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

T0 review · 1 major / 2 minor · reviewed 2026-06-28 · grok-4.3

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Pith's one-line read 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}.

desk verdict The paper gives a Pohozaev-neck upgrade from a preliminary decay rate to the sharp one for a conditional Harnack inequality in the critical p-Laplacian, but the main claim rests on two unproved inputs. read the letter →

arxiv 2606.05990 v1 pith:RWSNS4EU submitted 2026-06-04 math.AP

Guolin Qin , Yi Ru-Ya Zhang This is my paper · ORCID
classification math.AP
keywords PohozaevidentityneckanalysisHarnackinequalityp-Laplaciancriticalexponentblow-uplimitsAubin-Talentibubblesquasilinearellipticequations
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 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.

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.

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.

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Extended reading notes

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.

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.

Editorial extensions

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.

Reading between the lines

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.
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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 / 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 · score 0.0 of 10

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.

Assumptions & free parameters 0 free parameters · 0 assumptions · 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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Cite this review

Pith. "Pith review of A Pohozaev-type neck proof of a conditional Harnack inequality in the critical $p$-Laplacian setting." pith.science (2026). https://pith.science/paper/RWSNS4EU

@misc{pith2026260605990,
  author       = {Pith},
  title        = {Pith review of: A Pohozaev-type neck proof of a conditional Harnack inequality in the critical $p$-Laplacian setting},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/RWSNS4EU}},
  note         = {Machine review of arXiv:2606.05990}
}
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.

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