Recognition: no theorem link
A Volume-Growth Criterion for the p-Laplace Inequality on Weighted Graphs
Pith reviewed 2026-05-12 04:52 UTC · model grok-4.3
The pith
Nonnegative solutions to the p-Laplace inequality vanish on non-p-parabolic graphs when a certain weighted volume sum diverges.
A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.
Core claim
We prove a nonexistence result for nonnegative solutions of the quasi-linear elliptic inequality -Δ_p u ≥ u^σ on infinite locally finite connected weighted graphs, where 1 < p < ∞ and σ > p-1. Under the non-p-parabolic setting, every nonnegative solution is identically zero, provided the weighted ball volumes W_n = μ(B(o,n)) satisfy the sum from n=1 to ∞ of n^{pσ/(p-1)-1} / W_n^{(σ-p+1)/(p-1)} = ∞. This criterion recovers the known sharp pointwise critical volume-growth threshold and is strictly more flexible, since it allows irregular growth and does not require uniform upper bounds at every large radius.
What carries the argument
The adapted finite-network current method for the p-Laplace setting, which combines path decomposition with one-dimensional Hardy estimates, p-parallel-sum bounds across metric cuts, and the global p-Green function provided by non-p-parabolicity.
Load-bearing premise
The graph is non-p-parabolic so that a positive p-Green function exists, and σ is strictly larger than p-1.
What would settle it
Finding a counterexample graph where the sum diverges but a positive solution to the inequality exists, or where the sum converges yet only the zero solution exists despite non-p-parabolicity.
read the original abstract
We prove a nonexistence result for nonnegative solutions of the quasi-linear elliptic inequality \[ -\Delta_p u\ge u^\sigma \] on infinite locally finite connected weighted graphs, where $1<p<\infty$ and $\sigma>p-1$. Under the non-$p$-parabolic setting, we show that every nonnegative solution is identically zero, provided the weighted ball volumes $W_n=\mu(B(o,n))$ satisfy \[ \sum_{n=1}^{\infty} \frac{n^{\frac{p\sigma}{p-1}-1}} {W_n^{\frac{\sigma-p+1}{p-1}}} =\infty . \] This criterion recovers the known sharp pointwise critical volume-growth threshold and is strictly more flexible, since it allows irregular growth and does not require uniform upper bounds at every large radius. The proof adapts the finite-network current method to the $p$-Laplace setting, combining a path decomposition with one-dimensional Hardy estimates, $p$-parallel-sum bounds across metric cuts, and the global $p$-Green function furnished by non-$p$-parabolicity.
Editorial analysis
A structured set of objections, weighed in public.
Referee Report
Summary. The manuscript proves a nonexistence result for nonnegative solutions of the quasi-linear elliptic inequality −Δ_p u ≥ u^σ (1 < p < ∞, σ > p−1) on infinite locally finite connected weighted graphs. Under the non-p-parabolic assumption, every such solution is identically zero whenever the weighted ball volumes W_n = μ(B(o,n)) satisfy the divergence condition ∑_{n=1}^∞ n^{pσ/(p−1)−1} / W_n^{(σ−p+1)/(p−1)} = ∞. The argument adapts the current method to the p-Laplacian via path decomposition, one-dimensional Hardy estimates on paths, p-parallel-sum bounds on metric cuts, and the global p-Green function furnished by non-p-parabolicity.
Significance. If correct, the result supplies a flexible volume-growth criterion that recovers known sharp thresholds while permitting irregular growth without uniform radius bounds. The combination of the global Green function with the adapted current method on graphs is a technical strength that cleanly handles the infinite setting and extends prior work on Liouville-type theorems for nonlinear inequalities.
minor comments (2)
- [§2] §2: The precise definition of non-p-parabolicity and the existence of the positive p-Green function should be recalled explicitly, including the normalization or the point o at which it is evaluated, to aid readers unfamiliar with the discrete setting.
- [§4] §4 (path decomposition step): Clarify how the p-parallel-sum bound is applied across the metric cuts when the weights are non-uniform; a short remark on the absence of any bounded-degree or uniform ellipticity assumption would strengthen the presentation.
Simulated Author's Rebuttal
We thank the referee for the positive evaluation of our manuscript and for recommending minor revision. The referee's summary accurately reflects the main theorem and the key elements of the proof strategy.
Circularity Check
No significant circularity
full rationale
The paper presents a direct analytic proof of nonexistence for nonnegative solutions to the p-Laplace inequality under the stated volume-growth divergence condition, relying on the external structural hypothesis of non-p-parabolicity (which supplies an independent global p-Green function) together with standard adaptations of path decomposition, one-dimensional Hardy inequalities, and p-parallel-sum estimates on cuts. None of these steps defines the target nonexistence statement in terms of itself, renames a fitted quantity as a prediction, or reduces the central claim to a self-citation chain; the volume-growth sum is an independent, externally verifiable criterion rather than a tautological re-expression of the solution property.
Axiom & Free-Parameter Ledger
axioms (3)
- domain assumption The graph is infinite, locally finite, connected, and weighted.
- domain assumption The graph is non-p-parabolic.
- standard math Standard properties of the p-Laplacian and one-dimensional Hardy inequalities hold on the graph.
Reference graph
Works this paper leans on
-
[1]
Saloff-Coste,Some inequalities for superharmonic functions on graphs, Potential Anal.6(1997), no
L. Saloff-Coste,Some inequalities for superharmonic functions on graphs, Potential Anal.6(1997), no. 2, 163–181
work page 1997
-
[2]
Characterizations of $p$-Parabolicity on Graphs
A. Adriani, F. Fischer, and A. G. Setti,Characterizations ofp-parabolicity on graphs, arXiv:2507.13696, 2025
work page internal anchor Pith review Pith/arXiv arXiv 2025
-
[3]
A. T. Duong, Y. Liu, N. C. Minh, D. T. Quyet, and Y. Sun, Liouville type results for quaslinear elliptic inequalities involving gradient terms on weighted graphs, arXiv: 2604.21145, preprint
work page internal anchor Pith review Pith/arXiv arXiv
-
[4]
Q. Gu, X. Huang, and Y. Sun,Semi-linear elliptic inequalities on weighted graphs, Calc. Var. Partial Differential Equations62(2023), no. 2, Paper No. 42, 14 pp
work page 2023
-
[5]
Q. Gu, L. Hao, X. Huang, and Y. Sun,Sharp criteria for the existence of positive solutions to Lane– Emden-type inequalities on weighted graphs, arXiv: 2604.24932, preprint
work page internal anchor Pith review Pith/arXiv arXiv
-
[6]
Y. Ge and L. Wang,p-Laplace elliptic inequalities on the graph, Commun. Pure Appl. Anal.24(2025), no. 3, 389–411. Department of mathematics, Nanjing University, Nanjing 210093, P. R. China Email address:qingsonggu@nju.edu.cn Universit¨at Bielefeld, F akult ¨at f ¨ur Mathematik, Postfach 100131, D-33501, Bielefeld, Germany Email address:lhao@math.uni-biele...
work page 2025
discussion (0)
Sign in with ORCID, Apple, or X to comment. Anyone can read and Pith papers without signing in.