KAM tori for higher dimensional beam equation with a fixed constant potential

In this paper, we consider the higher dimensional nonlinear beam equation: u _tt + Δ² u + σu + f(u) = 0 with periodic boundary conditions, where the nonlinearity f(u) is a real-analytic function of the form f(u) = u ³ + h.o.t near u = 0 and σ is a positive constant. It is proved that for any fixed σ > 0, the above equation admits a family of small-amplitude, linearly stable quasi-periodic solutions corresponding to finite dimensional invariant tori of an associated infinite dimensional dynamical system.