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

• DOI: 10.14232/ejqtde.2024.1.10
• Publication Date: 2024-02-07T09:02:08Z
• Authors: Li Wang; Jun Wang; Jixiu Wang
• Keywords: QA Mathematics / matematika; penalization method; variational methods; QA1-939; schrödinger–bopp–podolsky system; 0101 mathematics; 01 natural sciences; Mathematics

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.