We prove an existence result for the Poisson equation on non-compact Riemannian manifolds satisfying weighted Poincaré 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 Poincaré inequality and on the behavior of the Ricci curvature at infinity. We do not require any curvature or spectral assumptions on the manifold. In comparison with previous works, we can deal with a more general setting on the curvature bounds and without any spectral assumption.

我们证明了紧集上满足加权Poincaré不等式的非紧致黎曼流形上的Poisson方程的存在性结果。我们的结果适用于一大类流形，例如，所有具有最小正格林函数在无穷大时消失的非抛物线形。在源函数上，我们根据Poincaré不等式中出现的权重以及无穷远处Ricci曲率的行为来假设逐点急剧衰减。我们不需要在歧管上有任何曲率或频谱假设。与以前的工作相比，我们可以在曲率边界上处理更通用的设置，而无需任何频谱假设。