Boundary geometry and concentration points for attractive Bose-Einstein condensates with a ring-shaped potential

Abstract

In this paper, we study the ground states of attractive Bose-Einstein condensates in a bounded domain $\Omega \subset {\mathbb{R}}^2$ under the ring-shaped trapping potential $$V(x)=(|x|-1{)}^2.$$ The boundary geometry of $\Omega$ may allow the ring $\Gamma =\{x\in {\mathbb{R}}^2:|x|=1\}$ of global minima for $V(x)$ to meet both the interior and the boundary of $\Omega$. As the interaction strength approaches the critical value, we prove that if $\Gamma \cap \Omega \ne \varnothing$, the mass of ground states concentrates at a point $x_0\in \Gamma \cap \Omega$ that has maximal distance from $\partial \Omega$. Our method relies on exponential upper and lower bounds for the difference between the bounded domain and whole space minimal energies. The essential lower bound follows from a uniform quadratic-growth estimate near the ground state manifold of the whole space problem, which is established by combining the nondegeneracy of the optimizer for the Gagliardo-Nirenberg inequality with a Lyapunov-Schmidt reduction.