Critical quasilinear equations on Riemannian manifolds

Linlin Sun; Youde Wang

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Critical quasilinear equations on Riemannian manifolds

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Pith's one-line read Positive weak solutions of the critical p-Laplace equation on complete noncompact manifolds with nonnegative Ricci curvature, under one of three growth or decay assumptions, force the manifold to be Euclidean and the solution to be an…

desk verdict Genuinely new tools and a mostly sound strategy, but a load-bearing regularity gap in Theorem 3.1 and an exponent error in Theorem II make this a major-revision paper, not a finished result. read the letter →

arxiv 2502.08495 v2 pith:TFFZ432N submitted 2025-02-12 math.DG math.AP

Linlin Sun , Youde Wang This is my paper · ORCID

classification math.DGmath.AP MSC 35J9235B3358J0553C21

keywords criticalp-LaplaceequationLiouvilleclassificationrigidityRiccicurvaturenonlinearKatoinequalityAubin-TalentibubbleCheng-Yaugradientestimate

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

This paper tries to settle a rigidity dichotomy for the critical $p$-Laplace equation $-\Delta_p u = u^{np/(n-p)-1}$: on a complete, connected, noncompact Riemannian manifold with nonnegative Ricci curvature, a positive weak solution should be nothing more than an Aubin-Talenti bubble living on flat Euclidean space. The authors show this under three broad enough hypotheses on $p$ or on the solution's growth at infinity, and they prove an analogous statement for the quasilinear Liouville equation $-\Delta_n u = e^{nu}$. These are classification results with a geometric payoff: if one such solution exists, the entire manifold must be isometric to $\mathbb{R}^n$. This matters because the same mechanism also rules out Sobolev minimizers on nonflat manifolds, a question at the heart of sharp Sobolev inequalities.

What carries the argument

The engine is the triple $(w,f,E)$: the auxiliary function $w$, the P-function $f=\Delta_p w$, and the trace-free endomorphism $E(W)=\nabla W - (\mathrm{div}\,W / n)\,g$ built from $W=|\nabla w|^{p-2}\nabla w$. The load-bearing identity is the Bochner-type formula $\mathrm{div}(w^{1-n}E(W)) = w^{1-n}\mathrm{tr}\,E^2 + w^{1-n}\mathrm{Ric}(W,W)$, combined with the sharp nonlinear Kato inequality $\mathrm{tr}\,E^2 \ge \frac{n(p-1)}{n-1} \cdot \frac{\langle E(W), A^{-1}(E(W))\rangle}{|W|^2}$, where $A(X)=X+(p-2)|\nabla w|^{-2}\langle X,\nabla w\rangle\nabla w$. This pair upgrades nonnegative Ricci curvature into a pointwise lower bound on a divergence, which after integration against test functions yields the dichotomy: either $f$ is constant or certain annulus integrals diverge at least like $R^2$. The Cheng–Yau type gradient estimate, proved by Moser iteration, is the supporting mechanism that converts polynomial decay of $u$ into the required upper bound on these integrals.

What would settle it

Build a complete rotational-symmetric metric of nonnegative Ricci curvature and solve the critical p-Laplace equation radially: if a positive solution with nonconstant P-function satisfies the growth bound in condition (B) or the decay bound in condition (C), then the annulus integral $\int_{B_R\setminus B_{R/2}} f^{-\alpha}w^{1-n}|\nabla w|^{2p-2}$ would be $O(R^2)$, contradicting its predicted divergence and falsifying the rigidity claim.

