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arxiv: 2606.05990 · v1 · pith:RWSNS4EUnew · submitted 2026-06-04 · 🧮 math.AP

A Pohozaev-type neck proof of a conditional Harnack inequality in the critical p-Laplacian setting

Guolin Qin , Yi Ru-Ya Zhang This is my paper

Pith reviewed 2026-06-28 00:41 UTC · model grok-4.3

classification 🧮 math.AP
keywords Pohozaev identityneck analysisHarnack inequalityp-Laplaciancritical exponentblow-up limitsAubin-Talenti bubblesquasilinear elliptic equations
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The pith

Under blow-up classification and neck control, solutions of the critical p-Laplace equation obey the bound (sup u in B_R) times (inf u in B_{2R})^{p-1} at most C R^{p-n}.

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

The paper proves a conditional Harnack inequality for positive weak solutions of the critical p-Laplace equation under a global Sobolev growth condition and monotonicity of s^{-(p^*-1)} g(s). It assumes two external inputs: that every bounded positive entire blow-up limit is an Aubin-Talenti p-bubble, and that normalized necks already satisfy a preliminary singular upper bound on their rate. A Pohozaev identity applied across those necks then upgrades the decay from the weaker singular rate |x|^{-(n-p)/p} to the sharp fundamental-solution rate |x|^{-(n-p)/(p-1)}. A reader would care because the usual Kelvin-transform and moving-sphere techniques that work for the semilinear case p=2 are unavailable once p differs from 2.

Core claim

Under the classification that all bounded positive entire blow-up limits are Aubin-Talenti p-bubbles together with a preliminary singular-rate upper bound on normalized necks, any positive weak solution u of -Δ_p u = g(u) in the ball B_{3R} satisfies (sup_{B_R} u) (inf_{B_{2R}} u)^{p-1} ≤ C R^{p-n}. The argument uses the Pohozaev identity on annular neck regions to improve the pointwise decay rate on the necks to the sharp p-harmonic rate.

What carries the argument

The Pohozaev-neck argument, which applies the Pohozaev identity in annular transition regions to upgrade the decay rate of the solution between bubbles.

If this is right

  • The stated Harnack bound holds for all such solutions once the two hypotheses are granted.
  • The pointwise decay on the necks reaches exactly the fundamental-solution exponent (n-p)/(p-1).
  • The same neck argument replaces conformal methods for any p between 1 and n.
  • The inequality is available under the given monotonicity assumption on g without further structural restrictions.

Where Pith is reading between the lines

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

  • If the classification of blow-up limits can be proved independently, the Harnack inequality becomes unconditional in this setting.
  • The neck technique may apply to other quasilinear equations once a comparable classification result is known.
  • The improved decay rate supplies a new tool for analyzing isolated singularities of critical p-Laplace equations.
  • Similar annular Pohozaev identities could be tested on higher-order or anisotropic operators with comparable scaling.

Load-bearing premise

That every bounded positive entire blow-up limit is an Aubin-Talenti p-bubble and that the normalized necks already obey a preliminary singular-rate upper bound.

What would settle it

A positive weak solution in some ball B_{3R} for which (sup in B_R) times (inf in B_{2R})^{p-1} exceeds every constant times R^{p-n}, or a bounded positive entire solution that is not an Aubin-Talenti p-bubble.

read the original abstract

We prove a conditional Schoen-type Harnack inequality for positive weak solutions of the critical $p$-Laplace equation $$ -\Delta_p u=g(u),\qquad 1<p<n, $$ under a global critical Sobolev growth assumption and the monotonicity condition that $s^{-(p^*-1)}g(s)$ is nonincreasing. The result is conditional on two inputs, the classification of bounded positive entire blow-up limits as Aubin--Talenti $p$-bubbles and a preliminary singular-rate upper control on the normalized necks. Under these two hypotheses, solutions in $B_{3R}$ satisfy $$ \Big(\sup_{B_R}u\Big)\Big(\inf_{B_{2R}}u\Big)^{p-1} \le C R^{p-n}. $$ The main point is a Pohozaev-neck argument which upgrades the preliminary singular decay rate $|x|^{-(n-p)/p}$ to the sharp $p$-harmonic fundamental rate $|x|^{-(n-p)/(p-1)}$. The argument replaces the Kelvin-transform and moving-sphere methods available in the conformally invariant semilinear case $p=2$, but unavailable for the general $p$-Laplacian.

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.

