Elliptic gradient estimate for the \(p \)-Laplace operator on the graph (Q6643134)

Let \(G(E,V)\) be a connected locally finite graph, and consider the \(p\)-Laplace operator, \(\Delta_p(\cdot)\) defined on \(G\), where \(G\) satisfies the curvature dimension \(CD^\psi_p(m,-K)\), with \(p\ge 2\), \(m>0\), \(K\ge 0\), for \(C^1\) concave function \(\psi:(0,+\infty)\to (0,+\infty)\). By definition the \(p\)-Laplace operator (\(p>1\)) on \(G\) is given by \N\[\N\Delta_p u(x) = \frac{1}{\mu(x)}\sum_{y\backsim x}w_{xy}|u(y)-u(x)|^{p-2}(u(y)-u(x))\N\]\Nfor any function \(u:V\to \mathbb{R}\), where \(\mu: V\to (0,+\infty)\) is a measure on \(V\) satisfying \(\sup_{x\in V} \mu(x)<+\infty\).\N\NA nonlinear eigenvalue problem of the form\N\begin{align*}\N\Delta_pu=-\lambda_p|u|^{p-2} u\N\end{align*}\Nis studied on \(G(V,E)\) for elliptic gradient estimates on positive solutions \(u\). Gradient estimates and their various applications have been widely studied on Riemannian manifolds under suitable curvature conditions since the seminal work of \textit{P. Li} and \textit{S. T. Yau} [Acta Math. 156, 154--201 (1986; Zbl 0611.58045)]. Finally, in the paper under review two Liouville type theorems and a Harnack inequality are derived as applications of the established gradient estimate.