# American Institute of Mathematical Sciences

April  2019, 12(2): 347-374. doi: 10.3934/dcdss.2019024

## 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

Received  April 2017 Revised  December 2017 Published  August 2018

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.

Citation: Nikolaos S. Papageorgiou, Calogero Vetro, Francesca Vetro. Multiple solutions for (p, 2)-equations at resonance. Discrete & Continuous Dynamical Systems - S, 2019, 12 (2) : 347-374. doi: 10.3934/dcdss.2019024
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