  In this paper, by using Schwarz rearrangement and isoperimetric inequalities,
we prove comparison results for the solutions of Poisson equations on complete
Riemannian manifolds with $Ric\geq (n-1)\kappa$, $\, \kappa\geq 0$, which
extends the results in \cite{ANT-Talenti-a}. Furthermore, as applications of
our comparison results, we obtain the Saint-Venant inequality and Bossel-Daners
inequality for Robin Laplacian.
