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

Abstract

In this paper, we study the following quasi-linear elliptic inequality ${\Delta }_{m}u+u^p|\nabla u{|}^q⩽0$ 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.

37 pages,2 figures