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. Alves1 , P.C. Carrião2 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

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).