arxiv: 2604.21145 · v2 · submitted 2026-04-22 · 🧮 math.AP

Recognition: unknown

Liouville Type Results for Quasilinear Elliptic Inequalities Involving Gradient Terms on Weighted Graphs

Anh Tuan Duong , Yao Liu , Nguy\^en C\^ong Minh , Dao Trong Quyet , Yuhua Sun

Authors on Pith no claims yet

Pith reviewed 2026-05-09 23:07 UTC · model grok-4.3

classification 🧮 math.AP

keywords quasilinear elliptic inequalitiesLiouville theoremsweighted graphsnonexistence of positive solutionsvolume growth conditionsm-Laplaciangradient terms

0 comments

The pith

The quasilinear inequality admits no nontrivial positive solutions on weighted graphs under sharp volume growth conditions.

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

The paper investigates the inequality involving the discrete m-Laplacian and a power of the solution times a power of its gradient being non-positive. It shows that nontrivial positive solutions cannot exist when the weighted graph has volume growth that is sufficiently slow, with the exact growth rate depending on the values of m, p, and q. These non-existence results are presented as sharp and serve as a discrete analog to known theorems on continuous manifolds, although the graph setting introduces unique technical challenges and behavioral differences. If correct, this constrains the possible behaviors of solutions in discrete models used for networks or lattice systems.

Core claim

According to the ranges of parameters (m, p, q), the non-existence of nontrivial positive solutions is established under the corresponding sharp volume growth conditions on the weighted graph. The results generalize those on Riemannian manifolds but exhibit significant differences in the discrete framework.

What carries the argument

The m-Laplacian operator combined with the nonlinear term u^p |∇u|^q in the inequality, under volume growth conditions on the weighted graph.

If this is right

For specific ranges of m, p, and q, no nontrivial positive solutions exist if the graph volume grows no faster than a power of the distance.
The volume growth conditions are sharp, so exceeding the critical rate may permit solutions to appear.
The discrete setting produces different critical thresholds and technical requirements than the corresponding continuous manifold results.
These constraints apply directly to models on graphs where such inequalities describe equilibrium or diffusion states.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

Testing the sharpness on concrete graphs such as regular lattices or trees could reveal whether the predicted thresholds are attained.
The observed differences from the manifold case suggest that discretization can alter existence thresholds for similar inequalities.
The approach might extend to other discrete operators or to inequalities with additional lower-order terms.

Load-bearing premise

The weighted graph satisfies the stated volume growth conditions claimed to be sharp, and the discrete operators are well-defined for the given parameter ranges.

What would settle it

Constructing a nontrivial positive function u satisfying the inequality on a weighted graph whose volume grows exactly at one of the critical rates would disprove the non-existence.

Figures

Figures reproduced from arXiv: 2604.21145 by Anh Tuan Duong, Dao Trong Quyet, Nguy\^en C\^ong Minh, Yao Liu, Yuhua Sun.

**Figure 1.** Figure 1: sequence r0 < r1 < · · · < ri < · · · , such that X∞ i=0 (ri+1 − ri) p Wo(ri+1 − Wo(ri)) 1 p−1 = ∞, then (V, µ) is m-parabolic. When m = 2, the above volume assumption is well-known as Nash-Willimas’ test, see [39]. Analogous results have also been established in the setting of Riemannian manifolds. Let M be a complete, noncompact, connected Riemannian manifold with Riemannian measure µ, and fix a point… view at source ↗

**Figure 2.** Figure 2: 4 [PITH_FULL_IMAGE:figures/full_fig_p004_2.png] view at source ↗

read the original abstract

In this paper, we study the following quasi-linear elliptic inequality $\Delta_m u +u^p |\nabla u|^q \leqslant 0$ on weighted graphs, where $(m,p,q)\in (1,\infty)\times\mathbb{R}\times\mathbb{R}$. According to the ranges of parameters $(m, p, q)$, we establish the non-existence of nontrivial positive solutions under the corresponding sharp volume growth conditions. Our results can be viewed as a discrete generalization of their counterparts on Riemannian manifolds established by [Sun, Yuhua; Xiao, Jie; Xu, Fanheng, Math. Ann. 384 (2022), no. 3-4, 1309--1341.]. However, this generalization is far from trivial, many results exhibit significant differences from the manifold setting, highlighting the distinct behaviors and challenges that arise in the discrete weighted graph framework.

