A Volume-Growth Criterion for the p-Laplace Inequality on Weighted Graphs

TL;DR

This paper establishes a new volume-growth criterion for the nonexistence of nonnegative solutions to a p-Laplace inequality on weighted graphs, extending known sharp thresholds and allowing irregular growth.

Contribution

It introduces a flexible volume-growth criterion for nonexistence results on weighted graphs, utilizing an adapted finite-network current method for the p-Laplace operator.

Findings

Nonnegative solutions are trivial under the new volume-growth condition.

The criterion generalizes and improves upon existing sharp thresholds.

The proof combines path decomposition, Hardy estimates, and p-Green functions.

Abstract

We prove a nonexistence result for nonnegative solutions of the quasi-linear elliptic inequality \[ -\Delta_p u\ge u^\sigma \] on infinite locally finite connected weighted graphs, where $1<p<\infty$ and $\sigma>p-1$. Under the non-$p$-parabolic setting, we show that every nonnegative solution is identically zero, provided the weighted ball volumes $W_n=\mu(B(o,n))$ satisfy \[ \sum_{n=1}^{\infty} \frac{n^{\frac{p\sigma}{p-1}-1}} {W_n^{\frac{\sigma-p+1}{p-1}}} =\infty . \] This criterion recovers the known sharp pointwise critical volume-growth threshold and is strictly more flexible, since it allows irregular growth and does not require uniform upper bounds at every large radius. The proof adapts the finite-network current method to the $p$-Laplace setting, combining a path decomposition with one-dimensional Hardy estimates, $p$-parallel-sum bounds across metric cuts, and the global $p$-Green function furnished by non-$p$-parabolicity.