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 rotating magnetic trap. give sharp condition on global existence blowup in mass-critical case. further stability of such systems via variational method. determine that at critical exponent $$p=1+4/n$$ , mass Q, ground state for zero is threshold both finite time orbital instability. Moreover, we prove theorem rotational inhomogeneous nonlinearity. The analysis relies as well virial identity associated kinetic-magnetic operator.