Threshold for Blowup and Stability for Nonlinear Schrödinger Equation with Rotation

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.