Nonlinear Dirichlet problems with a crossing reaction

Abstract

We consider a nonlinear Dirichlet problem driven by the sum of a p-Laplacian (p > 2) and a Laplacian, with a reaction that is (p-1)-sublinear and exhibits an asymmetric behavior near ∞ and -∞, crossing ^λ1 > 0, the principal eigenvalue of (-Δp,W01,p(Ω)) (crossing nonlinearity). Resonance with respect to ^λ1(p) > 0 can also occur. We prove two multiplicity results. The first for a Caratheodory reaction producing two nontrivial solutions and the second for a reaction C1 in the x-variable producing three nontrivial solutions. Our approach is variational and uses also the Morse theory.

Document Type

Article

DOI

https://doi.org/10.3934/cpaa.2014.13.2749

Keywords

nonlinear regularity, critical groups., resonance, nonlinear maximum principle

Publication Date

2014

Journal Title

Communications on Pure & Applied Analysis

Link to Full Text

Share

COinS