Existence of a ground state and scattering for a nonlinear Schroedinger   equation with critical growth
  We study the energy-critical focusing nonlinear Schr\"odinger equation with an energy- subcritical perturbation. We show the existence of a ground state in the four or higher dimensions. Moreover, we give a sufficient and necessary condition for a solution to scatter, in the spirit of Kenig-Merle [16]. 
Takafumi Akahori, Slim Ibrahim, Hiroaki Kikuchi and Hayato Nawa
math.AP