Multiple solutions for (p, 2)-equations at resonance

Nikolaos S. Papageorgiou; Calogero Vetro; Francesca Vetro

doi:10.3934/dcdss.2019024

Article Contents

2019, Volume 12, Issue 2: 347-374. Doi: 10.3934/dcdss.2019024

This issue Previous Article Regularity of extremal solutions of a Liouville system Next Article Critical Schrödinger-Hardy systems in the Heisenberg group

Multiple solutions for (p, 2)-equations at resonance

1.
Department of Mathematics, National Technical University, Zagrafou Campus, Athens, 15780, Greece
2.
Department of Mathematics and Computer Science, University of Palermo, Via Archirafi 34, 90123 Palermo, Italy
3.
Department of Energy, Information Engineering and Mathematical Models, University of Palermo, Viale delle Scienze, 90128 Palermo, Italy

* Corresponding author: Nikolaos Papageorgiou
* Corresponding author: Nikolaos Papageorgiou

Received: March 31, 2017

Revised: November 30, 2017

Published: April 2019

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Abstract

We consider a nonlinear nonhomogeneous Dirichlet problem driven by the sum of a $p$-Laplacian and a Laplacian and a reaction term which is $(p-1)$-linear near $\pm \infty$ and resonant with respect to any nonprincipal variational eigenvalue of $(-\Delta_p,W^{1,p}_0(\Omega))$. Using variational tools together with truncation and comparison techniques and Morse Theory (critical groups), we establish the existence of six nontrivial smooth solutions. For five of them we provide sign information and order them.

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

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