  We prove an existence result for the Poisson equation on non-compact
Riemannian manifolds satisfying weighted Poincar\'e inequalities outside
compact sets. Our result applies to a large class of manifolds including, for
instance, all non-parabolic manifolds with minimal positive Green's function
vanishing at infinity. On the source function we assume a sharp pointwise decay
depending on the weight appearing in the Poincar\'e inequality and on the
behavior of the Ricci curvature at infinity. We do not require any curvature or
spectral assumptions on the manifold.
