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Critical $p$-Laplace equations with monotone coefficients: Liouville classification and a Schoen-type Harnack inequality

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Pith's one-line read This paper proves that every positive entire solution of the critical p-Laplace equation with a bounded, nonincreasing coefficient is an explicit Aubin-Talenti profile, and derives a scale-invariant Harnack inequality from that…

desk verdict A promising new mechanism for the critical p-Laplace classification, but the central drift computation in Proposition 3.13 does not close as printed and needs a correction before the main theorems can stand. read the letter →

arxiv 2608.09113 v1 pith:6HTMCJXZ submitted 2026-08-10 math.AP math.DG

Yi Ru-Ya Zhang This is my paper · ORCID
classification math.APmath.DG MSC 35J9235B0835B33
keywords Criticalp-LaplacianLiouvilletheoremHarnackinequalityAubin-TalentiprofilemonotonecoefficientPohozaevidentityinvarianttensorfirst-contactargument
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 studies positive weak solutions of $-\Delta_p u = u^{p^*-1} h(u)$ in $\mathbb{R}^n$ for $1

What carries the argument

The proof transforms the unknown by $w=V^{-1/a}$ and works with the nonlinear stress $X_w=|\nabla w|^{p-2}\nabla w$; the transformed equation becomes $w\,\mathrm{div}\,X_w=b_p|\nabla w|^p+c_\tau(w)$. Monotonicity of $h$ enters through two nonnegative quantities: the Pohozaev defect $nG(s)-asg(s)\geq 0$ and the Stieltjes measure $X_w\cdot D(e_\tau(w))\geq 0$, where $e_\tau$ is the singular part of the transformed coefficient. The heart of the classification is a weak Bochner identity for the modified $P$-function $Q_w=\mathrm{div}\,X_w-e_\tau(w)$: $L_w Q_w$ equals a nonnegative trace-free tensor term plus the favorable Stieltjes defect. A multiplier $\phi_{\ell,m}$ constructed from a backwards ODE converts this identity into a pointwise coercive inequality on superlevel sets, yielding the global bound $Q_w\leq c_{\tau,0}$; a tangent analysis at minima and a punctured-ball barrier then upgrade this to equality, forcing the tensor defect to vanish and producing the explicit profile. The Harnack half uses a first-contact blow-down, identifies the blow-down limit as an exact $p$-harmonic pole through the Pohozaev sign and comparison arguments, and obtains the necessary de-concentration via nonlinear potential estimates.

What would settle it

The theorem predicts that a positive entire solution exists only if $h$ is constant on $(0,M]$, where $M$ is the solution's maximum. A concrete check would be to solve the radial ODE for $-\Delta_p u=u^{p^*-1}h(u)$ with $h(s)=e^{-s}$, which is positive, bounded, and nonincreasing on $(0,\infty)$: existence of any global positive solution with finite maximum would contradict Corollary 1.4, as would a numerical radial solution that does not match the explicit Aubin-Talenti formula.

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Extended reading notes

Core claim

The central claim is that monotonicity of the coefficient $h$ restores complete rigidity for the critical $p$-Laplace equation. For every $\tau\in[0,\infty]$ and every normalized solution $V$ of $-\Delta_p V=g_\tau(V)$ with $0<V\leq 1$ and $V(0)=1$, where $g_\tau(s)=s^{p^*-1}h(\tau s)$, the solution must be $V(x)=(1+\gamma_\tau|x|^{p'})^{-a}$ with the explicit constant $\gamma_\tau$ given in the paper. For finite $\tau$, the existence of such a profile also forces $h$ to be constant on the whole interval $(0,\tau]$, so the equation becomes a pure power on the amplitude range the solution actually attains. This classification is then applied to prove the scale-invariant Harnack estimate of Theorem 1.2, and Corollary 1.4 removes all auxiliary hypotheses: every positive entire solution is an Aubin-Talenti profile. For the purely critical equation with $h\equiv\lambda$, the paper shows that the Liouville classification and the Schoen-type Harnack inequality are equivalent.

Load-bearing premise

The proof relies on $h$ being nonincreasing: that monotonicity makes the Pohozaev defect $nG(s)-asg(s)$ and the Stieltjes measure $X_w\cdot D(e_\tau(w))$ nonnegative, and without those signs the classification and the Harnack proof do not go through.

Editorial extensions

If this is right

  • Every bounded normalized blow-up profile in the compact family $g_\tau$ is explicit, so blow-up analysis for monotone coefficients can proceed without assuming the limiting equation is a pure power.
  • The scale-invariant estimate $(\sup_{B_R}u)(\inf_{B_{2R}}u)^{p-1}\leq C R^{p-n}$ holds for all nonnegative solutions in $B_{3R}$, with $C$ depending only on $n$, $p$, and the equation.
  • Any positive entire solution is an explicit Aubin-Talenti profile, so no other shapes occur even with no decay, growth, or energy assumptions.
  • An entire solution can exist only if $h$ is constant on $(0,M]$, where $M$ is the solution's maximum; in particular, strictly decreasing nonincreasing coefficients admit no positive entire solution.
  • For $h\equiv\lambda$, the Liouville classification and the Schoen-type Harnack inequality are equivalent within the framework developed in the paper.

