Multiple solutions for the Schrödinger equations with sign-changing potential and Hartree nonlinearity

In this paper, we study the following Schrödinger equation −Δu+Vλ(x)u+μϕ|u|p−2u=f(x,u)+β(x)|u|ν−2u, in R3,(−Δ)α2ϕ=μ|u|p, in R3,where μ≥0 is a parameter, α∈(0,3), ν∈(1,2) and p∈[2,3+2α). Vλ is allowed to be sign-changing and ϕ|u|p−2u is a Hartree-type nonlinearity. We require that Vλ=λV+−V− with V+ having a bounded potential well Ω whose depth is controlled by λ. Under some mild conditions on Vλ(x) and f(x,u), we prove that the above system has at least two nontrivial solutions. Specially, our results cover the general Schrödinger equations and the Schrödinger-Poisson equations.