Existence and multiplicity results for the nonlinear Schrödinger–Poisson systems

In this paper, we study the existence and multiplicity results for the nonlinear Schrödinger–Poisson systems equation(∗){−Δu+V(x)u+K(x)ϕ(x)u=f(x,u),in R3,−Δϕ=K(x)u2,in R3. Under certain assumptions on VV, KK and ff, we obtain at least one nontrivial solution for (∗)(∗) without assuming the Ambrosetti and Rabinowitz condition by using the mountain pass theorem, and obtain infinitely many high energy solutions when f(x,⋅)f(x,⋅) is odd by using the fountain theorem.