  We consider a partially overdetermined problem for anisotropic $N$-Laplace
equations in a convex cone $\Sigma$ intersected with the exterior of a bounded
domain $\Omega$ in $\mathbb{R}^N$, $N\geq 2$. Under a prescribed logarithmic
condition at infinity, we prove a rigidity result by showing that the existence
of a solution implies that $\Sigma\cap\Omega$ must be the intersection of the
Wulff shape and $\Sigma$. Our approach is based on a Pohozaev-type identity and
the characterization of minimizers of the anisotropic isoperimetric inequality
inside convex cones.
