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

We prove a nonexistence result for nonnegative solutions of the quasi-linear elliptic inequality $$-{\mathrm{\Delta }}_{p}u\ge \sigma (x)u^q$$ on infinite locally finite connected weighted graphs, where $1<p<\mathrm{\infty }$ and $q>p-1$, $\sigma$ 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 $${\int }_1^{\mathrm{\infty }}\frac{r^{\frac{pq}{p-1}-1}}{\nu (B_{\rho }(o,r){)}^{\frac{q-p+1}{p-1}}}\text{dd}r=\mathrm{\infty },$$ 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.