A remark on an overdetermined problem in riemannian geometry

Let (M, g) be a Riemannian manifold with a distinguished point O and assume that the geodesic distance d from O is an isoparametric function. Let Ω ⊂ M be a bounded domain, with O ∈ Ω, and consider the problem Δpu = −1 in Ω with u = 0on∂Ω, where Δp is the p-Laplacian of g. We prove that if the normal derivative ∂νu of u along the boundary of Ω is a function of d satisfying suitable conditions, then Ω must be a geodesic ball. In particular, our result applies to open balls of Rn equipped with a rotationally symmetric metric of the form g = dt2 + ρ2 (t) gS, where gS is the standard metric of the sphere.