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.