AI 中文总结
该研究针对加权图上的Lane–Emden不等式组，建立了非对称与对称情形下的尖锐积分不存在判据，结合流分解与非线性检验完成证明，且通过例子验证了判据的尖锐性。
AI 中文摘要
我们在任意无限、连通、局部有限的加权图上，建立了Lane–Emden不等式组\\( -\Delta u\ge v^p,\\ -\Delta v\ge u^q \\)（\\( p,q>0 \\)，\\( pq>1 \\)）的尖锐积分不存在判据。在非对称情形\\( p\ne q \\)下，设\\( P=\max\{p,q\} \\)，若对某根节点\\( o\in V \\)，级数\\( \sum_{n=2}^{\infty} \frac{n^{2pq+2P-1}}{\mu(B(o,n))^{pq-1}}=\infty \\)，则所有非负解\\( (u,v) \\)满足\\( u\equiv v\equiv0 \\)。证明结合了有限Green流的流分解与非线性检验。在对称情形\\( p=q>1 \\)下，Liouville问题通过和\\( u+v \\)简化为标量判据\\( \sum_{n=2}^{\infty} \frac{n^{2p-1}}{\mu(B(o,n))^{p-1}}=\infty \\)。加权半直线例子表明，非对称结果中的临界对数端点是尖锐的。
英文摘要
We establish a sharp integral nonexistence criterion for the Lane--Emden system of inequalities \[ -Δu\ge v^p,\qquad -Δv\ge u^q, \qquad p,q>0,\quad pq>1, \] on arbitrary infinite, connected, locally finite weighted graphs. In the asymmetric case $p\ne q$, set $P=\max\{p,q\}$. If, for some root $o\in V$, \[ \sum_{n=2}^{\infty} \frac{n^{2pq+2P-1}}{μ(B(o,n))^{pq-1}}=\infty, \] 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 \[ \sum_{n=2}^{\infty} \frac{n^{2p-1}}{μ(B(o,n))^{p-1}}=\infty. \] Weighted half-line examples show that the critical logarithmic endpoint in the asymmetric result is sharp.
