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

Abstract

We are interested in the attractive Gross–Pitaevskii (GP) equation in , where the external potential vanishes on m disjoint bounded domains and as , that is, the union of these 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 , the ground states concentrate and blow up at the center of the incircle of some with the largest inradius. Moreover, under some further conditions on , we show that the ground states of the GP equations are unique and radially symmetric at least for almost every .