Reading between the lines

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

  • If the same first-contact scheme works for coefficients with a one-sided defect of controlled sign, the Harnack estimate may extend beyond monotone $h$ to a larger class of nonlinearities.
  • Because the proof replaces moving spheres with potential-theoretic de-concentration, the mechanism may be adaptable to variable-exponent or anisotropic $p$-Laplace operators, where no Kelvin transform exists.
  • The flatness conclusion suggests a testable dichotomy: for any $h$ that is strictly decreasing on $(0,M)$, numerical shooting for the radial equation should find no bounded entire solution, which would corroborate the sharpness of Corollary 1.4.
  • Theorem 1.2 is stated with $C=C(n,p,g)$ and does not claim uniformity over all nonincreasing $h$ with the same bounds $L$ and $\Lambda$; upgrading to such uniformity would require tracking constants through the compactness and de-concentration steps.
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Editorial analysis

A structured set of objections, weighed in public.

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

Referee Report

1 major / 5 minor

Summary. The paper studies positive weak solutions of the critical p-Laplace equation -Δ_p u = u^{p^*-1} h(u), where h is positive, bounded, continuous, and nonincreasing. Theorem 1.1 classifies all normalized, bounded, positive entire solutions in the compact family determined by h as Aubin–Talenti profiles and shows that, for finite amplitude, h is necessarily flat on the range of the profile. The proof introduces a transformed variable w = u^{-1/a}, a modified P-function Q_w, a weak Bochner-type identity with a nonnegative Stieltjes defect, a specially constructed multiplier solving a one-dimensional ODE, and a final tangent/barrier argument that upgrades the scalar bound Q_w ≤ c_{τ,0} to tensor rigidity. Theorem 1.2 establishes a scale-invariant sup-inf Harnack estimate, and Corollary 1.4 derives a fully unrestricted Liouville theorem.

Significance. If the proof is repaired, this is a substantial contribution: it extends the classification-to-Harnack correspondence to the full range 1 < p < n for a natural class of monotone critical nonlinearities, without finite-energy, growth, or prescribed-decay assumptions. The weak Bochner identity with a singular monotone defect, the target-dependent multiplier, and the first-contact de-concentration mechanism based on the Kilpeläinen–Malý potential estimates are original and structurally important. The paper is also careful about regularity, the BV/Stieltjes decomposition, and the use of the corrected strict-comparison statement. These strengths justify serious consideration once the algebra issue below is resolved.

major comments (1)
  1. [Section 3.3, Eq. (3.52)] Equation (3.52) does not follow from the printed definition of b in (3.41). From (3.44) and the definition of ζ in (3.41), one has X_w·∇M = φ|∇w|^p w^{-1} ζ and X_w·∇Q = |∇w|^p w^{-1}(ζ + θ Q). Substituting the vector field b from (3.41) into b·∇M therefore gives b·∇M = 2(p-1)θ φ w^{1-n} |∇w| Q/(n-1) ζ, which contains a residual factor |∇w| and has no term -(Q-k_τ)/b_p. The printed (3.52), by contrast, has no |∇w| and includes -(Q-k_τ)/b_p. Consequently, the exact cancellation of the linear-in-ζ terms between (3.51) and (3.52) is not justified, and the key differential inequality (3.42) of Proposition 3.13 is unproven as written. Since Proposition 3.17 relies on (3.42) to obtain the global bound Q_w ≤ c_{τ,0}, and that bound is the foundation of Theorem 1.1, the central classification is not established by the printed argument. The defect appears local and repairable: for example, replacing the drift in (3.41) by b = 2(p-1)θ w^{2-n}|∇w|^{-p}[Q/(n-1) - (Q-k_τ)/b_p] X_w makes the cancellation in (3.52) true. The authors should correct the definition of b, the displayed computation of b·∇M, and re-verify the subsequent estimates that use the drift.
minor comments (5)
  1. [Equation (4.35)] The string 'Lsq≤g_j(s)≤Λs^q' should read 'L s^q ≤ g_j(s) ≤ Λ s^q'; as printed, 'Lsq' is ambiguous.
  2. [Section 4.5, Step 1] After (4.40), the identity ϵ_j^a W_j = ρ_j^{-ap'} ρ_j^α v_j(ρ_j x) is correct because ap' = α, but the chain is easy to misread; writing ρ_j^{-α}ρ_j^α v_j(ρ_j x) = v_j(ρ_j x) explicitly would improve clarity.
  3. [Proof of Theorem 1.2] The passage 'Multiplying by A_jσ_j^{n-p} and using (4.66) yields ...' is printed twice in succession; one of the two identical computations should be deleted.
  4. [Proposition 4.5, Step 5] The symbol 'C K M' in (4.54) and 'CKM' in the surrounding text should be typeset consistently as C_{KM} or C_KM.
  5. [Section 4.2, opening] The sentence beginning 'Theproofisinspiredbythefirst-crossingconstruction...' has missing spaces; this should be corrected in the final version.

