Existence and non-existence results for a logistic-type equation on manifolds

TL;DR

This paper investigates the existence and non-existence of positive solutions to a generalized logistic equation on Riemannian manifolds, linking solution behavior to geometric properties like volume growth.

Contribution

It provides new criteria based on manifold geometry and coefficient interactions for when solutions exist or do not exist.

Findings

Conditions for existence depend on volume growth of geodesic balls.

Conditions for non-existence relate to coefficient sizes and manifold geometry.

Results connect geometric analysis with nonlinear PDE solutions.

Abstract

We study the steady state solutions of a generalized logistic type equation on a complete Riemannian manifold. We provide sufficient conditions for existence, respectively non-existence of positive solutions, which depend on the relative size of the coefficients and their mutual interaction with the geometry of the manifold, which is mostly taken into account by means of conditions on the volume growth of geodesic balls.