Abstract and Applied Analysis
Volume 3 (1998), Issue 1-2, Pages 191-201
doi:10.1155/S1085337598000517

Multiple solutions for a problem with resonance involving the p-Laplacian

C. O. Alves,1 P. C. Carrião,2 and O. H. Miyagaki3

1Departamento de matemática e Estatística, Universidade Federal da Paraíba, Campina Grande 58109-970, (PB), Brazil
2Departamento de Matemática, Universidade Federal de Minas Gerais, Belo Horizonte 31270-010, (MG), Brazil
3Departamento de Matemática, Universidade Federal de Viçosa, Viçosa 36571-000, (MG), Brazil

Received 18 March 1998

Copyright © 1998 C. O. Alves et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Abstract

In this paper we will investigate the existence of multiple solutions for the problem (P)                                                         −Δpu+g(x,u)=λ1h(x)|u|p−2u,     in     Ω,    u∈H01,p(Ω) where Δpu=div(|∇u|p−2∇u) is the p-Laplacian operator, Ω⫅ℝN is a bounded domain with smooth boundary, h and g are bounded functions, N≥1 and 1<p<∞. Using the Mountain Pass Theorem and the Ekeland Variational Principle, we will show the existence of at least three solutions for (P).