Existence and Asymptotic Behavior of Minimizers for Rotating Bose-Einstein Condensations in Bounded Domains

Abstract

This paper is concerned with the existence and mass concentration behavior of minimizers for rotating Bose-Einstein condensations (BECs) with attractive interactions in a bounded domain $\mathcal{D}\subset {\mathbb{R}}^2$. It is shown that, there exists a finite constant $a^{\ast }$, denoting mainly the critical number of bosons in the system, such that the least energy $e(a)$ admits minimizers if and only if $0<a<a^{\ast }$, no matter the trapping potential $V(x)$ rotates at any velocity $\Omega \ge 0$. This is quite different from the rotating BECs in the whole plane case, where the existence conclusions depend on the value of $\Omega$ (cf. \cite[Theorem 1.1]{GLY}). Moreover, by establishing the refined estimates of the rotation term and the least energy, we also analyze the mass concentration behavior of minimizers in a harmonic potential as $a\nearrow a^{\ast }$.