Localized nodal solutions of higher topological type for nonlinear Schrödinger–Poisson system. (September 2020)

• Record Type: Journal Article
• Authors: Chen, Zhi; Qin, Dongdong; Zhang, Wen
• Is Part Of: Nonlinear analysis. Volume 198(2020)
• Publication Date: 2020-09
• Subjects: 35J20 -- 35J60; Schrödinger–Poisson system -- Semiclassical states -- Sign-changing solutions -- Concentration behavior; Mathematical analysis -- Periodicals; Functional analysis -- Periodicals; Nonlinear theories -- Periodicals; Analyse mathématique -- Périodiques; Analyse fonctionnelle -- Périodiques; Théories non linéaires -- Périodiques; Functional analysis; Mathematical analysis; Nonlinear theories; Periodicals; Electronic journals; 515.7248
• DOI: 10.1016/j.na.2020.111896
• Languages: English
• ISSNs: 0362-546X

Abstract

Abstract: In this paper, we focus our attention on following Schrödinger–Poisson system: − ε 2 Δ u + V ( x ) u + λ ψ u = | u | p − 2 u, x ∈ R 3, − ε 2 Δ ψ = u 2, u ∈ H 1 ( R 3 ) . where 4 < p < 6 and ε, λ > 0 are small positive parameters. Under a local condition imposed on the potential V, we study above system and obtain an infinite sequence of localized sign-changing solutions by applying the symmetric mountain pass theorem. Precisely, these solutions are constructed as higher topological type critical points of the energy functional and concentrated at a local minimum set of the potential V . Our method follows the same spirit of Chen and Wang's method (Chen and Wang, 2017) which does not need any non-degeneracy condition of the limiting equations, but we cannot use it directly due to the presence of nonlocal term ψ ( x ) = 1 4 π ∫ R 3 u 2 ( y ) | x − y | d y . We employ some new analytical skills to overcome the obstacles caused by the nonlocal term, our results improve and extend some related ones in the literature.