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

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).