We prove an abstract Nash–Moser implicit function theorem with parameters which covers the applications to the existence of finite dimensional, differentiable, invariant tori of Hamiltonian PDEs with merely differentiable nonlinearities. The main new feature of the abstract iterative scheme is that the linearized operators, in a neighborhood of the expected solution, are invertible, and satisfy the “tame” estimates, only for proper subsets of the parameters. As an application we show the existence of periodic solutions of nonlinear wave equations on Riemannian Zoll manifolds. A point of interest is that, in presence of possibly very large “clusters of small divisors”, due to resonance phenomena, it is more natural to expect solutions with only Sobolev regularity.
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M. Berti, P. Bolle, M. Procesi, An abstract Nash–Moser theorem with parameters and applications to PDEs. Ann. Inst. H. Poincaré Anal. Non Linéaire 27 (2010), no. 1, pp. 377–399DOI 10.1016/J.ANIHPC.2009.11.010