Existence and multiplicity of positive solutions for nonlinear boundary value problems driven by the scalar p-Laplacian

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December 2004, 3(4): 729-756. doi: 10.3934/cpaa.2004.3.729

Existence and multiplicity of positive solutions for nonlinear boundary value problems driven by the scalar $p$-Laplacian

Michael E. Filippakis ^1, and Nikolaos S. Papageorgiou ^1,

1.	Department of Mathematics, National Technical University, Zografou Campus, Athens 15780, Greece, Greece

Received January 2004 Revised August 2004 Published September 2004

Abstract
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We study nonlinear Dirichlet problems driven by the scalar $p$-Laplacian with a nonsmooth potential. First for the so-called "sublinear problem", under nonuniform nonresonance conditions, we establish the existence of at least one strictly positive solution. Then we prove two multiplicity results for positive solutions. The first concerns the "superlinear problem" and the second is for the sublinear problem. The method of proof is variational based on the nonsmooth critical point theory for locally Lipschitz functions. Our results complement the ones obtained by De Coster (Nonlin.Anal.23 (1995)).

Keywords: multiple strictly positive solutions., generalized subdifferential, Mountain Pass Theorem, nonsmooth critical point theory, nonsmooth Palais-Smale condition, Locally Lipschitz function.

Mathematics Subject Classification: 34B18, 34B1.

Citation: Michael E. Filippakis, Nikolaos S. Papageorgiou. Existence and multiplicity of positive solutions for nonlinear boundary value problems driven by the scalar $p$-Laplacian. Communications on Pure & Applied Analysis, 2004, 3 (4) : 729-756. doi: 10.3934/cpaa.2004.3.729

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