Persistence of lower dimensional degenerate invariant tori with prescribed frequencies in Hamiltonian systems with small parameter

In this paper we develop some KAM techniques to prove the persistence of lower dimensional elliptic-type degenerate invariant tori with prescribed frequencies in Hamiltonian systems. The proof is based on a formal KAM theorem, which allows us to solve the equation of equilibrium points and choose the parameter of small divisors after the KAM iteration, instead of in each KAM step. The proof is also based on the Leray-Schauder continuation theorem, which insures the existence of a path of real roots of an approximating odd-order real polynomial which depends continuously on parameters. This result is very important for us to tackle the Melnikov condition in the elliptic-type degenerate case.