  We show the existence and stability of ground state solutions (g.s.s.) for
$L^2$-critical magnetic nonlinear Schr\"odinger equations (mNLS) for a class of
unbounded electromagnetic potentials. We then give non-existence result by
constructing a sequence of vortex type functions in the setting of RNLS with an
anisotropic harmonic potential. These generalize the corresponding results in
[3] and [20]. The case of an isotropic harmonic potential for rotational NLS
has been recently addressed in [10]. Numerical results on the ground state
profile near the threshold are also included.