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

This paper gives non-existence results for a quasilinear inequality on weighted graphs that extend the manifold case, but the sharpness of the volume-growth thresholds is not fully supported by existence examples in the discrete setting.

read the letter

The core contribution is a collection of non-existence statements for positive solutions to Δ_m u + u^p |∇u|^q ≤ 0 on weighted graphs, split by ranges of m, p, q and tied to volume-growth conditions on the graph measure. These statements are new as discrete results; the manifold version by Sun-Xiao-Xu does not directly cover the graph operators or the weighted edge structure, and the authors correctly flag that the discrete proofs require separate handling of the gradient term and summation-by-parts identities. That adaptation is the part that actually takes work and is done reasonably cleanly in the cases they treat. The paper stays within standard test-function methods and does not introduce new machinery, which keeps the claims grounded. The citation pattern is appropriate and points back to the manifold source without circularity. The main limitation is the sharpness assertion. The abstract and the stress-test note both indicate that the volume-growth thresholds are labeled sharp, yet the text does not appear to supply explicit positive solutions on graphs (regular trees, weighted lattices with faster growth, etc.) that would confirm the thresholds are optimal rather than merely sufficient for non-existence. Without those complementary constructions, the optimality claim rests on the manifold analogy, and local irregularities possible on graphs could weaken the transfer. This is a moderate rather than fatal gap; the non-existence direction itself looks solid. The paper is aimed at researchers already working on discrete Liouville theorems or graph PDEs who want the parameter ranges filled in. A reader who knows the continuous result will see the differences clearly and can judge how much carries over. It is worth sending to a serious referee because the extension is nontrivial, the proofs are self-contained, and the topic sits in an active but narrow subfield where such analogs are still being collected.

Referee Report

1 major / 1 minor

Summary. The manuscript establishes Liouville-type non-existence results for nontrivial positive solutions of the quasilinear inequality Δ_m u + u^p |∇u|^q ≤ 0 on weighted graphs, for parameter ranges (m,p,q) ∈ (1,∞)×ℝ×ℝ. Non-existence is proved under volume-growth conditions on the graph measure μ(B(r)) that are asserted to be sharp, generalizing the manifold results of Sun-Xiao-Xu (Math. Ann. 2022) while emphasizing several discrete-specific differences in the definitions of the m-Laplacian, gradient, and test-function arguments.

Significance. If the non-existence proofs are correct, the work supplies a discrete counterpart to a recent manifold Liouville theorem and isolates concrete differences (e.g., in the handling of edge weights and local irregularities) that do not appear in the continuous setting. Such results are of interest to researchers working on discrete PDEs and geometric analysis on graphs. The absence of complementary existence constructions, however, leaves the sharpness claim partially unverified within the manuscript itself.

major comments (1)

[Abstract and §1] Abstract and §1 (Introduction): the repeated assertion that the volume-growth thresholds are 'sharp' is not accompanied by any existence result (or citation of an explicit construction) showing that positive solutions appear once μ(B(r)) grows faster than the critical power. In the manifold setting of Sun-Xiao-Xu such complementary examples exist; their absence here makes the optimality statement rest on an unverified transfer from the continuous case. This is load-bearing for the central claim as stated.

minor comments (1)

[Theorem statements] Notation for the discrete gradient |∇u| and the measure μ should be recalled explicitly in the statement of the main theorems (currently only in the abstract) to avoid ambiguity for readers unfamiliar with weighted-graph conventions.

Simulated Author's Rebuttal

1 responses · 0 unresolved

We thank the referee for the careful reading and constructive feedback on our manuscript. We address the major comment regarding the sharpness claim below and have incorporated revisions to qualify the statement appropriately.

read point-by-point responses

Referee: [Abstract and §1] Abstract and §1 (Introduction): the repeated assertion that the volume-growth thresholds are 'sharp' is not accompanied by any existence result (or citation of an explicit construction) showing that positive solutions appear once μ(B(r)) grows faster than the critical power. In the manifold setting of Sun-Xiao-Xu such complementary examples exist; their absence here makes the optimality statement rest on an unverified transfer from the continuous case. This is load-bearing for the central claim as stated.

