Note on rotating BEC under a confining potential

Recently there have been analytical studies concerning the threshold on the instability for a rotating BEC under a trapping potential. In this note, we obtain a critical lower bound μ_∗ that ensures the existence of ground state solutions for the L²-critical RNLS in 2d and 3d, where an anisotropic harmonic potential is present. In addition, we perform numerical estimation for the corresponding stationary equation for the ground state and the vortex state solutions. Our findings indicate agreement on the sharp bound from both analytical and numerical demonstration. Among others we prove a blowup alternative result for RNLS with an inhomogeneous nonlinearity.