Minimax methods in critical point theory with applications to differential equations}. \newblock Published for the Conference Board of the Mathematical Sciences, Washington, DC, 1986. \bibitem{Tr} N. Trudinger. \newblock On {H}arnack type inequalities and their application to quasilinear elliptic equations. \newblock {\em Comm. Pure Appl. Math.}, 20:721--747, 1967. \end{thebibliography} \noindent\textsc{Kanishka Perera} \\ Department of Mathematical Sciences\\ Florida Institute of Technology\\ Melbourne, FL 32901, USA\\ e-mail: kperera@fit.edu\quad http://my.fit.edu/$\sim$kperera/ \end{document}