Existence and stability of standing waves for nonlinear Schrödinger equations with a critical rotational speed

We study the existence and stability of standing waves associated with the Cauchy problem for the nonlinear Schrödinger equation (NLS) with a critical rotational speed and an axially symmetric harmonic potential. This equation arises as an effective model describing the attractive Bose–Einstein condensation in a magnetic trap rotating with an angular velocity. By viewing the equation as NLS with a constant magnetic field and with (or without) a partial harmonic confinement, we establish the existence and orbital stability of prescribed mass standing waves for the equation with mass-subcritical, mass-critical, and mass-supercritical nonlinearities. Our result extends a recent work of Bellazzini et al. (Commun Math Phys 353(1):229–251, 2017), where the existence and stability of standing waves for the supercritical NLS with a partial confinement were established.