  We consider rotationally symmetric $p$-harmonic maps from the unit disk
$D^2\subset\real^2$ to the unit sphere $S^2\subset\real^3$, subject to
Dirichlet boundary conditions and with $1<p<\infty$. We show that the
associated energy functional admits a unique minimizer which is of class
$C^{\infty}$ in the interior and $C^1$ up to the boundary. We also show that
there exist infinitely many global solutions to the associated Euler-Lagrange
equation and we completely characterize them.