Referee Report

1 major / 2 minor

Summary. The manuscript claims a conditional Schoen-type Harnack inequality for positive weak solutions of -Δ_p u = g(u) (1 < p < n) under global critical Sobolev growth and the monotonicity condition that s^{-(p^*-1)}g(s) is nonincreasing. Under the two explicit hypotheses that bounded positive entire blow-up limits are Aubin-Talenti p-bubbles and that normalized necks satisfy a preliminary upper bound of order |x|^{-(n-p)/p}, solutions in B_{3R} obey (sup_{B_R} u)(inf_{B_{2R}} u)^{p-1} ≤ C R^{p-n}. The proof consists of a Pohozaev-neck argument that upgrades the preliminary singular decay to the sharp p-harmonic fundamental solution rate |x|^{-(n-p)/(p-1)}.

Significance. If the two stated hypotheses hold, the Pohozaev-neck upgrade supplies a method for obtaining sharp Harnack control in the critical p-Laplacian setting where Kelvin-transform and moving-sphere techniques are unavailable. The argument is logically self-contained once the inputs are granted and isolates the precise rate-improvement step needed for the inequality.

major comments (1)
  1. [Abstract] Abstract (paragraph beginning 'The result is conditional on two inputs'): the claimed inequality follows from the Pohozaev-neck upgrade only after the two external hypotheses are granted; the manuscript supplies neither a proof nor a citation establishing the bubble classification or the preliminary neck bound, so the central claim remains conditional on work outside the present scope.
minor comments (2)
  1. [Introduction] The introduction should contain a short paragraph summarizing the logical dependence on the two inputs and indicating where in the literature (or in forthcoming work) each hypothesis is expected to be verified.
  2. Notation for the normalized neck quantities and the precise statement of the preliminary rate |x|^{-(n-p)/p} should be introduced before the Pohozaev identity is applied, to make the upgrade step easier to follow.

Simulated Author's Rebuttal

1 responses · 0 unresolved

We thank the referee for the careful reading and the recommendation of minor revision. The manuscript is explicitly conditional on the two stated hypotheses, as already indicated in the abstract and introduction; we address the comment below.

read point-by-point responses

  1. Referee: [Abstract] Abstract (paragraph beginning 'The result is conditional on two inputs'): the claimed inequality follows from the Pohozaev-neck upgrade only after the two external hypotheses are granted; the manuscript supplies neither a proof nor a citation establishing the bubble classification or the preliminary neck bound, so the central claim remains conditional on work outside the present scope.

    Authors: We agree that the result is conditional on the bubble classification and the preliminary neck bound, which are stated as explicit hypotheses both in the abstract and in the body of the paper. The contribution of the manuscript is the Pohozaev-neck upgrade step that improves the decay rate once these inputs are granted; the paper does not claim to prove the classification or the preliminary bound. To address the concern we will revise the abstract to make the conditional nature even more prominent and will add citations to existing literature where the classification of bounded entire solutions as p-bubbles and preliminary neck estimates have been established (or are treated as standard assumptions) for the critical p-Laplacian. A self-contained proof of the full classification lies outside the scope of this work. revision: partial

Circularity Check

0 steps flagged

No circularity detected; derivation is conditional on explicitly external inputs with independent upgrade step

full rationale

The paper states its Harnack inequality is conditional on two external inputs (classification of blow-up limits as Aubin-Talenti p-bubbles; preliminary neck rate control) and supplies a Pohozaev-neck argument to upgrade the preliminary rate to the sharp one. No quoted step reduces the final bound to these inputs by definition, by fitting, or by a self-citation chain that itself lacks independent verification. The derivation chain therefore adds content given the stated assumptions rather than reproducing them. This matches the default non-circular case.

Axiom & Free-Parameter Ledger

0 free parameters · 0 axioms · 0 invented entities

Abstract-only review supplies no explicit free parameters, axioms, or invented entities; the result rests on two external classification hypotheses whose status is not detailed here.

