Multi-bump solutions for the nonlinear magnetic Choquard equation with deepening potential well

Abstract In this paper, using variational methods, we study multiplicity of multi-bump solutions for the following nonlinear magnetic Choquard equation { − ( ∇ + i A ( x ) ) 2 u + ( λ V ( x ) + 1 ) u = ( 1 | x | μ ⁎ | u | p ) | u | p − 2 u x ∈ R N , u ∈ H 1 ( R N , C ) , where N ≥ 2 , λ > 0 is a real parameter, 0 μ 2 , i is the imaginary unit, p ∈ ( 2 , 2 ⁎ ( 2 ( N − μ ) 2 N ) ) , where 2 ⁎ = 2 N N − 2 if N ≥ 3 , 2 ⁎ = + ∞ , if N = 2 . The magnetic potential A ∈ L l o c 2 ( R N , R N ) and V : R N → R is a nonnegative continuous function. We show that if the zero set of V has several isolated connected components Ω 1 , ⋯ , Ω k such that the interior of Ω j is non-empty and ∂ Ω j is smooth, then for λ > 0 large enough, the above equation has at least 2 k − 1 multi-bump solutions.