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

We consider the nonlinear Schrödinger equation in higher dimension with Dirichlet boundary conditions and with a nonlocal smoothing nonlinearity. We prove the existence of small amplitude periodic solutions. In the fully resonant case we find solutions which at leading order are wave packets, in the sense that they continue linear solutions with an arbitrarily large number of resonant modes. The main difficulty in the proof consists in a “small divisor problem” which we solve by using a renormalisation group approach.