  We obtain existence of minimizers for the $p$-capacity functional defined
with respect to a centrally symmetric anisotropy for $1 < p<\infty$, including
the case of a crystalline norm in $\mathbb R^N$. The result is obtained by a
characterization of the corresponding subdifferential and it applies for
unbounded domains of the form $\mathbb R^N \setminus \overline{\Omega}$ under
mild regularity assumptions (Lipschitz-continuous boundary) and no convexity
requirements on the bounded domain $\Omega$. If we further assume an interior
ball condition (where the Wulff shape plays the role of a ball), then any
minimizer is shown to be Lipschitz continuous.
