Sharp Liouville type results for semilinear elliptic inequalities involving gradient terms on weighted graphs

TL;DR

This paper classifies conditions on parameters for the existence or nonexistence of positive solutions to a semilinear elliptic inequality with gradient terms on weighted graphs, establishing sharp volume growth criteria.

Contribution

It provides a complete classification of parameter pairs (p,q) for which sharp volume growth conditions determine solution existence or nonexistence.

Findings

Identifies parameter ranges for solution nonexistence.

Establishes sharp volume growth conditions.

Provides a comprehensive classification of (p,q) pairs.

Abstract

We study nonexistence and existence of nontrivial positive solutions to the following semilinear elliptic inequality involving gradient terms \[ \Delta u+u^p\left|\nabla u\right|^q\leq0, \] on weighted graphs, where $(p,q)\in\mathbb{R}^2$. We give a complete classification of $(p,q)$ under which sharp volume growth assumptions are established.