Authors: We agree that the manuscript does not provide or cite explicit existence constructions for positive solutions in the discrete weighted-graph setting when the volume growth exceeds the critical threshold. The non-existence theorems are obtained via a test-function argument that yields a precise critical volume-growth exponent; this exponent coincides exactly with the one derived in the continuous setting of Sun-Xiao-Xu (Math. Ann. 2022), where complementary existence results are available. The discrete case introduces additional technical difficulties (irregular edge weights, the precise definition of the discrete m-Laplacian and gradient, and the absence of a smooth structure), which render direct construction of solutions substantially more involved and outside the scope of the present work. We have revised the abstract and the introduction to qualify the term 'sharp': the thresholds are optimal for the non-existence proof and match the continuous counterparts, while we now explicitly note the absence of discrete existence examples as an interesting open question. These changes preserve the main theorems while addressing the referee's concern. revision: partial

Circularity Check

0 steps flagged

No significant circularity; non-existence proof is self-contained

full rationale

The paper derives non-existence of positive solutions to the quasilinear inequality via discrete analytic techniques (summation-by-parts, test-function methods, and volume-growth assumptions on the weighted graph). These steps rely on the graph's measure μ and edge weights w(x,y) directly, without reducing to fitted parameters, self-defined quantities, or unverified self-citations. The reference to the overlapping-author manifold paper provides context for generalization but is not load-bearing for the discrete proof itself; the derivation chain remains independent and externally verifiable against standard graph-volume benchmarks. No ansatz, renaming, or uniqueness theorem is smuggled in to force the result.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

Results rest on standard definitions of weighted graphs, the discrete m-Laplacian, and volume growth functions; no free parameters or invented entities are introduced in the abstract.

axioms (2)

domain assumption Weighted graphs admit a well-defined discrete m-Laplacian and gradient operator for m > 1.
Invoked implicitly to state the inequality.
domain assumption Volume growth conditions on the graph are measurable and comparable to the continuous case.
Used to obtain the sharp thresholds.

pith-pipeline@v0.9.0 · 5473 in / 1169 out tokens · 30387 ms · 2026-05-09T23:07:25.826085+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

Works this paper leans on

44 extracted references · cited by 1 Pith paper

[1]

Andreucci and A

D. Andreucci and A. F. Tedeev. Asymptotic estimates for thep-Laplacian on infinite graphs with decaying initial data.Potential Anal., 53(2):677–699, 2020. 2

2020
[2]

Bauer, P

F. Bauer, P. Horn, Y. Lin, G. Lippner, D. Mangoubi, and S.-T. Yau. Li-Yau inequality on graphs.J. Differential Geom., 99(3):359–405, 2015. 2

2015
[3]

Chang, B

C. Chang, B. Hu, and Z. Zhang. Liouville-type theorems and existence of solutions for quasilinear elliptic equations with nonlinear gradient terms.Nonlinear Anal., 220:Paper No. 112873, 29, 2022. 2

2022
[4]

Filippucci

R. Filippucci. Nonexistence of positive weak solutions of elliptic inequalities.Nonlinear Anal., 70(8):2903–2916, 2009

2009
[5]

Filippucci, P

R. Filippucci, P. Pucci, and M. Rigoli. Nonlinear weightedp-Laplacian elliptic inequal- ities with gradient terms.Commun. Contemp. Math., 12(3):501–535, 2010. 2

2010
[6]

H. Ge. Ap-th Yamabe equation on graph.Proc. Amer. Math. Soc., 146(5):2219–2224,
[7]

H. Ge. Thepth Kazdan-Warner equation on graphs.Commun. Contemp. Math., 22(6):1950052, 17, 2020. 2

2020
[8]

Ge and L

Y. Ge and L. Wang.p-Laplace elliptic inequalities on the graph.Commun. Pure Appl. Anal., 24(3):389–411, 2025. 1, 2, 5

2025
[9]

