Periodic solutions of some semilinear wave equations on balls and on spheres (Q1310498)

Radially symmetric solutions of the problem \[ u_{tt}- \Delta u+ g(u)= f(t,x), \qquad (t,x)\in \mathbb{R}\times B_ a^ n, \] \[ u(t,x)=0, \quad (t,x)\in\mathbb{R} \times S_ a^{n-1}, \qquad u(t+T,x)= u(t,x), \quad (t,x)\in \mathbb{R}\times B_ a^ n \] are considered where \(S_ a^{n-1}= \{x\in\mathbb{R}^ n\): \(\| x\|=a\}\), \(B_ a^ n= \{x\in \mathbb{R}^ n\): \(\| x\|<a\}\). In the case of even \(n\) it is supposed that \(g(t,x,u)\) satisfies nonresonance conditions of non-uniform type with respect to \(t\) and \(x\). In the case of odd \(n\) using recent existence theorems of the authors, a result of Smiley is improved by providing sharp conditions on the monotonicity constants of \(g(t,x,u)\) which insure the existence and uniqueness of the solution.