Global Well-Posedness, Blow-Up and Stability of Standing Waves for Supercritical NLS with Rotation

We consider the focusing mass supercritical nonlinear Schrodinger equation with rotation iu(t) = -1/2 Delta u + 1/2V(x)u - vertical bar u vertical bar(p-1)u + L(Omega)u, (x, t) is an element of R-N x R, where N = 2 or 3 and V(x) is an anisotropic harmonic potential. Here L-Omega is the quantum mechanical angular momentum operator. We establish conditions for global existence and blow-up in the energy space. Moreover, we prove strong instability of standing waves under certain conditions on the rotation and the frequency of the wave. Finally, we construct orbitally stable standing waves solutions by considering a suitable local minimization problem. Those results are obtained for nonlinearities which are L-2-supercritical.