On the Double Phase Variational Problems Without Ambrosetti–Rabinowitz Condition

We are concerned with the existence and multiplicity of nontrivial solutions to the following double phase problems: −div(|∇u|p−2∇u + α(x)|∇u|q−2∇u) + V(x)|u|γ−2u = f (x, u), in , u = 0, on ∂ , applying the mountain pass theorem and fountain theorem. The Ambrosetti— Rabinowitz condition as well as the monotonicity of f (x, t)/|t|q−1 are not assumed.