On the stability of ground states in 4D and 5D nonlinear Schrödinger equation including subcritical cases

We consider a class of nonlinear Schrödinger equation in four and five space dimensions with an attractive potential. The nonlinearity is local but rather general encompassing for the first time both subcritical and supercritical (in L) nonlinearities. We show that the center manifold formed by localized in space periodic in time solutions (bound states) is an attractor for all solutions with a small initial data. The proof hinges on dispersive estimates that we obtain for the time dependent, Hamiltonian, linearized dynamics around a one parameter family of bound states that “shadows” the nonlinear evolution of the system. The methods we employ are an extension to higher dimensions, hence different linear dispersive behavior, and to rougher nonlinearities of our previous results [10, 11, 7].