Multiplicity of solutions for a class of resonant p-Laplacian Dirichlet problems

Evgenia H. Papageorgiou and Nicolaos S. Papageorgiou

Vol. 241 (2009), No. 2, 309–328

DOI: 10.2140/pjm.2009.241.309

Abstract

We consider nonlinear Dirichlet problems driven by the p-Laplacian, which are resonant at +∞ with respect to the principal eigenvalue. Using a variational approach based on the critical point theory, we show that the problem has three nontrivial smooth solutions, two of which have constant sign (one positive, the other negative). In the semilinear case, assuming stronger regularity on the nonlinear perturbation f(z,⋅) and using Morse theory, we show that the problem has at least four nontrivial smooth solutions, two of constant sign.

Keywords

p-Laplacian, resonant problem, mountain pass theorem, second deformation theorem, Morse theory

Mathematical Subject Classification 2000

Primary: 35J65, 58E05

Milestones

Received: 5 October 2008

Revised: 27 November 2008

Accepted: 11 December 2008

Published: 1 June 2009