Nonlinear Dirichlet problems with double resonance

Sergiu Aizicovici; Nikolaos S. Papageorgiou; Vasile Staicu

doi:10.3934/cpaa.2017056

Article Contents

2017, Volume 16, Issue 4: 1147-1168. Doi: 10.3934/cpaa.2017056

This issue Previous Article Existence of multiple solutions for a semilinear problem and a counterexample by E. N. Dancer Next Article Existence, nonexistence and uniqueness of positive solutions for nonlinear eigenvalue problems

Nonlinear Dirichlet problems with double resonance

1.
Department of Mathematics, Ohio University, Athens, OH 45701, USA
2.
Department of Mathematics, National Technical University, Zografou Campus, Athens 15780, Greece
3.
CIDMA and Department of Mathematics, University of Aveiro, 3810-193 Aveiro, Portugal

^* Corresponding author
^* Corresponding author

Received: May 31, 2016

Revised: January 31, 2017

Published: May 2017

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Abstract

We study a nonlinear Dirichlet problem driven by the sum of a $p-$Laplacian ($p>2$) and a Laplacian and which at $\pm\infty$ is resonant with respect to the spectrum of $\left(-\triangle_{p}, W_{0}^{1, p}\left(\Omega\right)\right) $ and at zero is resonant with respect to the spectrum of $\left(-\triangle, H_{0}^{1}\left(\Omega\right) \right) $ (double resonance). We prove two multiplicity theorems providing three and four nontrivial solutions respectivelly, all with sign information. Our approach uses critical point theory together with truncation and comparison techniques and Morse theory.

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Mathematics Subject Classification: Primary: 35J20, 35J60; Secondary: 58E05.

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