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Works this paper leans on

43 extracted references · 2 canonical work pages · 2 internal anchors

  1. [1]

    Bianchi,Non-existence of positive solutions to semilinear elliptic equations onR n orR n + through the method of moving planes, Comm

    G. Bianchi,Non-existence of positive solutions to semilinear elliptic equations onR n orR n + through the method of moving planes, Comm. Partial Differential Equations 22 (1997), no. 9-10, 1671–1690

    1997

  2. [2]

    Brezis, Y

    H. Brezis, Y. Y. Li, and I. Shafrir,A sup + inf inequality for some nonlinear elliptic equations involving exponential nonlinearities, J. Funct. Anal. 115 (1993), no. 2, 344–358

    1993

  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

    1989

  4. [4]

    L. ´A. Caffarelli, T. Jin, Y. Sire, and J. Xiong,Local analysis of solutions of fractional semi-linear elliptic equations with isolated singularities, Arch. Ration. Mech. Anal. 213 (2014), no. 1, 245–268

    2014

  5. [5]

    Catino, D

    G. Catino, D. D. Monticelli, and A. Roncoroni,On the criticalp-Laplace equation, Adv. Math. 433 (2023), 109331

    2023

  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

    1991

  7. [7]

    Chen and C

    W. Chen and C. Li,A sup + inf inequality nearR= 0, Adv. Math. 220 (2009), no. 1, 219–245

    2009

  8. [8]

    Chen and C.-S

    C.-C. Chen and C.-S. Lin,Local behavior of singular positive solutions of semilinear elliptic equations with Sobolev exponent, Duke Math. J. 78 (1995), no. 2, 315–334

    1995

  9. [9]

    Chen and C.-S

    C.-C. Chen and C.-S. Lin,Estimates of the conformal scalar curvature equation via the method of moving planes, Comm. Pure Appl. Math. 50 (1997), no. 10, 971–1017. A POHOZAEV-NECK PROOF OF A CRITICAL P-LAPLACIAN HARNACK INEQUALITY 25

    1997

  10. [10]

    Chen and C.-S

    C.-C. Chen and C.-S. Lin,A sharp sup + inf inequality for a nonlinear elliptic equation inR 2, Comm. Anal. Geom. 6 (1998), no. 1, 1–19

    1998

  11. [11]

    Chen and C.-S

    C.-C. Chen and C.-S. Lin,Estimates of the conformal scalar curvature equation via the method of moving planes. II, J. Differential Geom. 49 (1998), no. 1, 115–178

    1998

  12. [12]

    L. Chen, W. Dai, C. Gui, and Y. Luo,Liouville theorems forp-Laplacian equations in convex cones without finite-energy condition, arXiv:2605.29281

    work page internal anchor Pith review Pith/arXiv arXiv

  13. [13]

    Classification results for bounded positive solutions to the critical $p$-Laplace equation

    G. Ciraolo, M. Gatti,Classification results for bounded positive solutions to the criticalp-Laplace equation, arXiv:2510.23243

    work page internal anchor Pith review Pith/arXiv arXiv

  14. [14]

    Ciraolo, A

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

    2020

  15. [15]

    T. H. Colding and W. P. Minicozzi II,Width and finite extinction time of Ricci flow, Geom. Topol. 12 (2008), no. 5, 2537–2586

    2008

  16. [16]

    Damascelli, S

    L. Damascelli, S. Merch´ an, L. Montoro, and B. Sciunzi,Radial symmetry and applications for a problem involving the−∆ p(·)operator and critical nonlinearity inR N, Adv. Math. 265 (2014), 313–335

    2014

  17. [17]

    Da Lio, T

    F. Da Lio, T. Rivi` ere, and D. Schlagenhauf,Morse index stability for Sacks–Uhlenbeck approx- imations for harmonic maps into a sphere, Nonlinear Anal. 264 (2026), Paper No. 113987, 39 pp

    2026

  18. [18]

    Esposito,A classification result for the quasi-linear Liouville equation, Ann

    P. Esposito,A classification result for the quasi-linear Liouville equation, Ann. Inst. H. Poincar´ e C Anal. Non Lin´ eaire 35 (2018), no. 3, 781–801

    2018

  19. [19]

    Esposito and M

    P. Esposito and M. Lucia,Harnack inequalities and quantization properties for then-Liouville equation, Calc. Var. Partial Differential Equations 63 (2024), no. 6, Paper No. 159, 17 pp

    2024

  20. [20]

    Gidas, W

    B. Gidas, W. M. Ni, and L. Nirenberg,Symmetry and related properties via the maximum prin- ciple, Comm. Math. Phys. 68 (1979), no. 3, 209–243

    1979

  21. [21]

    Jin and J

    T. Jin and J. Xiong,Asymptotic symmetry and local behavior of solutions of higher order confor- mally invariant equations with isolated singularities, Ann. Inst. H. Poincar´ e C Anal. Non Lin´ eaire 38 (2021), no. 4, 1167–1216

    2021

  22. [22]

    Kichenassamy and L

    S. Kichenassamy and L. V´ eron,Singular solutions of thep-Laplace equation, Math. Ann. 275 (1986), no. 4, 599–615

    1986

  23. [23]

