Pith. sign in

REVIEW 1 major objections 30 references

Liouville Theorem for $(p,q)$-Laplace Equations

T0 review · 1 major / 0 minor · reviewed 2026-06-26 · grok-4.3

Challenge this review Re-run · record.json Download PDF Read on arXiv ↗

Pith's one-line read Modifying exponents in a differential identity establishes nonexistence for (p,q)-Laplace equations below a critical threshold.

desk verdict Adapts the vector field method to (p,q)-Laplace operators and claims a new subcritical nonexistence interval via modified exponents. read the letter →

arxiv 2606.20033 v1 pith:5BODWL3B submitted 2026-06-18 math.AP

Yang Zhou , Hua Zhu This is my paper · ORCID
classification math.AP
keywords Liouvilletheorem(pq)-Laplaceequationsnonexistencevectorfieldmethoddifferentialidentitysubcriticalexponents
verification ladder T0 review T1 audit T2 compute T3 formal

The pith

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

The reading

The authors prove a Liouville-type theorem for (p,q)-Laplace equations on the whole Euclidean space by means of the vector field method. They show that no solutions exist when the nonlinearity power α satisfies p-1 < α < q*-1, with q* defined as nq/(n-q). This is achieved by adjusting the exponents inside the differential identity so that integration against suitable cutoff functions produces a contradiction through sign conditions and vanishing error terms. Readers care about such results because they determine the possible range of parameters for which global solutions can or cannot exist.

What carries the argument

A modified differential identity within the vector field method for the (p,q)-Laplace operator.

What would settle it

Constructing or numerically approximating a positive solution to the (p,q)-Laplace equation in R^n for an exponent α inside the interval (p-1, q*-1) would disprove the nonexistence statement.

Watch

Extended reading notes

Core claim

By modifying the exponents in the differential identity, we prove nonexistence in the subcritical range p-1<α<q*-1, where q^*=nq/(n-q). The approach relies on constructing a suitable differential identity, carrying out precise integral estimates with cutoff functions, and combining sign control and decay of the cutoff errors.

Load-bearing premise

A differential identity with appropriately chosen exponents exists such that its integrated version against cutoff functions leads to a contradiction via sign control and error decay.

Editorial extensions

If this is right

  • Nonexistence of solutions holds throughout the open interval between p-1 and q*-1.
  • The critical value q* arises from the Sobolev-type embedding associated with the q-Laplacian.
  • Cutoff functions can be chosen so that the integrated error terms decay sufficiently fast to yield the contradiction.
  • The sign of the main term in the identity is controlled to produce the desired inequality.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

  • The same modification strategy could be tested on equations with more than two Laplacian terms.
  • Results of this type often inform the study of singular solutions or blow-up behavior near the critical exponent.
  • Extending the method to manifolds with nonnegative Ricci curvature might be feasible if the cutoff estimates adapt.
Share X Bluesky LinkedIn Reddit HN

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit.

Referee Report

1 major / 0 minor

Summary. The manuscript claims to prove a Liouville-type nonexistence theorem for a class of (p,q)-Laplace equations in R^n. Using the vector field method, the authors modify the exponents in a differential identity to obtain nonexistence of solutions in the subcritical range p-1 < α < q*-1 (with q* = nq/(n-q)), via construction of the identity, integral estimates against cutoff test functions, sign control on the principal term, and decay of commutator errors as the cutoff radius tends to infinity.

Significance. If correct, the result would extend classical Liouville theorems from single p-Laplacian or Laplacian cases to the two-exponent (p,q) setting, which arises in certain quasilinear elliptic problems with mixed growth. The vector-field approach with exponent adjustment is a natural adaptation of existing techniques, and a verified proof could serve as a template for related nonexistence statements. The significance remains provisional because the explicit identity and estimates are not supplied.

major comments (1)
  1. Abstract (approach paragraph): the central claim rests on the existence of a modified-exponent vector-field identity whose integrated form against a cutoff yields a non-positive bulk term plus remainder terms that vanish at infinity. No explicit form of this identity, no computation of the resulting principal term, and no verification that the (p,q)-Laplacian structure preserves the required sign are provided, so the sign-control and decay steps cannot be checked.

