Existence of a nontrivial solution for the (p, q) Laplacian in RN without the Ambrosetti–Rabinowitz condition

Abstract

In this paper we prove the existence of at least one nonnegative nontrivial weak solution in D1,p (R N ) ∩ D1,q (R N ) for the equation −∆pu − ∆qu + a(x)|u| p−2 u + b(x)|u| q−2 u = f(x, u), x ∈ R N , where 1 < q < p < q ⋆ := Nq N−q , p < N, ∆mu := 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.