Scattering above energy norm of solutions of a loglog energy-supercritical Schrödinger equation with radial data

We prove global well-posedness and scattering of -solutions of the loglog energy-supercritical Schrödinger equation , 0<c<cn, n={3,4}, with radial data , . This is achieved, roughly speaking, by extending Bourgain's argument in Bourgain (1999) [1] (see also Grillakis, 2000 [5]) and Tao's argument in Tao (2005) [10] in high dimensions.