  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.
