Self-Bound vortex states in nonlinear Schrödinger equations with LHY correction

Abstract

We study the cubic-quartic nonlinear Schrödinger 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. In 2D we also obtain a sharp lower bound on their mass. Even though these vortex states are known to be unstable in general, we prove that they are indeed orbitally stable (as a set) under the flow associated to the NLS with repulsive inverse-square potential in 2D.