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

Abstract

We prove a nonexistence result for nonnegative solutions of the quasi-linear elliptic inequality \[ -\Delta_p u\ge \sigma(x)u^q \] on infinite locally finite connected weighted graphs, where $1 p-1$, $\sigma$ is a nonnegative Radon measure. Under the non-$p$-parabolic setting, we show that every nonnegative solution is identically zero, provided the volume of intrinsic balls satisfy \[ \int_1^\infty \frac{r^{\frac{pq}{p-1}-1}} {\nu(B_\rho(o,r))^{\frac{q-p+1}{p-1}}} \dd r =\infty, \] 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.