On the existence of nontrivial solutions for p-harmonic equations on unbounded domains

In this paper, we consider the existence of nontrivial solutions for the pp-harmonic problem {Δ(|Δu|p−2Δu)+a(x)|u|p−2u=f(x,u),u∈W2,p(RN), where p≥2p≥2 is a constant, N>2pN>2p, p