Periodic solutions for the Schrodinger equation with nonlocal smoothing nonlinearities in higher dimension

We consider the nonlinear Schr¨dinger equation in higher dimension with Dirichlet boundary conditions and with a nonlocal smoothing nonlinearity. We prove the existence of small amplitude periodic solutions. In certain cases such solutions at leading order are wave packets. The main difficulty in the proof consists in solving a “small divisor problem” which we do by using a renormalization group approach.