A symmetry result for the p-Laplacian in a punctured manifold

Let M be a Riemannian manifold such that its geodesic spheres centered at a point a ∈ M are isoperimetric and the distance function dist ( ⋅ , a ) is isoparametric, and let Ω ⊂ M be a bounded domain. We prove that if there exists a lower bounded nonconstant function u which is p-harmonic ( 1 < p ⩽ n ) in the punctured domain Ω ∖ { a } such that both u and ∂ u ∂ ν are constant on ∂Ω, then u is radial and ∂Ω is a geodesic sphere. The proof hinges on a combination of maximum principles, isoparametricity and the isoperimetric inequality.