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 ) { − Δ p u = λ f ( x , u ) , x ∈ Ω , u = 0 , x ∈ ∂ Ω where p > 1 , λ ∈ R 1 , Ω ⊂ R N is a bounded domain and Δ p u = d i v ( | ∇ u | p − 2 ∇ u ) is the p -Laplacian of u . f ∈ C 0 ( Ω × R 1 , R 1 ) 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] .