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

Abstract

We study positive solutions of the superlinear Lane-Emden inequality \(-Δu\ge σu^q\), \(q>1\), on infinite locally finite weighted graphs and connected domains of such graphs. We first prove that solvability is equivalent to the pointwise test \[ G_Ω(σg_Ω(o,\cdot)^q)(x)\le Cg_Ω(o,x) \] for every fixed pole \(o\inΩ\). We also prove sharp existence criteria under \textnormal{(VD)}, \textnormal{(PI)}, and \textnormal{(P$_0$)}, and 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 \[ \sum_{n\ge1}\frac{n^{2q-1}}{μ(B(o,n))^{q-1}}=\infty, \] then every nonnegative solution of \(-Δu\ge u^q\) is identically zero. The proof combines a flow decomposition with Hardy estimates along paths. For general positive $σ$, an intrinsic-metric version is obtained.