The existence of a nontrivial solution to a nonlinear elliptic boundary value problem of pp-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 pp-Laplacian type: equation((P)λ(P)λ){−Δpu=λf(x,u),x∈Ω,u=0,x∈∂Ω where p>1p>1, λ∈R1λ∈R1,Ω⊂RNΩ⊂RN is a bounded domain and Δpu=div(|∇u|p−2∇u)Δpu=div(|∇u|p−2∇u) is the pp-Laplacian of uu. f∈C0(Ω̄×R1,R1) is pp-superlinear at t=0t=0 and subcritical at t=∞t=∞. We prove that under suitable conditions for all λ>0λ>0, the problem ((P)λ)((P)λ) has at least one nontrivial solution without assuming the Ambrosetti–Rabinowitz condition. Our main result extends a result for ((P)λ)((P)λ) for when p=2p=2 given by Miyagaki and Souto (2008) in [8] to the general problem ((P)λ)((P)λ) where p>1p>1. Meanwhile, our result is stronger than a similar result of Li and Zhou (2003) given in [15].