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arXiv · 2412.15420

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

Lu Hao·Yuhua Sun
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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} -\Delta u\geq 0, \Delta(|\Delta u|^{p-2}\Delta u)\geq 0, \end{array} \right. \end{equation*} on weighted graphs, where $1\leq p< \infty$, and $(p, q)$ are H\"{o}lder 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.

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Lu Hao, Yuhua Sun. 2024-12-19. On the equivalence of Lp-parabolicity and Lq-liouville property on weighted graphs. https://arxiv.org/abs/2412.15420

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