  We study the cubic-quartic nonlinear Schr\"odinger equation (NLS) in two and
three spatial dimension. This equation arises in the mean-field description of
Bose-Einstein condensates with Lee-Huang-Yang correction. We first prove global
existence of solutions in natural energy spaces which allow for the description
of self-bound quantum droplets with vorticity. Existence of such droplets,
described as central vortex states in 2D and 3D, is then proved using an
approach via constrained energy minimizers. A natural connection to the NLS
with repulsive inverse-square potential in 2D arises, leading to an orbital
stability result under the corresponding flow.
