Multi-bump bound states for a nonlinear Schrِdinger system with electromagnetic fields

In this paper, we are concerned with the following nonlinear Schrödinger system with electromagnetic fields (Sλ) { − ( ∇ + i A ( x ) ) 2 u ( x ) + λ V ( x ) u ( x ) = 2 α α + β | u ( x ) | α − 2 | v ( x ) | β u ( x ) , x ∈ R N , − ( ∇ + i B ( x ) ) 2 v ( x ) + λ W ( x ) v ( x ) = 2 β α + β | u ( x ) | α | v ( x ) | β − 2 v ( x ) , x ∈ R N , | u ( x ) | → 0 , | v ( x ) | → 0 as  | x | → ∞ for sufficiently large λ , where i is the imaginary unit, α > 1 , β > 1 , α + β < Θ and Θ = 2 N N − 2 for N ≥ 3 , Θ = + ∞ for N = 1 , 2 . A ( x ) and B ( x ) are real-valued electromagnetic vector potentials. V ( x ) and W ( x ) are real-valued continuous nonnegative functions on R N . By modifying the nonlinearity and using the decay flow we show that if Ω : = int  V − 1 ( 0 ) ∩ int  W − 1 ( 0 ) has several isolated connected components Ω 1 , Ω 2 , … , Ω k such that the interior of Ω i is not empty and ∂ Ω i is smooth for all i ∈ { 1 , 2 , … , k } , then for any non-empty subset J ⊂ { 1 , 2 , … , k } there exists a solution ( u λ , v λ ) of ( S λ ) for λ > 0 large. Moreover for any sequence λ n → ∞ , up to a subsequence ( u λ n , v λ n ) converges in Ω j ( j ∈ J ) to a least energy solution of the following limit problem (DΩj) { − ( ∇ + i A ( x ) ) 2 u ( x ) = 2 α α + β | u ( x ) | α − 2 | v ( x ) | β u ( x ) , x ∈ Ω j , − ( ∇ + i B ( x ) ) 2 v ( x ) = 2 β α + β | u ( x ) | α | v ( x ) | β − 2 v ( x ) , x ∈ Ω j , ( u ( x ) , v ( x ) ) ∈ H A , B 0 , 1 ( Ω j ) and outside of ⋃ j ∈ J Ω j to ( 0 , 0 ) .