Properties of ground states of attractive Gross–Pitaevskii equations with multi-well potentials

We are interested in the attractive Gross–Pitaevskii (GP) equation in R 2 , where the external potential V ( x ) vanishes on m disjoint bounded domains Ω i ⊂ R 2 ( i = 1 , 2 , ⋯ , m ) and V ( x ) → ∞ as | x | → ∞ , that is, the union of these Ω i is the bottom of the potential well. By establishing some delicate estimates on the associated energy functional of the GP equation, we prove that when the interaction strength a approaches some critical value a ∗ , the ground states concentrate and blow up at the center of the incircle of some Ω j with the largest inradius. Moreover, under some further conditions on V ( x ) , we show that the ground states of the GP equations are unique and radially symmetric at least for almost every a ∈ ( 0 , a ∗ ) .