Elliptic gradient estimate for the $p$−Laplace operator on the graph

Let $G(V,E)$ be a connected locally finite graph. In this paper we consider the elliptic gradient estimate for solutions to the equation\[\Delta_p u - \lambda_p {\lvert u \rvert}^{p-2} u\]on $G$ with the $\mathrm{CD}^\psi_p (m,-K)$ condition, where $p \geq 2$, $m \gt 0$, $K \geq 0$, and $\Delta_p$ denotes the $p\textrm{-}$Laplacian. As applications, we can derive Liouville theorems and the Harnack inequality.

Keywords

graph, $p$-Laplacian, $\mathrm{CD}^\psi_p (m,K)$ condition, Liouville theorem

2010 Mathematics Subject Classification

05C99, 53C21

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This paper is supported by the Natural of Science Foundation of Nantong City, Jiangsu Province (JC2023071).

Received 21 March 2021

Accepted 19 July 2023

Published 7 August 2024