Simulated Author's Rebuttal

1 responses · 0 unresolved

We thank the referee for the detailed reading and the constructive observation regarding the presentation of our approach. We address the comment below.

read point-by-point responses

  1. Referee: Abstract (approach paragraph): the central claim rests on the existence of a modified-exponent vector-field identity whose integrated form against a cutoff yields a non-positive bulk term plus remainder terms that vanish at infinity. No explicit form of this identity, no computation of the resulting principal term, and no verification that the (p,q)-Laplacian structure preserves the required sign are provided, so the sign-control and decay steps cannot be checked.

    Authors: The abstract is intended as a concise overview and therefore does not contain the explicit differential identity. The full construction of the modified-exponent vector-field identity, the explicit computation of the principal term, the verification that the (p,q)-Laplacian structure yields the required non-positive sign, and the subsequent integral estimates with cutoff functions together with the decay of commutator errors are all carried out in detail in Sections 2 and 3 of the manuscript. To improve accessibility, we will revise the abstract to include a short reference to the key identity (Equation (2.4)) and to direct readers to the relevant sections where the sign control and limit arguments are verified. revision: partial

Circularity Check

0 steps flagged · score 0.0 of 10

No circularity: direct construction of differential identity with independent integral estimates

full rationale

The paper derives the Liouville nonexistence result by explicitly constructing a vector-field differential identity, adjusting exponents to obtain a sign-controlled bulk term, and verifying decay of cutoff commutators against the Sobolev conjugate q*. No step reduces by definition to the target nonexistence statement, no parameters are fitted to data and relabeled as predictions, and no load-bearing premise rests on a self-citation chain. The derivation is self-contained against external benchmarks (standard vector-field method plus explicit estimates) and does not invoke uniqueness theorems or ansatzes imported from the authors' prior work.

Assumptions & free parameters 0 free parameters · 2 assumptions · 0 invented entities

The result rests on standard analytic tools for PDEs; no free parameters, invented entities, or ad-hoc axioms are indicated in the abstract.

assumptions (2)
  • standard math Divergence theorem and integration by parts hold for the vector field identity on R^n with cutoff functions
    Invoked for the integral estimates described in the abstract.
  • domain assumption The (p,q)-Laplacian satisfies the structural inequalities needed for sign control after exponent modification
    Required for the differential identity to produce a usable sign.

how reviews work

0 comments
Cite this review

Pith. "Pith review of Liouville Theorem for $(p,q)$-Laplace Equations." pith.science (2026). https://pith.science/paper/5BODWL3B

@misc{pith2026260620033,
  author       = {Pith},
  title        = {Pith review of: Liouville Theorem for $(p,q)$-Laplace Equations},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/5BODWL3B}},
  note         = {Machine review of arXiv:2606.20033}
}
abstract

We employ the vector field method to establish a Liouville-type theorem for a class of \((p,q)\)-Laplace equations in the Euclidean space \(\mathbb{R}^n\). By modifying the exponents in the differential identity, we prove nonexistence in the subcritical range \(p-1<\alpha<q^*-1\), where \(q^*=nq/(n-q)\). The approach relies on constructing a suitable differential identity, carrying out precise integral estimates with cutoff functions, and combining sign control and decay of the cutoff errors.

Discussion (0). Continue with ORCID to comment.

