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

Michael E. Filippakis; Nikolaos S. Papageorgiou

doi:10.3934/cpaa.2004.3.729

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2004, Volume 3, Issue 4: 729-756. Doi: 10.3934/cpaa.2004.3.729

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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

Received: December 31, 2003

Revised: July 31, 2004

Published: November 2004

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Abstract

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)).

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Mathematics Subject Classification: 34B18, 34B15.

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