On the critical Schrödinger–Bopp–Podolsky system with general nonlinearities. (June 2020) - Chen, Sitong; Tang, Xianhua
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.