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

TL;DR

This paper characterizes the existence of positive solutions to Lane-Emden inequalities on weighted graphs using Green potential criteria, resolving a volume-growth conjecture and providing sharp existence and nonexistence conditions.

Contribution

It establishes a graph analogue of Green-kernel criteria for superlinear inequalities, resolving a key conjecture without requiring volume doubling or Poincaré inequalities.

Findings

Nonexistence of positive solutions under certain volume-growth conditions.

Sharp existence criteria under volume doubling and Poincaré inequalities.

Determination of Serrin-type critical exponents on ^d and related domains.

Abstract

We study positive solutions of Lane--Emden-type inequalities on infinite, connected, locally finite weighted graphs. For arbitrary connected domains (with Dirichlet boundary when present), we establish the equivalence between \[ -\Delta u \ge \sigma u^q \] and the associated Green potential inequality. In particular, existence of positive solutions is characterized by the pointwise condition \[ G_{\Omega}\big(\sigma g_{\Omega}(o,\cdot)^q\big)(x) \le C\, g_{\Omega}(o,x). \] This yields a graph analogue of Green-kernel criteria for superlinear elliptic inequalities, without requiring a separate weak maximum principle. Our main result resolves a volume-growth conjecture: for any infinite, connected, locally finite weighted graph, if \[ \sum_{n=1}^{\infty} \frac{n^{2q-1}}{\mu(B(o,n))^{q-1}} = \infty, \] then every nonnegative solution of \( -\Delta u \ge u^q \) is trivial. This nonexistence result requires no volume doubling, Poincar\'e inequality, or ellipticity assumptions. The proof combines finite-network Green-function methods, a unit-current decomposition, and a Hardy-type estimate. We also derive sharp existence criteria under \textnormal{(VD)}, \textnormal{(PI)}, and \textnormal{(P$_0$)}, and, under the \textnormal{(3G)} condition, obtain a criterion via Green level sets. As applications, we determine Serrin-type critical exponents on \(\mathbb{Z}^d\), including half-spaces and orthants with Dirichlet boundary conditions.