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
\[
-\Delta_p u\ge \sigma(x)u^q
\]
on infinite locally finite connected weighted graphs, where $1<p<\infty$ 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
\[
\int_1^\infty
\frac{r^{\frac{pq}{p-1}-1}}
{\nu(B_\rho(o,r))^{\frac{q-p+1}{p-1}}}
\dd r
=\infty,
\]
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.