Multiple solutions for nonlinear nonhomogeneous resonant coercive problems

Diego Averna; Nikolaos S. Papageorgiou; Elisabetta Tornatore

doi:10.3934/dcdss.2018010

Article Contents

2018, Volume 11, Issue 2: 155-178. Doi: 10.3934/dcdss.2018010

This issue Previous Article Preface Next Article One-dimensional nonlinear boundary value problems with variable exponent

Multiple solutions for nonlinear nonhomogeneous resonant coercive problems

1.
Dipartimento di Matematica e Informatica, Università degli studi di Palermo, Via Archirafi, 90123 -Palermo, Italy
2.
Department of Mathematics, National Technical University, Zografou Campus, Athens 15780, Greece

* Corresponding author: E. Tornatore
* Corresponding author: E. Tornatore

Received: November 30, 2016

Revised: March 31, 2017

Published: April 2018

The authors Averna, Papageorgiou and Tornatore were partially supported by INdAM -GNAMPA Project 2016.

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Abstract

We consider a nonlinear, nonhomogeneous Dirichlet problem driven by the sum of a $p$-Laplacian ($2<p$) and a Laplacian. The reaction term is a Carathéodory function $f(z,x)$ which is resonant with respect to the principal eigenvalue of ($-\Delta_p,\, W^{1,p}_0(\Omega)$). Using variational methods combined with truncation and comparison techniques and Morse theory (critical groups) we prove the existence of three nontrivial smooth solutions all with sign information and under three different conditions concerning the behavior of $f(z,\cdot)$ near zero. By strengthening the regularity of $f(z,\cdot)$, we are able to generate a second nodal solution for a total of four nontrivial smooth solutions.

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Mathematics Subject Classification: 35J20, 35J60, 58E05.

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