  We consider the focusing NLS with an angular momentum and a harmonic
potential, which models Bose-Einstein condensate under a rotating magnetic
trap. We give a sharp condition on the global existence and blowup in the
mass-critical case. We further consider the stability of such systems via
variational method. We determine that at the critical exponent $p=1+4/n$, the
mass of $Q$, the ground state for the NLS with zero potential, is the threshold
for both finite time blowup and orbital instability. Moreover, we prove a sharp
threshold theorem for the rotational NLS with an inhomogeneous nonlinearity.
The analysis relies on the existence of ground state as well as a virial
identity for the associated kinetic-magnetic operator.
