  In this work, we study some mathematical features for the action ground
states of the defocusing nonlinear Schr\"odinger equation with possible
rotation. Main attention is paid to characterizing the relation between the
action ground states and the energy ground states. Theoretical equivalence and
non-equivalence results have been established. Asymptotic behaviours of the
physical quantities are derived in some limiting parameter regimes. Numerical
evidence of non-equivalence is observed and numerical explorations for vortices
phenomena in action ground states are done.
