Existence and Non-existence of Ground State Solutions for Magnetic NLS

We show the existence and stability of ground state solutions (g.s.s.) for \(L^2\) -critical magnetic nonlinear Schrödinger 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 Arbunich et al. (Letters in Mathematical Physics, 109(6):1415–1432, 2019) and Van Duong Dinh (J Dyn Diff Equat, 2022). The case of an isotropic harmonic potential for rotational NLS has been recently addressed in Basharat et al. (Annales Henri Poincaré, 24(4):1377–1416, 2023). Numerical results on the ground state profile near the threshold are also included.