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 ohm subset of M be a bounded domain, with O is an element of ohm, and consider the problem Delta(p)u = - 1 in O with u = 0 on partial derivative ohm, where Delta(p) is the p-Laplacian of g. We prove that if the normal derivative partial derivative(v)u of u along the boundary of ohm is a function of d satisfying suitable conditions, then ohm must be a geodesic ball. In particular, our result applies to open balls of R-n equipped with a rotationally symmetric metric of the form g = dt(2) + rho(2)(t) gs, where gS is the standard metric of the sphere.