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A Multiplicity Result for the p-Laplacian Involving a Parameter

Authors:

Friedemann Brock Leonelo Iturriaga Pedro Ubilla

Institution:

(1) Department of Mathematics, American University of Beirut, P.O. Box 11-0236, Beirut, Lebanon;(2) Instituto de Alta Investigación, Universidad de Tarapacá, Casilla 7 D, Arica, Chile;(3) Departamento de Matemáticas y C. C, Universidad de Santiago de Chile, Casilla 307 Correo 2, Santiago, Chile

Abstract:

We study existence and multiplicity of positive solutions for the following problem

$\left\{ {\begin{array}{*{20}l} { - \Delta _p u = \lambda f(x,u)} & {in\,\Omega ,} \\ {u = 0} & {on\,\partial \Omega } \\ \end{array} } \right.$

, where λ is a positive parameter, Ω is a bounded and smooth domain in ${\mathbb{R}^N}, p \in (1, N), f(x,t)$ behaves, for instance, like $o(|t|^{p-1})$ near 0 and +∞, and satisfies some further properties. In particular, our assumptions allow us to consider both positive and sign changing nonlinearitites f, the latter describing logistic as well as reaction–diffusion processes. By using sub- and supersolutions and variational arguments, we prove that there exists a positive constant $\bar{\lambda}$ such that the above problem has at least two positive solutions for $\lambda > \bar{\lambda}$ , at least one positive solution for $\lambda = \bar{\lambda}$ and no solution for $\lambda < \bar{\lambda}$ . An important r?le plays the fact that local minimizers of certain functionals in the C ¹-topology are also minimizers in $W^{1,p}_0 (\Omega)$ . We give a short new proof of this known result. Friedemann Brock: Supported by FONDECYT N^o 1050412 Leonelo Iturriaga: Partially supported by FONDECYT N^o 3060061, FONDAP Matemáticas aplicadas and Convenio de Desempe?o UTA-MECESUP 2 Pedro Ubilla:Supported by FONDECYT N^o 1040990 Submitted: November 8, 2007. Accepted: May 15, 2008.

Keywords:

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