Equivalences among parabolicity, comparison principle and capacity on complete Riemannian manifolds

Abstract

In this work we establish new equivalences for the concept of pparabolic Riemannian manifolds. We define a concept of comparison principle for elliptic PDE's on exterior domains of a complete Riemannian manifold M and prove that M is p-parabolic if and only if this comparison principle holds for the p-Laplace equation. We show also that the p-parabolicity of M implies the validity of this principle for more general elliptic PDS's and, in some cases, these results can be extended for non p-parabolic manifolds or unbounded solutions, provided that some growth of these solutions are assumed.