On the critical Schrödinger–Bopp–Podolsky system with general nonlinearities. (June 2020)

• Record Type: Journal Article
• Authors: Chen, Sitong; Tang, Xianhua
• Is Part Of: Nonlinear analysis. Volume 195(2020)
• Publication Date: 2020-06
• Subjects: 35J48 -- 35J50 -- 35Q60; Schrödinger–Bopp–Podolsky system -- 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.2019.111734
• Languages: English
• ISSNs: 0362-546X

Abstract

Abstract: This paper deals with the following nonlinear Schrödinger–Bopp–Podolsky system − Δ u + V ( x ) u + ϕ u = μ f ( u ) + u 5 − 2 Δ ϕ + a 2 Δ 2 ϕ = 4 π u 2 in R 3, where a > 0, V ∈ C ( R 3, [ 0, ∞ ) ) with V ∞ ≔ lim | y | → ∞ V ( y ) ≥ sup x ∈ R 3 V ( x ) > 0, and f ∈ C ( R, R ) satisfying ∫ 0 t f ( s ) d s ≥ t p with p ∈ ( 2, 6 ) for all t ≥ 0 . By using some new analytic techniques and new inequalities, we prove that the above system admits ground state solutions for all μ > 0 if p ∈ ( 4, 6 ) ; for all μ > μ 0 if p ∈ ( 2, 4 ], where μ 0 is a positive constant determined by a, V ∞ and p . Our results extend and complement the existence results in d'Avenia and G. Siciliano (2019) which only dealt with the subcritical case.