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

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

Scattering Below the Ground State for the 2\u3csub\u3ed\u3c/sub\u3e Radial Nonlinear Schrödinger Equation

We revisit the problem of scattering below the ground state threshold for the mass-supercritical focusing nonlinear Schrödinger equation in two space dimensions. We present a simple new proof that treats the case of radial initial data. The key ingredient is a localized virial/Morawetz estimate; the radial assumption aids in controlling the error terms resulting from the spatial localization