Sharp Criteria for the existence of positive solutions to Lane-Emden-type inequalities on weighted graphs

We study positive solutions of the superlinear Lane-Emden inequality $-\mathrm{\Delta }u\ge \sigma u^q$, $q>1$, on infinite locally finite weighted graphs and connected domains. We first prove that solvability is equivalent to the pointwise test $$G_{\mathrm{\Omega }}(\sigma g_{\mathrm{\Omega }}(o,\cdot {)}^q)(x)\le C{g}_{\mathrm{\Omega }}(o,x)$$ for each fixed pole $o\in \mathrm{\Omega }$. We also prove sharp existence criteria under $\text{(VD)}$, $\text{(PI)}$, and $\text{(P}{}_0\text{)}$, with applications giving the Serrin-type exponents on ${\mathbb{Z}}^d$ and orthant domains, including half-spaces. Our main result resolves the volume-growth conjecture for arbitrary weighted graphs: if $$\sum _{n\ge 1}\frac{n^{2q-1}}{\mu (B(o,n){)}^{q-1}}=\mathrm{\infty },$$ then every nonnegative solution of $-\mathrm{\Delta }u\ge u^q$ is identically zero. The proof combines a flow decomposition with Hardy estimates along paths. For general positive $\sigma$, an intrinsic-metric version is obtained.