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

Abstract

In this paper, we study the following Schrödinger equation − Δ u + V λ ( x ) u + μ ϕ | u | p − 2 u = f ( x , u ) + β ( x ) | u | ν − 2 u , in R 3 , ( − Δ ) α 2 ϕ = μ | u | p , in R 3 , where μ ≥ 0 is a parameter, α ∈ ( 0 , 3 ) , ν ∈ ( 1 , 2 ) and p ∈ [ 2 , 3 + 2 α ) . V λ is allowed to be sign-changing and ϕ | u | p − 2 u is a Hartree-type nonlinearity. We require that V λ = λ V + − V − with V + having a bounded potential well Ω whose depth is controlled by λ . Under some mild conditions on V λ ( 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.