# American Institute of Mathematical Sciences

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

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

 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.
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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