Multiple solutions for the Schrödinger equations with sign-changing potential and Hartree nonlinearity. (July 2018)

• Record Type: Journal Article
• Authors: Che, Guofeng; Chen, Haibo
• Is Part Of: Applied mathematics letters. Volume 81(2018)
• Page Start: 21
• Page End: 26
• Publication Date: 2018-07
• Subjects: Schrödinger equations -- Sign-changing potential -- Hartree-type nonlinearity -- Variational methods; Applied mathematics -- Periodicals; 519.05
• DOI: 10.1016/j.aml.2017.12.014
• Languages: English
• ISSNs: 0893-9659

Abstract

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.