Existence of a countable set of periodic, spherically symmetric solutions of a nonlinear wave equation

Under suitable conditions countable solvability of the problem $-u_{tt}+\Delta u-g(u,r,t)=h(r,t)$ in $B_\pi$, $u(x,t)=u(x,t+T)$, $T>0$, $u(\partial B_\pi,t)=0$, where $B_\pi\subset\mathbf R^N$ is a ball of radius $\pi$, is proved.