Existence and Asymptotic Behavior of Ground States for Rotating Bose-Einstein Condensates

We study ground states of two-dimensional Bose-Einstein condensates with repulsive ($a>0$) or attractive ($a<0$) interactions in a trap $V (x)$ rotating at the velocity $\Omega$. It is known that there exist critical parameters $a^*>0$ and $\Omega ^*:=\Omega^*(V(x))>0$ such that if $\Omega>\Omega^*$, then there is no ground state for any $a\in\R$; if $0\le \Omega <\Omega ^*$, then ground states exist if and only if $a\in(-a^*,+\infty)$. As a completion of the existing results, in this paper, we focus on the critical case where $0<\Omega=\Omega^*<+\infty$ and classify the existence and nonexistence of ground states for $a\in\R$. Moreover, for a suitable class of radially symmetric traps $V(x)$, employing the inductive symmetry method, we prove that up to a constant phase, the ground states must be real-valued, unique and free of vortices as $\Omega \searrow 0$, no matter whether the interactions of the condensates are repulsive or not.