KAM tori for higher dimensional beam equations with constant potentials∗

In this paper, we consider the higher dimensional nonlinear beam equations utt + u + σu + f (u) = 0, with periodic boundary conditions, where the nonlinearity f (u) is a real– analytic function nearu = 0 with f (0) = f ′(0) = 0 and σ is a real parameter in an interval I ≡ [σ1, σ2]. It is proved that for ‘most’ positive parameters σ lying in the finite interval I, the above equations admit a family of small-amplitude, linearly stable quasi-periodic solutions corresponding to a Cantor family of finite dimensional invariant tori of an associated infinite dimensional dynamical system. The proof is based on an infinite dimensional KAM theorem, modified from (Geng and You 2006 Commun. Math. Phys. 262 343–72) and (Xu J et al 1996 Sci. China Ser. A 39 372–83, 383–94) with weaker non-degeneracy conditions. Mathematics Subject Classification: 37K55, 35B10, 35J10, 35Q40, 35Q55