Existence of a nontrivial solution for the [formula omitted]-Laplacian in [formula omitted] without the Ambrosetti–Rabinowitz condition

Abstract

In this paper we prove the existence of at least one nonnegative nontrivial weak solution in D1,p(RN)∩D1,q(RN) for the equation −Δpu−Δqu+a(x)|u|p−2u+b(x)|u|q−2u=f(x,u),x∈RN, where 1<q<p<q⋆:=NqN−q,p<N,Δmu:=div(|∇u|m−2∇u) is the m-Laplacian operator, the coefficients a and b are continuous, coercive and positive functions, and the nonlinearity f is a Carathéodory function satisfying some hypotheses which do not include the Ambrosetti–Rabinowitz condition.