Energy estimates and symmetry breaking in attractive Bose-Einstein condensates with ring-shaped potentials

This paper is concerned with the properties of $L^2$-normalized minimizers of the Gross-Pitaevskii (GP) functional for a two-dimensional Bose-Einstein condensate with attractive interaction and ring-shaped potential. By establishing some delicate estimates on the least energy of the GP functional, we prove that symmetry breaking occurs for the minimizers of the GP functional as the interaction strength $a>0$ approaches a critical value $a^*$, each minimizer of the GP functional concentrates to a point on the circular bottom of the potential well and then is non-radially symmetric as $a\nearrow a^*$. However, when $a>0$ is suitably small we prove that the minimizers of the GP functional are unique, and this unique minimizer is radially symmetric.