  We consider the Cauchy problem for (energy-subcritical) nonlinear
Schr\"odinger equations with sub-quadratic external potentials and an
additional angular momentum rotation term. This equation is a well-known model
for superfluid quantum gases in rotating traps. We prove global existence (in
the energy space) for defocusing nonlinearities without any restriction on the
rotation frequency, generalizing earlier results given in [11,12]. Moreover, we
find that the rotation term has a considerable influence in proving finite time
blow-up in the focusing case.
