Mass concentration near the boundary for attractive Bose–Einstein condensates in bounded domains

We are devoted to studying the ground states of trapped attractive Bose–Einstein condensates in a bounded domain $\Omega \subset R^2$, which can be described by minimizers of $L^2$-critical constraint Gross–Pitaevskii energy functional. It has been shown that there is a threshold $a^{\ast }>0$ such that minimizers exist if and only if the interaction strength $a$ satisfies $a<a^{\ast }$. In present paper, we prove that when the trapping potential $V\left(x\right)$ attains its flattest global minimum only at the boundary of $\Omega$, the mass of minimizers must concentrate near the boundary of $\Omega$ as $a\nearrow a^{\ast }$. This result extends the work of Luo and Zhu (2019).