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

In this paper, we study the existence of a nontrivial solution to the following nonlinear elliptic boundary value problem of p-Laplacian type: {-Delta(p)u = lambda f(x, u), x is an element of Omega, u = 0, x is an element of partial derivative Omega ((P)(lambda) where p > 1, lambda is an element of R(1), Omega subset of R(N) is a bounded domain and Delta(p)u = div(vertical bar del u vertical bar(p) 2 del u) is the p-Laplacian of u.f is an element of C(0) ((Omega) over bar x R(1), R(1)) is p-superlinear at t = 0 and subcritical at t = infinity. We prove that under suitable conditions for all lambda > 0, the problem ((P)(lambda)) has at least one nontrivial solution without assuming the Ambrosetti-Rabinowitz condition. Our main result extends a result for ((P)(lambda)) for when p = 2 given by Miyagaki and Souto (2008) in [8] to the general problem ((P)(lambda)) where p > 1. Meanwhile, our result is stronger than a similar result of Li and Zhou (2003) given in [15]. (C) 2010 Elsevier Ltd. All rights reserved.