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May  2009, 8(3): 1031-1051. doi: 10.3934/cpaa.2009.8.1031

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, Greece
2. 

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

Received  June 2008 Revised  November 2008 Published  February 2009

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.
Keywords: p–superlinear perturbation, truncation techniques, p–linear perturbation, upper–lower solutions, resonance, critical point theory., concave–convex nonlinearities, Concave nonlinearity.
Mathematics Subject Classification: 35J65, 35J7.
Citation: Sophia Th. Kyritsi, Nikolaos S. Papageorgiou. Pairs of positive solutions for $p$--Laplacian equations with combined nonlinearities. Communications on Pure & Applied Analysis, 2009, 8 (3) : 1031-1051. doi: 10.3934/cpaa.2009.8.1031
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