Remarks on nonlinear Schrödinger equations arising on rotational Bose–Einstein condensates

In this paper, we study the Cauchy problem for nonlinear Schrödinger equations arising as an effective model of the Bose–Einstein condensate in a magnetic trap rotating with an angular velocity. We first establish sufficient conditions showing the existence of global-in-time and finite time blow-up solutions to the equation. We next derive sharp thresholds for global existence versus finite time blow-up in the mass-critical and mass-supercritical cases. We also study the existence and strong instability of ground state standing waves related to the equation. Finally we prove the existence, non-existence, and orbital stability of prescribed mass standing waves when the rotational speed is smaller than a critical value.