Bambusi D., Berti M.
A Birkhoff--Lewis type theorem for some Hamiltonian PDEs
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ABSTRACT. In this paper we give an extension of the Birkhoff--Lewis theorem to
some semilinear PDEs. Accordingly we prove existence of infinitely
many periodic orbits with large period accumulating at the
origin. Such periodic orbits bifurcate from resonant finite
dimensional invariant tori of the fourth order normal form of the
system. Besides standard nonresonance and nondegeneracy assumptions,
our main result is obtained assuming a regularizing property of the
nonlinearity. We apply our main theorem to a semilinear beam equation
and to a nonlinear Schr\"odinger equation with smoothing nonlinearity.