The Existence Of Nontrivial Solution To A Nonlinear Elliptic Boundary Value Problem Of P-Laplacian Type Without Ambrosetti-Rabinowitz Condition

In this paper, we study the existence of a nontrival solution to the following nonlinear elliptic boundary value problem of p-Laplacian type :where p > 1,Ω(?)R~N is a bounded domain, (?)is the p-Laplacian of u and f∈C~0((?)×R~1,R~1) is p-superlinear at t = 0 and subcritical at t =∞.We prove under suitable conditions that for allλ> 0, the problem (P)λhas at least one nontrivial solution without assuming the Ambrosetti-Rabinowitz condition. Our main result extends a result for (P)λwhen p = 2 by O. H. Miyagaki and M. A. S. Souto in [13] to the general problem (P)λwhere p > 1. In the mean time, our result is stronger than a similar result of G. B. Li and H. S. Zhou in [12].