We present an abstract Nash-Moser implicit function theorem with parameters which covers  applications to the existence of finite dimensional, 

differentiable, invariant tori  of Hamiltonian PDEs with merely differentiable nonlinearities.  The new feature with respect to the classical Nash-Moser theory

is that the linearized operators are invertible, and satisfy the ``tame" estimates, only for proper subsets of the parameters. 

As applications we show the existence of  Cantor families of small amplitude periodic solutions of  nonlinear wave and Schrodinger equations  on T^d, 

Riemannian Zoll manifolds, compact Lie groups and homogeneous spaces, with merely differentiable nonlinearities and  weak non-resonance conditions

(generalizing previous results by Craig, Wayne, Bourgain). The NLS equation on Lie groups and homogeneous spaces arises as a mean field approximation 

of condensates of many-body lattice problems.

The eigenvalues  of the Laplace Beltrami operator  on such manjfolds are highly degenerate. Interestingly, in these cases, due to resonance phenomena, 

it is more natural to expect  solutions with a low regularity,  and our abstract Nash-Moser theorem is very natural to find solutions in Sobolev scales. 

We shall also discuss  applications to some equations arising in fluid dynamics as the KdV and Benjaimin-Ono equations.

References. 

- Sobolev periodic solutions of nonlinear wave equations in higher spatial dimensions, M. Berti, P. Bolle, Archive for Rational Mechanics, 2009.

- An abstract Nash-Moser Theorem with parameter and applications to PDEs, M. Berti, P. Bolle, M.Procesi, preprint 2009.

- Nonlinear wave and Schr"odinger equations on compact Lie groups and homogeneous spaces, M. Berti, M. Procesi, preprint 2009.