An exterior overdetermined problem for Finsler N-Laplacian in convex cones

We consider a partially overdetermined problem for anisotropic N-Laplace equations in a convex cone Σ intersected with the exterior of a bounded domain Ω in RN, N≥ 2. Under a prescribed logarithmic condition at infinity, we prove a rigidity result by showing that the existence of a solution implies that Σ ∩ Ω must be the intersection of the Wulff shape and Σ. Our approach is based on a Pohozaev-type identity and the characterization of minimizers of the anisotropic isoperimetric inequality inside convex cones.