    J. L. Lewis,Regularity of the derivatives of solutions to certain degenerate elliptic equations, Indiana Univ. Math. J. 32 (1983), no. 6, 849–858

    1983

  24. [24]

    Li and Y

    A. Li and Y. Y. Li,On some conformally invariant fully nonlinear equations, Comm. Pure Appl. Math. 56 (2003), no. 10, 1416–1464

    2003

  25. [25]

    Li and Y.Y

    A. Li and Y.Y. Li,On some conformally invariant fully nonlinear equations, Part II: Liouville, Harnack and Yamabe, Acta Math. 195 (2005), 117–154

    2005

  26. [26]

    Y. Y. Li,Prescribing scalar curvature onS n and related problems. I, J. Differential Equations 120 (1995), no. 2, 319–410

    1995

  27. [27]

    Y. Y. Li,Prescribing scalar curvature onS n and related problems. II. Existence and compactness, Comm. Pure Appl. Math. 49 (1996), no. 6, 541–597

    1996

  28. [28]

    Li,Harnack type inequality: the method of moving planes, Comm

    Y.Y. Li,Harnack type inequality: the method of moving planes, Comm. Math. Phys. 200 (1999), no. 2, 421–444

    1999

  29. [29]

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

    2003

  30. [30]

    Obata,The conjecture on conformal transformations of Riemannian manifolds, J

    M. Obata,The conjecture on conformal transformations of Riemannian manifolds, J. Differential Geom. 6 (1971), 247–258

    1971

  31. [31]

    Ou,On the classification of entire solutions to the criticalp-Laplace equation, Math

    Q. Ou,On the classification of entire solutions to the criticalp-Laplace equation, Math. Ann. 392 (2025), no. 2, 1711–1729. 26 GUOLIN QIN, YI RU-YA ZHANG

    2025

  32. [32]

    Pucci and J

    P. Pucci and J. Serrin,The Maximum Principle, Progress in Nonlinear Differential Equations and their Applications, vol. 73, Birkh¨ auser, Basel, 2007

    2007

  33. [33]

    Saintier,Asymptotic estimates and blow-up theory for critical equations involving thep- Laplacian, Calc

    N. Saintier,Asymptotic estimates and blow-up theory for critical equations involving thep- Laplacian, Calc. Var. Partial Differential Equations 25 (2006), 299–331

    2006

  34. [34]

    Serrin,Local behavior of solutions of quasi-linear equations, Acta Math

    J. Serrin,Local behavior of solutions of quasi-linear equations, Acta Math. 111 (1964), 247–302

    1964

  35. [35]

    Serrin,Isolated singularities of solutions of quasi-linear equations, Acta Math

    J. Serrin,Isolated singularities of solutions of quasi-linear equations, Acta Math. 113 (1965), 219–240

    1965

  36. [36]

    Schoen,Courses at Stanford University, 1988, and New York University, 1989

    R. Schoen,Courses at Stanford University, 1988, and New York University, 1989

    1988

  37. [37]

    Schoen and D

    R. Schoen and D. Zhang,Prescribed scalar curvature on then-sphere, Calc. Var. Partial Differ- ential Equations 4 (1996), no. 1, 1–25

    1996

  38. [38]

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

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

    2016

  39. [39]

    Shafrir,A sup + inf inequality for the equation−∆u=V e u, C

    I. Shafrir,A sup + inf inequality for the equation−∆u=V e u, C. R. Acad. Sci. Paris S´ er. I Math. 315 (1992), no. 2, 159–164

    1992

  40. [40]

    N. S. Trudinger,On Harnack type inequalities and their application to quasilinear elliptic equa- tions, Comm. Pure Appl. Math. 20 (1967), 721–747

    1967

  41. [41]

    Tolksdorf,Regularity for a more general class of quasilinear elliptic equations, J

    P. Tolksdorf,Regularity for a more general class of quasilinear elliptic equations, J. Differential Equations 51 (1984), no. 1, 126–150

    1984

  42. [42]

    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 Equations 260 (2016), no. 2, 149–161

    2016

  43. [43]

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

    J. V´ etois,A note on the classification of positive solutions to the criticalp-Laplace equation in Rn, Adv. Nonlinear Stud. 24 (2024), no. 3, 543–552. State Key Laboratory of Mathematical Sciences, Academy of Mathematics and Systems Science, Chinese Academy of Sciences, Beijing 100190, China Email address:qinguolin18@mails.ucas.ac.cn Institute of Mathema...

    2024