ABSTRACT:

We prove existence of Cantor families of periodic solutions for nonlinear wave equations in higher spatial dimensions, generalizing previous results of Bourgain, in case of nonlinearities of class C^k and assuming weaker non-resonance conditions. The proof is based on a differentiable Nash-Moser scheme where we just need estimates of interpolation type for the inverse linearized operators. A point of interest is that it is easier to achieve such estimates using Sobolev norms, instead of analytic or Gevrey ones.