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

In this paper we prove the existence of at least one nonnegative nontrivial weak solution in D1,p(RN)∩D1,q(RN)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