A pair of positive solutions for $(p,q)$-equations with combined nonlinearities

Leszek Gasiński; Nikolaos S. Papageorgiou

doi:10.3934/cpaa.2014.13.203

Article Contents

2014, Volume 13, Issue 1: 203-215. Doi: 10.3934/cpaa.2014.13.203

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A pair of positive solutions for $(p,q)$-equations with combined nonlinearities

Leszek Gasiński^1, and
Nikolaos S. Papageorgiou^2,

1.
Jagiellonian University, Faculty of Mathematics and Computer Science, ul. Łojasiewicza 6, 30-348 Kraków
2.
Department of Mathematics, National Technical University of Athens, Zografou Campus, Athens 15780

Received: October 2012

Revised: April 2013

Published: January 2014

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Abstract

We consider a parametric nonlinear Dirichlet problem driven by the $(p,q)$-differential operator, with a reaction consisting of a ``concave'' term perturbed by a $(p-1)$-superlinear perturbation, which need not satisfy the Ambrosetti-Rabinowitz condition (problem with combined or competing nonlinearities). Using variational methods we show that for small values of the parameter the problem has at least two nontrivial positive smooth solutions.

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Mathematics Subject Classification: Primary: 35J20; Secondary: 35J60, 35J92.

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References

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