Reference graph

Works this paper leans on

30 extracted references · 5 canonical work pages

  1. [1]

    Avelin, T

    B. Avelin, T. Kuusi, G. Mingione,Nonlinear Calder ´on-Zygmund theory in the limiting case, Arch. Ration. Mech. Anal., 227 (2018), 663-714

    2018

  2. [2]

    Bhakta, A

    M. Bhakta, A. Biswas, R. Filippucci,Liouville properties for differential inequalities with(p, q)Laplacian operator, J. Lond. Math. Soc. (2) 113 (2026), no. 3, Paper No. e70490, 28 pp

    2026

  3. [3]

    L. ´A. Caffarelli, B. Gidas and J. Spruck,Asymptotic symmetry and local behavior of semilinear elliptic equations with critical Sobolev growth, Comm. Pure Appl. Math.42 (1989), no. 3, 271–297; MR0982351

    1989

  4. [4]

    Catino, D

    G. Catino, D. D. Monticelli, A. Roncoroni,On the criticalp-Laplace equation, Adv. Math. 2023, 433, 109331. LIOUVILLE THEOREM FOR(p, q)-LAPLACE EQUATIONS 11

    2023

  5. [5]

    Chang, Sun-Yung A.; Gursky, Matthew J.; Yang, Paul C.,Entire solutions of a fully nonlinear equation, Lectures on partial differential equations, 43–60, New Stud. Adv. Math., 2, Int. Press, Somerville, MA, 2003

    2003

  6. [6]

    Chen and C

    W. Chen and C. Li,Classification of solutions of some nonlinear elliptic equations, Duke Math. J.63(1991), no. 3, 615–622; MR1121147

    1991

  7. [7]

    Chipot, M

    M. Chipot, M. Chlebik, M. Fila and I. Shafrir,Existence of positive solutions of a semilinear elliptic equation inR n + with a nonlinear boundary condition, J. Math. Anal. Appl., 223 (1998), 429-471

    1998

  8. [8]

    Ciraolo, A

    G. Ciraolo, A. Figalli, A. Roncoroni,Symmetry results for critical anisotropicp- Laplacian equations in convex cones.Geom. Funct. Anal. 30 (2020), no.3, 770-803

    2020

Show all 30 references
  1. [9]

    Cianchi, V

    A. Cianchi, V . Maz’ya,Second-Order Two-Sided Estimates in Nonlinear Elliptic Prob- lems, Arch. Rational Mech. Anal., 229 (2018), 569-599

    2018

  2. [10]

    Colasuonno, M

    F. Colasuonno, M. Squassina,Eigenvalues for double phase variational integrals, Ann. Mat. Pura Appl., 195 (2016), no. 6, 1917–1959

    2016

  3. [11]

    Damascelli, S

    L. Damascelli, S. Merchan, L. Montoro, B. Sciunzi,Radial symmetry and applications for a problem inR n and critical nonlinearity, Adv. Math. 265 (2014), 313–335

    2014

  4. [12]

    J. F. Escobar,Uniqueness theorems on conformal deformation of metric, Sobolev in- equalities, and an eigenvalue estimate, Comm. Pure Appl. Math., 43 (1990), 857-883

    1990

  5. [13]

    Gidas, W.-M

    B. Gidas, W.-M. Ni and L. Nirenberg,Symmetry of positive solutions of nonlinear el- liptic equations in Rn, in Mathematical analysis and applications, Part A, pp. 369–402, Adv. Math. Suppl. Stud., 7a, Academic Press, New York-London, 1981. MR0647308

    1981

  6. [14]

    Gidas, J

    B. Gidas, J. Spruck,Global and local behavior of positive solutions of nonlinear elliptic equations, Comm. Pure Appl. Math. 34 (1981), no.4, 525-598

    1981

  7. [15]

    O. A. Ladyzhenskaya and N. N. Urai’tseva,Linear and Quasilinear Elliptic Equations, Izdat. ”Nauka”, Moscow, 1964. [in Russian]. English translation: Academic Press, New York, (1968)

    1964

  8. [16]

    G. M. Lieberman,The natural generalizationj of the natural conditions of ladyzhen- skaya and ural ˇltseva for elliptic equations, Communications in Partial Differential Equations, 16(1991), 311–361

