# The Poisson equation on Riemannian manifolds with weighted Poincar\'e inequality at infinity

## TL;DR

This paper establishes existence results for the Poisson equation on non-compact Riemannian manifolds with weighted Poincaré inequalities, covering a broad class of manifolds without curvature or spectral restrictions.

## Contribution

It provides the first existence theorem for the Poisson equation under weighted Poincaré inequalities at infinity without curvature or spectral assumptions.

## Key findings

- Existence of solutions under weighted Poincaré inequalities.
- Applicable to non-parabolic manifolds with Green's functions.
- Sharp decay conditions on source functions.

## Abstract

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.