Liouville Type Results for Quasilinear Elliptic Inequalities Involving Gradient Terms on Weighted Graphs

TL;DR

This paper investigates Liouville type non-existence results for quasilinear elliptic inequalities with gradient terms on weighted graphs, extending known manifold results to a discrete setting with significant differences.

Contribution

It provides the first sharp volume growth conditions for non-existence of positive solutions to these inequalities on weighted graphs, generalizing manifold results.

Findings

Established non-existence of positive solutions under sharp volume growth conditions.

Identified significant differences from the manifold setting in the discrete case.

Abstract

In this paper, we study the following quasi-linear elliptic inequality $\Delta_m u +u^p |\nabla u|^q \leqslant 0$ on weighted graphs, where $(m,p,q)\in (1,\infty)\times\mathbb{R}\times\mathbb{R}$. According to the ranges of parameters $(m, p, q)$, we establish the non-existence of nontrivial positive solutions under the corresponding sharp volume growth conditions. Our results can be viewed as a discrete generalization of their counterparts on Riemannian manifolds established by [Sun, Yuhua; Xiao, Jie; Xu, Fanheng, Math. Ann. 384 (2022), no. 3-4, 1309--1341.]. However, this generalization is far from trivial, many results exhibit significant differences from the manifold setting, highlighting the distinct behaviors and challenges that arise in the discrete weighted graph framework.