Existence of semiclassical solutions for some critical Schrödinger–Poisson equations with potentials. (September 2020)

• Record Type: Journal Article
• Authors: Yang, Minbo
• Is Part Of: Nonlinear analysis. Volume 198(2020)
• Publication Date: 2020-09
• Subjects: 35J20 -- 35J60 -- 35Q55; Schrödinger–Poisson equations -- Semiclassical solutions -- Critical Growth -- Variational methods; 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.2020.111874
• Languages: English
• ISSNs: 0362-546X

Abstract

Abstract: In this paper we are interested in the existence of semiclassical solutions for the critical Schrödinger–Poisson equations with multiple potentials. Under suitable assumptions on the potentials, we are able to prove the existence of semiclassical solutions ( u ε, ϕ ε ) with maximum points x ε of u ε concentrating at a special set C P characterized by some of the potentials. Moreover, for any sequence x ε → x 0 ∈ C P, by setting v ε ( x ) ≔ u ε ( ε x + x ε ), the ground solutions v ε convergence strongly in H 1 ( R 3 ) to a ground state solution v of − Δ v + V ( x 0 ) v = P ( x 0 ) g ( v ) + Q ( x 0 ) | v | 4 v, in R 3 .