arxiv: 2604.24932 · v1 · submitted 2026-04-27 · 🧮 math.AP

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

Qingsong Gu , Lu Hao , Xueping Huang , Yuhua Sun

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classification 🧮 math.AP

keywords Lane-Emden inequalitiesweighted graphsGreen functionpositive solutionsvolume growthnonexistence criteriadiscrete Laplaciancritical exponents

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The pith

A volume-growth condition on any infinite weighted graph implies that the Lane-Emden inequality -Δu ≥ u^q has only the trivial nonnegative solution.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The paper shows that on an arbitrary infinite connected locally finite weighted graph, divergence of the series summing n to the 2q-1 over the measure of the ball of radius n to the q-1 forces every nonnegative solution of -Δu ≥ u^q to be identically zero. This nonexistence result requires no volume doubling, Poincaré inequality or ellipticity and resolves a conjecture on volume growth. The proof first establishes equivalence between the differential inequality and a Green-potential bound on arbitrary domains, then estimates the potential from below on large balls using finite-network Green functions and a unit-current decomposition. The same machinery yields existence criteria under additional regularity and determines Serrin-type critical exponents on lattices and domains with boundary.

Core claim

For any infinite connected locally finite weighted graph, if ∑_{n=1}^∞ n^{2q-1}/μ(B(o,n))^{q-1} diverges then every nonnegative solution of -Δu ≥ u^q is trivial. The argument proceeds by proving that any such solution satisfies the integral inequality G( u^q ) ≤ C u, then obtaining a lower bound for the left-hand side on large balls via a unit-current flow from the origin together with a Hardy-type estimate, all without assuming a weak maximum principle.

What carries the argument

The Green potential inequality G_Ω(σ g_Ω(o,·)^q)(x) ≤ C g_Ω(o,x), shown equivalent to the original Lane-Emden inequality on arbitrary domains, together with the unit-current decomposition on finite subnetworks that produces the volume-based lower bound.

Load-bearing premise

The weighted graph is infinite, connected and locally finite so that the discrete Laplacian and the Green function on its finite subnetworks are defined.

What would settle it

An explicit positive function u on some infinite connected locally finite weighted graph where the volume series diverges yet -Δu ≥ u^q holds at every vertex.

read the original 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.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Desk Editor's Note private letter to a colleague

The paper settles the volume-growth conjecture for nonexistence of solutions to -Δu ≥ u^q on general weighted graphs via Green-potential equivalence, but the Hardy estimate from unit-current decomposition needs checking for arbitrary edge weights.

read the letter

The main takeaway is that this work turns the stated volume-growth conjecture into a theorem: on any infinite connected locally finite weighted graph, divergence of that sum forces every nonnegative solution of the superlinear inequality to be zero. They reach it by first equating the differential inequality to a Green-potential bound, then using finite-network approximations plus a unit-current decomposition to extract a Hardy-type pointwise control that kills the solution when the sum diverges. No volume doubling, Poincaré inequality, or ellipticity is assumed, which is the concrete advance over earlier graph results that needed those conditions. The applications to critical exponents on Z^d, half-spaces, and orthants with Dirichlet data are straightforward and useful once the main criterion is in hand. The Green-kernel characterization itself, without a separate weak-maximum-principle step, is also a tidy organizational move. The soft spot sits in the passage from the divergent sum to the triviality conclusion. The unit-current decomposition and resulting Hardy estimate are asked to deliver a uniform comparison between the Green kernel and the volume integral even when positive edge weights are completely unrestricted beyond local finiteness. If the weights can be arbitrarily unbalanced, it is not obvious that the pointwise bound stays strong enough everywhere; the stress-test note flags exactly this possibility. The abstract asserts the result holds without further regularity, so the proof must close that gap, but the estimate is load-bearing and would benefit from explicit verification in the text. This is a paper for people already working on discrete elliptic inequalities and graph Laplacians. A reader who cares about sharp nonexistence criteria or about moving continuous Green-function methods to the graph setting will find the organization and the conjecture resolution helpful. The thinking is coherent and the claims are stated in checkable form, so the manuscript deserves a serious referee even if the Hardy step requires tightening or extra examples.

Referee Report

1 major / 2 minor

Summary. The paper establishes an equivalence between the Lane-Emden inequality −Δu ≥ σ u^q and the Green-potential inequality G_Ω(σ g_Ω(o,·)^q)(x) ≤ C g_Ω(o,x) on arbitrary connected domains in weighted graphs. The central result is a nonexistence theorem: on any infinite connected locally finite weighted graph, divergence of ∑_{n=1}^∞ n^{2q−1}/μ(B(o,n))^{q−1} implies that every nonnegative solution of −Δu ≥ u^q is trivial. The proof proceeds via finite-network Green functions, unit-current decomposition, and a Hardy-type estimate. Additional results give sharp existence criteria under (VD), (PI), (P_0), and (3G) assumptions, together with applications determining Serrin-type critical exponents on ℤ^d, half-spaces, and orthants with Dirichlet conditions.

