Pairs of positive solutions for $p$--Laplacian equations with combined nonlinearities

Sophia Th. Kyritsi; Nikolaos S. Papageorgiou

doi:10.3934/cpaa.2009.8.1031

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2009, Volume 8, Issue 3: 1031-1051. Doi: 10.3934/cpaa.2009.8.1031

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Pairs of positive solutions for $p$--Laplacian equations with combined nonlinearities

Sophia Th. Kyritsi^1, and
Nikolaos S. Papageorgiou^2,

1.
Department of Mathematics, Hellenic Naval Academy, Piraeus 18539
2.
Department of Mathematics, National Technical University, Zografou Campus, Athens 15780

Received: May 31, 2008

Revised: October 31, 2008

Published: April 2009

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Abstract

We consider a nonlinear Dirichlet problem driven by the $p$--Laplacian differential operator, with a nonlinearity concave near the origin and a nonlinear perturbation of it. We look for multiple positive solutions. We consider two distinct cases. One when the perturbation is $p$--linear and resonant with respect to $\lambda_1>0$ (the principal eigenvalue of $(-\Delta_p,W^{1,p}_0(Z))$) at infinity and the other when the perturbation is $p$--superlinear at infinity. In both cases we obtain two positive smooth solutions. The approach is variational, coupled with the method of upper--lower solutions and with suitable truncation techniques.

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Mathematics Subject Classification: 35J65, 35J70.

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