The existence of a nontrivial solution to a nonlinear elliptic boundary value problem of p-Laplacian type without the Ambrosetti-Rabinowitz condition

Abstract

In this paper, we study the existence of a nontrivial solution to the following nonlinear elliptic boundary value problem of p-Laplacian type: ((P)λ){(- Δp u = λ f (x, u),, x ∈ Ω,; u = 0,, x ∈ ∂ Ω) where p > 1, λ ∈ R1,Ω ⊂ RN is a bounded domain and Δp u = d i v (| ∇ u |p - 2 ∇ u) is the p-Laplacian of u. f ∈ C0 (over(Ω, ̄) × R1, R1) is p-superlinear at t = 0 and subcritical at t = ∞. We prove that under suitable conditions for all λ > 0, the problem ((P)λ) has at least one nontrivial solution without assuming the Ambrosetti-Rabinowitz condition. Our main result extends a result for ((P)λ) for when p = 2 given by Miyagaki and Souto (2008) in [8] to the general problem ((P)λ) where p > 1. Meanwhile, our result is stronger than a similar result of Li and Zhou (2003) given in [15]. © 2010 Elsevier Ltd. All rights reserved.