Significance. If the central nonexistence claim holds, the work supplies a parameter-free, assumption-light criterion for triviality of nonnegative supersolutions that resolves a volume-growth conjecture on general weighted graphs. The reduction to Green-potential inequalities and the use of finite-network approximations constitute a technically clean approach that extends classical Euclidean Green-kernel criteria to the discrete setting without invoking doubling or Poincaré inequalities. The applications to critical exponents on lattices with boundary conditions further enhance the result's utility for discrete geometry and PDEs on graphs.

major comments (1)

[Proof of the main nonexistence result (finite-network approximation and unit-current decomposition)] The nonexistence theorem (stated in the abstract and proved via the strategy outlined there) rests on a Hardy-type pointwise bound obtained from the unit-current decomposition of the Green kernel. For graphs with arbitrarily varying positive edge weights, it is not immediate that this decomposition yields a uniform comparison between the Green potential and the volume integral that forces u ≡ 0 whenever the given sum diverges. The argument must explicitly verify that no additional ellipticity or bounded-weight-ratio assumption is hidden in the passage from the divergent-sum hypothesis to triviality.

minor comments (2)

[Abstract] The abstract refers to 'resolving a volume-growth conjecture' without citing the precise statement or reference of the conjecture being resolved.
[Introduction / Notation section] Notation for the Green function g_Ω and the associated potential operator G_Ω is introduced without an early self-contained definition or reference to the precise normalization used in the finite-network approximations.

Simulated Author's Rebuttal

1 responses · 0 unresolved

We are grateful to the referee for the thorough review and the encouraging evaluation of our results. The only major comment concerns the generality of the proof for the nonexistence theorem with respect to varying edge weights. We respond to this below and propose a minor clarification in the revised manuscript.

read point-by-point responses

Referee: [Proof of the main nonexistence result (finite-network approximation and unit-current decomposition)] The nonexistence theorem (stated in the abstract and proved via the strategy outlined there) rests on a Hardy-type pointwise bound obtained from the unit-current decomposition of the Green kernel. For graphs with arbitrarily varying positive edge weights, it is not immediate that this decomposition yields a uniform comparison between the Green potential and the volume integral that forces u ≡ 0 whenever the given sum diverges. The argument must explicitly verify that no additional ellipticity or bounded-weight-ratio assumption is hidden in the passage from the divergent-sum hypothesis to triviality.

Authors: We thank the referee for raising this important point about the robustness of our argument. The finite-network approximation and unit-current decomposition are developed in Section 3 of the manuscript using only the definition of the weighted Laplacian Δ on graphs with positive (but possibly unbounded) edge weights. Specifically, the unit flow is constructed via the Dirichlet principle on finite subgraphs, which does not require any ratio bounds on the weights. The resulting pointwise estimate for the Green potential is then combined with a discrete Hardy inequality that depends solely on the graph distance and the measure μ. The divergence of the given series then forces the supersolution to vanish by a standard iteration argument, again without ellipticity assumptions. We believe the proof is free of hidden conditions, as stated in the abstract. However, to make this explicit as requested, we will insert a clarifying sentence or short remark in the proof of Theorem 1.1 in the revised version. revision: partial

Circularity Check

0 steps flagged

No circularity: nonexistence follows from independent Green-function and decomposition constructions on arbitrary weighted graphs

full rationale

The central nonexistence theorem reduces the Lane-Emden inequality to a Green-potential comparison via finite-network approximations, then applies a unit-current decomposition and Hardy estimate to force triviality when the given volume sum diverges. These steps are built directly from the definitions of the weighted graph Laplacian, Green kernel, and current flows; none of the intermediate objects is defined in terms of the final conclusion or fitted to the target inequality. The divergent-sum test is an external, directly verifiable condition on the measure μ. No self-definitional loops, renamed empirical patterns, or load-bearing self-citations appear in the derivation chain.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

The central claims rest on the standard definition of the weighted graph Laplacian, the existence of the Green function on infinite graphs, and the validity of finite-network approximations; no free parameters or invented entities are introduced.

axioms (2)

domain assumption The graph is infinite, connected, and locally finite, allowing a well-defined positive Green function G_Ω
Invoked throughout the equivalence and the nonexistence argument.
standard math Finite-network Green functions converge to the infinite-graph Green function in the appropriate sense
Used in the proof strategy combining finite-network methods.

pith-pipeline@v0.9.0 · 5602 in / 1456 out tokens · 57610 ms · 2026-05-08T01:54:21.346025+00:00 · methodology

discussion (0)

Forward citations

Cited by 1 Pith paper

Reviewed papers in the Pith corpus that reference this work. Sorted by Pith novelty score.

A Volume-Growth Criterion for the p-Laplace Inequality on Weighted Graphs
math.AP 2026-05 unverdicted novelty 7.0

Nonnegative solutions to -Δ_p u ≥ u^σ on non-p-parabolic weighted graphs are zero whenever the divergent sum condition on weighted ball volumes holds.

Reference graph

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