Superlinear elliptic inequalities on manifolds

TL;DR

This paper investigates conditions for the existence of positive solutions to a semilinear elliptic inequality on complete non-compact Riemannian manifolds, providing explicit criteria based on Green functions and curvature assumptions.

Contribution

It establishes necessary and sufficient conditions for positive solutions using Green function estimates, including explicit criteria under Ricci curvature and 3G-inequality assumptions.

Findings

Derived criteria for solution existence based on Green function properties.

Provided explicit conditions for manifolds with nonnegative Ricci curvature.

Connected solution existence to geometric properties of the manifold.

Abstract

Let $M$ be a complete non-compact Riemannian manifold and let $\sigma $ be a Radon measure on $M$. We study the problem of existence or non-existence of positive solutions to a semilinear elliptic inequaliy \begin{equation*} -\Delta u\geq \sigma u^{q}\quad \text{in}\,\,M, \end{equation*} where $q>1$. We obtain necessary and sufficent criteria for existence of positive solutions in terms of Green function of $\Delta $. In particular, explicit necessary and sufficient conditions are given when $M$ has nonnegative Ricci curvature everywhere in $M$, or more generally when Green's function satisfies the 3G-inequality.