# American Institute of Mathematical Sciences

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

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

Received  January 2004 Revised  August 2004 Published  September 2004

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)).
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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