L^p-Parabolicity, the L^q-Liouville Property, and Nonlinear Elliptic Systems on Weighted Graphs

Abstract

We establish an equivalence between $L^p$-parabolicity, the $L^q$-Liouville property for nonnegative superharmonic functions, and the rigidity of nonnegative solutions to the elliptic system \[ \left\{ \begin{array}{rcl} -Δu\geq0,\\ Δ\bigl(|Δu|^{p-2}Δu\bigr)\geq0, \end{array} \right. \] where $1<p<\infty$ and $q=p/(p-1)$. More precisely, the three properties above are also equivalent to divergence of the $L^q$-Green kernel. For $1<p\le 2$, we obtain a volume-growth criterion under only a uniform positive lower bound on the vertex measure. Under $(\mathrm{VD})$, $(\mathrm{PI})$, and $(P_0)$, an auxiliary weighted graph whose associated random walk is aperiodic yields a two-sided Green-function estimate, an exact series criterion for $L^p$-parabolicity, and a critical logarithmic sufficient condition. Cayley graphs of polynomial growth illustrate the sharpness of the resulting threshold.