Existence of multiple periodic solutions to asymptotically linear wave equations in a ball

This paper is concerned with the Dirichlet problem of the asymptotically linear wave equation u(tt) - Delta u = g(t, x, u) in a n-dimensional ball with radius R, where n > 1 and g(t, x, u) is radially symmetric in x and T-periodic in time. An interesting feature is that the solvable of the problem depends on the space dimension n and the arithmetical properties of R and T. Based on the spectral properties of the radially symmetric wave operator, we use the saddle point reduction and variational methods to construct at least three radially symmetric solutions with time period T, when T is a rational multiple of R and g(t, x, u) satisfies some monotonicity and asymptotically linear conditions.