On the equivalence of Lp-parabolicity and Lq-liouville property on weighted graphs

Abstract

We study the equivalence between the $L^p$-parabolicity, the $L^q$-Liouville property of positive super-harmonic functions, and the existence of nonharmonic positive solutions to the following elliptic differential system \begin{equation*} \left\{ \begin{array}{lr} -Δu\geq 0, Δ(|Δu|^{p-2}Δu)\geq 0, \end{array} \right. \end{equation*} on weighted graphs, where $1\le p<\infty$, and $(p,q)$ are Hölder conjugate exponent pair. Furthermore, by refining a new technique on estimate of heat kernel, we can establish two-sided estimates of Green function on graph, and find the sharp volume growth criteria for the $L^q$-Liouville property on a large class of graphs. As an application, many non-trivial interesting examples are presented.