A sharp integral criterion for the Lane--Emden system of inequalities on weighted graphs

We establish a sharp integral nonexistence criterion for the Lane--Emden system of inequalities $$-\Delta u\ge v^p,\qquad -\Delta v\ge u^q, \qquad p,q>0,\quad pq>1,$$ on arbitrary infinite, connected, locally finite weighted graphs. In the asymmetric case $p\ne q$, set $P=\max\{p,q\}$. If, for some root $o\in V$, $$\sum_{n=2}^{\infty} \frac{n^{2pq+2P-1}}{\mu(B(o,n))^{pq-1}}=\infty,$$ then every nonnegative solution $(u,v)$ satisfies $u\equiv v\equiv0$. The proof combines flow decomposition of the finite Green current with nonlinear testing. In the symmetric case $p=q>1$, the Liouville problem reduces, via the sum $u+v$, to the scalar criterion $$\sum_{n=2}^{\infty} \frac{n^{2p-1}}{\mu(B(o,n))^{p-1}}=\infty.$$ Weighted half-line examples show that the critical logarithmic endpoint in the asymmetric result is sharp.