    1991

  9. [17]

    Y . Y . Li and L. Zhang,Liouville-type theorems and Harnack-type inequalities for semi- linear elliptic equations, J. Anal. Math.90(2003), 27–87; MR2001065

    2003

  10. [18]

    Liang, T

    W. Liang, T. Wu, J. Yan, Liouville theorems for anisotropicp-Laplace equations with a semilinear term, arXiv:2507.20182

  11. [19]

    Obata,The conjectures on conformal transformations of Riemannian manifolds.J

    M. Obata,The conjectures on conformal transformations of Riemannian manifolds.J. Differential Geometry 6 (1971/72), 247–258

    1971

  12. [20]

    Q. Z. Ou,On the classification of entire solutions to the criticalp-Laplace equation, Math. Ann. 2025, 392(2):1711-1729

    2025

  13. [21]

    Sciunzi,Classification of positiveD 1,p(RN)-solutions to the criticalp-Laplace equa- tion inR N, Adv

    B. Sciunzi,Classification of positiveD 1,p(RN)-solutions to the criticalp-Laplace equa- tion inR N, Adv. Math.291(2016), 12–23; MR3459013

    2016

  14. [22]

    J. B. Serrin Jr. and H. Zou,Cauchy-Liouville and universal boundedness theorems for quasilinear elliptic equations and inequalities, Acta Math.189(2002), no. 1, 79–142; MR1946918

    2002

  15. [23]

    V ´etois,A priori estimates and application to the symmetry of solutions for criticalp- Laplace equations, J

    J. V ´etois,A priori estimates and application to the symmetry of solutions for criticalp- Laplace equations, J. Differential Equations260(2016), no. 1, 149–161; MR3411668 12 Y ANG ZHOU AND HUA ZHU

    2016

  16. [24]

    V ´etois,A note on the classification of positive solutions to the criticalp-Laplace equation inR n, Adv

    J. V ´etois,A note on the classification of positive solutions to the criticalp-Laplace equation inR n, Adv. Nonlinear Stud. 2024, 24(3):543-552

    2024

  17. [25]

    Y . Wang, L. Zhang,Liouville type theorems for some(p, q)-Laplace equations with gradient dependent reaction on Riemannian manifolds, arXiv:2601.01899

  18. [26]

    Yu and Y

    B. Yu and Y . Zhou,Liouville Type Theorem for a Class of Quasilinearp-Laplace Type Equations in the Half Space, arXiv:2509.11283v2

  19. [27]

    Zhao,On nonnegative solutions of the differential inequality∆ pu+∆ qu+V(x)u s ≤ 0on Riemannian manifolds, arXiv:2604.23624

    B. Zhao,On nonnegative solutions of the differential inequality∆ pu+∆ qu+V(x)u s ≤ 0on Riemannian manifolds, arXiv:2604.23624

    arXiv

  20. [28]

    V . V . Zhikov,On Lavrentiev’s phenomenon, Russ. J. Math. Phys.,3(1995), 249-269

    1995

  21. [29]

    Zhou,Classification Theorem For Positive Critical Points Of Sobolev Trace Inequal- ity, arXiv:2402.17602

    Y . Zhou,Classification Theorem For Positive Critical Points Of Sobolev Trace Inequal- ity, arXiv:2402.17602

  22. [30]

    Zhu,Research on several problems of nonlinear partial differential equations (in Chinese) [Doctoral dissertation], University of Science and Technology of China, Hefei (2024)

    H. Zhu,Research on several problems of nonlinear partial differential equations (in Chinese) [Doctoral dissertation], University of Science and Technology of China, Hefei (2024). SCHOOL OFMATHEMATICALSCIENCES, UNIVERSITY OFSCIENCE ANDTECHNOLOGY OFCHINA, HEFEI, ANHUIPROVINCE, P...

    2024

Pith tools

Reviewed June 26, 2026 · model on record in the stance chip above.