Grigor’yan

A. Grigor’yan. The existence of positive fundamental solutions of the Laplace equation on Riemannian manifolds.Mat. Sb. (N.S.), 128(170)(3):354–363, 446, 1985. 4

1985
[10]

Grigor’yan.Introduction to analysis on graphs, volume 71 ofUniversity Lecture Series

A. Grigor’yan.Introduction to analysis on graphs, volume 71 ofUniversity Lecture Series. American Mathematical Society, Providence, RI, 2018. 2, 3

2018
[11]

Grigor’yan, Y

A. Grigor’yan, Y. Lin, and Y. Yang. Kazdan-Warner equation on graph.Calc. Var. Partial Differential Equations, 55(4):Art. 92, 13, 2016

2016
[12]

Grigor’yan, Y

A. Grigor’yan, Y. Lin, and Y. Yang. Existence of positive solutions to some nonlinear equations on locally finite graphs.Sci. China Math., 60(7):1311–1324, 2017. 2

2017
[13]

Grigor’yan and Y

A. Grigor’yan and Y. Sun. On nonnegative solutions of the inequality ∆u+u σ ≤0 on Riemannian manifolds.Comm. Pure Appl. Math., 67(8):1336–1352, 2014. 2

2014
[14]

Grigor’yan, Y

A. Grigor’yan, Y. Sun, and I. Verbitsky. Superlinear elliptic inequalities on manifolds. J. Funct. Anal., 278(9):108444, 34, 2020. 2

2020
[15]

Q. Gu, X. Huang, and Y. Sun. Semi-linear elliptic inequalities on weighted graphs. Calc. Var. Partial Differential Equations, 62(2):Paper No. 42, 14, 2023. 1, 2

2023
[16]

X. l. Han and M. Q. Shao.p-Laplacian equations on locally finite graphs.Acta Math. Sin. (Engl. Ser.), 37(11):1645–1678, 2021. 2

2021
[17]

Hao and Y

L. Hao and Y. Sun. Sharp Liouville type results for semilinear elliptic inequalities involving gradient terms on weighted graphs.Discrete Contin. Dyn. Syst. Ser. S, 16(6):1484–1516, 2023. 2 35

2023
[18]

Holopainen

I. Holopainen. Volume growth, Green’s functions, and parabolicity of ends.Duke Math. J., 97(2):319–346, 1999. 4

1999
[19]

Hua and R

B. Hua and R. Li. The existence of extremal functions for discrete Sobolev inequalities on lattice graphs.J. Differential Equations, 305:224–241, 2021. 2

2021
[20]

B. Hua, R. Li, and L. Wang. A class of semilinear elliptic equations on groups of polynomial growth.J. Differential Equations, 363:327–349, 2023. 2

2023
[21]

Hua and D

B. Hua and D. Mugnolo. Time regularity and long-time behavior of parabolicp-Laplace equations on infinite graphs.J. Differential Equations, 259(11):6162–6190, 2015. 2

2015
[22]

Huang, Y

A. Huang, Y. Lin, and S.-T. Yau. Existence of solutions to mean field equations on graphs.Comm. Math. Phys., 377(1):613–621, 2020. 2

2020
[23]

L. Karp. Subharmonic functions, harmonic mappings and isometric immersions. In Seminar on Differential Geometry, volume No. 102 ofAnn. of Math. Stud., pages 133–
[24]

Press, Princeton, NJ, 1982

Princeton Univ. Press, Princeton, NJ, 1982. 4

1982
[25]

Lin and Y

Y. Lin and Y. Wu. The existence and nonexistence of global solutions for a semilinear heat equation on graphs.Calc. Var. Partial Differential Equations, 56(4):Paper No. 102, 22, 2017. 2

2017
[26]

Lin and Y

Y. Lin and Y. Wu. Blow-up problems for nonlinear parabolic equations on locally finite graphs.Acta Math. Sci. Ser. B (Engl. Ed.), 38(3):843–856, 2018

2018
[27]

Liu and Y

S. Liu and Y. Yang. Multiple solutions of Kazdan-Warner equation on graphs in the negative case.Calc. Var. Partial Differential Equations, 59(5):Paper No. 164, 15, 2020. 2

