Flow Decomposition, Green Testing, and Lane--Emden Inequalities on Weighted Graphs

Abstract

We study positive solutions of the superlinear Lane--Emden inequality \[ -\Delta u\ge \sigma u^q,\qquad q>1, \] on infinite locally finite weighted graphs and connected domains. When the Dirichlet Green function is finite, the existence of a positive solution is equivalent to \[ G_\Omega\bigl(\sigma g_\Omega(o,\cdot)^q\bigr)(x) \le C g_\Omega(o,x) \] for some pole \(o\in\Omega\). Under Green function estimates, this yields sharp existence criteria and the Serrin-type exponents on \(\mathbb Z^d\) and orthant domains. For nonexistence, the principal method is flow decomposition. Its basic estimate bounds Green energy from below in terms of the relative capacities of intrinsic balls. %It yields annular conductance, capacity-to-infinity, and Nash--Williams cut-resistance criteria. For \(\sigma>0\), set \(\nu=\sigma\mu\). We show that if \(d_\rho\) is a complete \(\nu\)-adapted path metric and \[ \int_1^\infty \frac{r^{2q-1}}{\nu(B_{d_\rho}(o,r))^{q-1}}\,dr=\infty, \] then every nonnegative solution is identically zero. The proof combines a flow decomposition of the acyclic Green current, a pathwise Hardy estimate, and a relative capacity estimate. It requires none of (VD), (PI), (P$_0$), or the (3G) condition.