Multi-bump solutions of Schrödinger–Bopp–Podolsky system with steep potential well

Abstract

In this paper, we study the existence of multi-bump solutions for the following Schrödinger–Bopp–Podolsky system with steep potential well: { − Δ u + ( λ V ( x ) + V 0 ( x ) ) u + K ( x ) ϕ u = | u | p − 2 u , x ∈ R 3 , − Δ ϕ + a 2 Δ 2 ϕ = K ( x ) u 2 , x ∈ R 3 , where p ∈ ( 4 , 6 ) , a > 0 and λ is a parameter. We require that V ( x ) ≥ 0 and has a bounded potential well Ω = V − 1 ( 0 ) . Combining this with other suitable assumptions on Ω , V 0 and K , when λ is large enough, we obtain the existence of multi-bump-type solutions u λ by using variational methods.