Circularity Check

0 steps flagged · score 2.0 of 10

No significant circularity: the classification and Harnack proofs are self-contained, with the only self-citation to [35] acknowledged and non-load-bearing.

full rationale

The derivation chain is acyclic. Theorem 1.1 is proved without assuming the Aubin–Talenti conclusion: the transform w=V^{-1/a} and the stress X_w=|\nabla w|^{p-2}\nabla w lead to the transformed equation (3.4), the weak Bochner identity (3.21) (with the nonnegative singular defect coming from monotonicity of h), and a multiplier \phi_{\ell,m} defined by the linear ODE (3.31) whose coefficients are c_\tau, k_\tau, e_\tau — all determined by the given h, not by the target profile. The flatness conclusion h(s)=h(\tau) for s\le\tau is derived from e_\tau\equiv 0 after rigidity, so it is a consequence rather than an input. Theorem 1.2 uses Theorem 1.1 to identify the blow-up limit and then proves the first-contact Proposition 4.5 self-containedly via the Pohozaev sign, comparison principles, and the Kilpeläinen–Malý Wolff-potential bound; it does not cite [35] for any of these steps. Corollary 1.4 combines Theorem 1.2 with the finite-amplitude flatness and the external classification [12]; the dependency graph never loops back to the claim being proved. The only self-citation, [35], is explicitly described as conditional and is strengthened or removed rather than used as a black box; hence it is not load-bearing. The manuscript even flags its own limitations (Remark 1.3: no uniformity in g; Appendix B: no simpler p=2 argument), which further indicates the claims are not being forced by definition or by self-citation. The score of 2 reflects the acknowledged minor self-citation, not an actual circular step.

Assumptions & free parameters 0 free parameters · 5 assumptions · 0 invented entities

The paper introduces the transformed function w=V^{-1/a}, the modified P-function Q_w, and the ODE-designed multiplier φ_{ℓ,m}, but these are mathematical objects constructed from the solution, not new postulated entities. The coefficient h is an input, not a fitted parameter; the constants in the results depend on n,p,g but are not tuned to data.

assumptions (5)
  • domain assumption h ∈ C((0,∞)) is positive, bounded and nonincreasing, with limits h_0 and h_∞ (1.4).
    Stated as the main hypothesis; used in Lemma 2.1 and throughout to control signs of Pohozaev defects and compactness of the family g_τ.
  • standard math Kilpeläinen-Malý potential estimates (truncated Wolff potentials) for p-superharmonic functions (Theorem 1.6 in [26]).
    Used in Proposition 2.4 to prove local boundedness and in Proposition 4.5 Step 5 to obtain the Morrey bound (4.54).
  • standard math Kichenassamy-Véron isolated singularity expansion for p-harmonic functions, with the erratum [25].
    Used in Proposition 4.3 to expand blow-down limits near isolated poles and compute Pohozaev flux.
  • standard math Ciraolo-Gatti classification [12, Theorem 1.2] of bounded positive solutions to the pure critical p-Laplace equation with sharp infimum decay.
    Used at the end of Corollary 1.4 to identify the pure-power solution after h is shown flat.
  • standard math Standard p-Laplace regularity: Tolksdorf C^{1,γ} estimates, Vázquez strong minimum principle, Harnack inequality for equations with bounded coefficients.
    Used in Section 2 to bootstrap regularity and in Lemmas 2.6-2.7 and Proposition 4.5.

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Pith. "Pith review of Critical $p$-Laplace equations with monotone coefficients: Liouville classification and a Schoen-type Harnack inequality." pith.science (2026). https://pith.science/paper/6HTMCJXZ

@misc{pith2026260809113,
  author       = {Pith},
  title        = {Pith review of: Critical $p$-Laplace equations with monotone coefficients: Liouville classification and a Schoen-type Harnack inequality},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/6HTMCJXZ}},
  note         = {Machine review of arXiv:2608.09113}
}
abstract

We investigate positive weak solutions of the critical $p$-Laplace equation $$ -\Delta_p u = u^{p^*-1} h(u), \qquad 1 < p < n, $$ where $h$ is a positive, bounded, continuous, and nonincreasing function. Our first main result is a complete classification of normalized, bounded, positive entire solutions for every equation in a compact family determined by $h$: Any such solution must coincide with an Aubin--Talenti profile. Moreover, the existence of an Aubin--Talenti profile as a solution implies that $h$ is constant on the entire interval of values attained by that profile. Subsequently, applying this classification result, we establish the following scale-invariant Schoen-type estimate $$ \left(\sup_{B_R} u\right)\left(\inf_{B_{2R}} u\right)^{p-1} \le C R^{p-n} $$ for nonnegative weak solutions defined in $B_{3R}$. As a direct corollary, we obtain a fully unrestricted Liouville theorem: Every positive entire solution must coincide with an Aubin--Talenti profile. For the purely critical equation, we also show that the corresponding Liouville classification is equivalent to a Schoen-type Harnack inequality. The arguments developed in this work constitute quasilinear counterparts of the Kelvin transform and the method of moving spheres, which are classically available only in the semilinear framework, and furthermore provide alternative proofs of the corresponding classical significant results.

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