Rotationally symmetric p-harmonic maps from D^2 to S^2

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.