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

We consider rotationally symmetric $p$-harmonic maps from the unit disk $D^2\subset {\mathrm{\Re }}^2$ to the unit sphere $S^2\subset {\mathrm{\Re }}^3$, subject to Dirichlet boundary conditions and with $1<p<\mathrm{\infty }$. We show that the associated energy functional admits a unique minimizer which is of class $C^{\mathrm{\infty }}$ 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.