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 $-{\mathrm{\Delta }}_{g_{\mathcal{M}}}u=f(r)$ in a model manifold $\mathcal{M}=[0,S){\times }_h{\mathbb{S}}^{N-1}$ 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 \begin{array}{ll} -Δ_{g_\mathcal{M}} {u}=f(r) &in $Ω$, u=\varphi(r) &on $\partial Ω$, \frac{\partial u}{\partial ν} = κ(r) &on $\partial Ω$, \end{array} where $\mathrm{\Omega }\subset \mathcal{M}$ is a bounded domain containing the point $O\in \mathcal{M}$ corresponding to $r=0$, $\nu$ is the exterior unit normal vector on $\mathrm{\partial }\mathrm{\Omega }$, and $f$, $\phi$, $\kappa$ 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 $\mathrm{\Omega }\setminus {\bar{B}}_{R_0}(O)$, where $B_{R_0}(O)$ denotes the geodesic ball centered at $O$ with radius $R_0$, within the class of functions that vanish on $\mathrm{\partial }{B}_{R_0}(O)$. In both cases, we give conditions on $f$, $\phi$ and $\kappa$ implying that the solution $u$ is radial and $\mathrm{\Omega }$ is a geodesic ball centered at $O$. Our results apply in particular to the three space forms ${\mathbb{R}}^N$, ${\mathbb{H}}^N$ and ${\mathbb{S}}^N$.