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 $\Omega_i\subset \R^2\ (i=1,2,\cdots,m)$ and $V(x)\to\infty$ as $|x|\to\infty$, that is, the union of these $\Omega_i$ is the bottom of the potential well. By making some delicate estimates on the 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 $\Omega_j$ which has the largest inradius. Moreover, under some further conditions on $V(x)$ we show that the ground states of GP equations are unique and radially symmetric at leat for almost every $a \in (0, a^*)$.