Overdetermined problems for the rotationally invariant Poisson equation in model manifolds

We present rigidity results for overdetermined problems associated to the rotationally invariant Poisson equation (X) in a model manifold (Y) with warping function h. The variable r ranges in the interval [0, S), whose endpoint S is positive and possibly infinite. The first part of the paper deals with the problem (Z), where (W) is a bounded domain containing the point zero to M, corresponding to r = 0, ni is the exterior unit normal vector on delta omega, and f, fi, k are three prescribed functions. In the second part of the paper, we consider a similar overdetermined problem for the exterior Bernoulli problem in a domain (A), where (B) denotes the geodesic ball centered at O with radius R0, within the class of functions that vanish on (C). In both cases, we give conditions on f, fi and k implying that the solution u is radial and omega is a geodesic ball centered at O. Our results apply in particular to the three space forms Rn, Hn and Sn.