  This paper is devoted to the study of periodic solutions for a radially
symmetric semilinear wave equation in an $n$-dimensional ball. By combining the
variational methods and saddle point reduction technique, we prove there exist
at least three periodic solutions for arbitrary space dimension $n$. The
structure of the spectrum of the linearized problem plays an essential role in
the proof, and the construction of a suitable working space is devised to
overcome the restriction of space dimension.