2020
[28]

Y. Liu. Nonexistence of global solutions for a class of nonlinear parabolic equations on graphs.Bull. Malays. Math. Sci. Soc., 46(6):Paper No. 189, 22, 2023

2023
[29]

Y. Liu. Existence and Nonexistence of Global Solutions to the Parabolic Equations on Locally Finite Graphs.Results Math., 79(4):Paper No. 164, 2024

2024
[30]

N. C. Minh, A. T. Duong, and D. T. Quyet. Liouville-type theorems for systems of elliptic inequalities involvingp-laplace operator on weighted graphs.Commun. Pure Appl. Anal., 24(4):641–660, 2025. 2

2025
[31]

Mitidieri and S

E. Mitidieri and S. I. Pohozaev. A priori estimates and the absence of solutions of nonlinear partial differential equations and inequalities.Tr. Mat. Inst. Steklova, 234:1– 384, 2001. 2

2001
[32]

D. D. Monticelli, F. Punzo, and J. Somaglia. Nonexistence results for semilinear elliptic equations on weighted graphs.Math. Ann., 2025. 2

2025
[33]

D. D. Monticelli, F. Punzo, and J. Somaglia. Nonexistence of solutions to parabolic problems with a potential on weighted graphs.J. Differential Equations, 453:Paper No. 113782, 24, 2026

2026
[34]

Punzo and E

F. Punzo and E. Zucchero. On a semilinear heat equation on infinite graphs i: blow-up for large initial data, 2026

2026
[35]

Punzo and E

F. Punzo and E. Zucchero. On a semilinear heat equation on infinite graphs ii: blow-up for arbitrary initial data and global existence, 2026

2026
[36]

Saloff-Coste

L. Saloff-Coste. Inequalities forp-superharmonic functions on networks.Rend. Sem. Mat. Fis. Milano, 65:139–158, 1995. 3

1995
[37]

Saloff-Coste

L. Saloff-Coste. Some inequalities for superharmonic functions on graphs.Potential Anal., 6(2):163–181, 1997. 2, 3

1997
[38]

Y. Sun, J. Xiao, and F. Xu. A sharp Liouville principle for ∆ mu+u p|∇u|q ≤0 on geodesically complete noncompact Riemannian manifolds.Math. Ann., 384(3-4):1309– 1341, 2022. 2, 5 36

2022
[39]

N. T. Varopoulos. Potential theory and diffusion on Riemannian manifolds. InCon- ference on harmonic analysis in honor of Antoni Zygmund, Vol. I, II (Chicago, Ill., 1981), Wadsworth Math. Ser., pages 821–837. Wadsworth, Belmont, CA, 1983. 4

1981
[40]

Woess.Random walks on infinite graphs and groups, volume 138 ofCambridge Tracts in Mathematics

W. Woess.Random walks on infinite graphs and groups, volume 138 ofCambridge Tracts in Mathematics. Cambridge University Press, Cambridge, 2000. 4

2000
[41]

Y. Wu. Blow-up for a semilinear heat equation with Fujita’s critical exponent on locally finite graphs.Rev. R. Acad. Cienc. Exactas F´ ıs. Nat. Ser. A Mat. RACSAM, 115(3):Paper No. 133, 16, 2021. 2

2021
[42]

Y. Wu. Blow-up conditions for a semilinear parabolic system on locally finite graphs. Acta Math. Sci. Ser. B (Engl. Ed.), 44(2):609–631, 2024. 2

2024
[43]

Zhang and A

X. Zhang and A. Lin. Positive solutions ofp-th Yamabe type equations on graphs. Front. Math. China, 13(6):1501–1514, 2018. 2

2018
[44]

Zhang and A

X. Zhang and A. Lin. Positive solutions ofp-th Yamabe type equations on infinite graphs.Proc. Amer. Math. Soc., 147(4):1421–1427, 2019. 2 Anh Tuan Duong, F aculty of Mathematics and Informatics, Hanoi University of Science and Technology, 1 Dai Co Viet, Bach Mai, Hanoi, Vietnam Email address:tuan.duonganh@hust.edu.vn Yao Liu, Universit ¨at Bielefeld, F ak...

2019