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Extended reading notes

Core claim

The paper's central claim is that the critical $p$-Laplace equation is rigid on complete noncompact manifolds with nonnegative Ricci curvature. The proof introduces the auxiliary function $w$ and the P-function $f=\Delta_p w$, where $w = ((n-p)/p)^{(p-1)/p} u^{-p/(n-p)}$ for the $p$-Laplace equation and $w=e^{-u}$ for the $n$-Laplace Liouville equation. The load-bearing Bochner-type identity $\mathrm{div}(w^{1-n}E(W)) = w^{1-n}\mathrm{tr}\,E^2 + w^{1-n}\mathrm{Ric}(W,W)$, with $W=|\nabla w|^{p-2}\nabla w$ and $E$ the trace-free part of $\nabla W$, combines with a new sharp nonlinear Kato inequality to give a dichotomy: if the P-function is not constant, certain annulus integrals grow faster than $R^2$. Each hypothesis in Theorem I—an upper range of $p$, finite weighted $L^q$ growth, or polynomial decay faster than a stated rate—supplies a matching $O(R^2)$ upper bound, forcing $f$ to be constant. A constant $f$ makes $E(W)$ a homothetic vector field, which by a classical geometric theorem forces the metric to be Euclidean; existing Euclidean classifications then identify the solution as an Aubin-Talenti bubble. Theorem II repeats this scheme for $-\Delta_n u = e^{nu}$ and yields the standard logarithmic solution on $\mathbb{R}^n$.

Load-bearing premise

The argument rests on treating the weak solution as smooth enough away from a measure-zero critical set and on the claimed integrability of the weight functions, since the sharp Kato inequality and Bochner identity are proved in the smooth regime and then applied to weak solutions.

Editorial extensions

If this is right

The first part of Theorem I covers the range $p_n<p<n$, and for $p=2$ in dimensions $n=3,4,5$ it recovers known rigidity for the semilinear critical Laplace equation.
Finite potential energy forces the flat model, so any complete noncompact nonflat manifold with nonnegative Ricci curvature has no Sobolev minimizer for the best constant $S_p(M^n)$, for every $1<p<n$.
Rigidity extends to infinite-energy solutions whose polynomial decay is faster than the stated threshold, supplementing the finite-energy classification.
For the $n$-Laplace Liouville equation, the same rigidity holds whenever $u(x)\ge -\frac{n}{n-1}\ln(rF^{1/2}(r))-C$ for a positive nondecreasing $F$ with $\int^\infty ds/(sF(s))=\infty$, and then the solution is the standard logarithmic one on $\mathbb{R}^n$.

Reading between the lines

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

If the regularity barrier at the critical set could be removed, the same divergence-versus-growth contradiction would likely push condition (A) down toward all $1<p<n$, since the threshold $p_n$ appears mainly from the Moser-iteration step.
The same Bochner-Kato mechanism should apply to anisotropic $p$-Laplace operators, because the $A$-endomorphism already encodes anisotropy through the term $(p-2)|\nabla w|^{-2}\langle X,\nabla w\rangle\nabla w$.
The decay threshold in condition (C) is probably not sharp: the comparison-principle lower bound $(n-p)/(p-1)$ lies above it whenever $n>2$, leaving a window in which the true borderline could be lower.
The $n$-Laplace Liouville theorem suggests a conformally invariant classification in which the lower-bound condition with $F\equiv 1$ gives $u(x)\ge -\frac{n}{n-1}\ln r - C$, and any slower allowed growth would break rigidity.

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Editorial analysis

A structured set of objections, weighed in public.

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

Desk Editor's Note

read the letter

Read the paper. The real story: new sharp nonlinear Kato inequality, clean algebraic identities, and a coherent integral-estimate strategy; but the weak-regularity step in Theorem 3.1 is asserted rather than proved, and the proof of Theorem II has an exponent error for n=2,3. The main theorems are likely true, but the write-up is not there yet.

What's genuinely good: the Kato inequality (2.4) is new and sharp, with a nice verification via the example; the Bochner-type formula (2.15) is the right frame. The integral estimates of Theorems 3.1–3.2 are the engine, and the overall strategy—force f constant, then pull a homothetic vector field out of E(W)=0—is sound. The exponent bookkeeping in Theorem 5.1 and the growth-range arguments in Theorem 5.2 look correct. The disclosure about Nobili–Violo is transparent.

The problems: the stress-test is on target. Lemma 2.4 and the Kato inequality are proven for C^3 w with ∇w ≠ 0; for the weak solutions in Theorem 3.1 the paper cites regularity and then says the divergence inequality holds in the sense of distributions and uses f^{-α}η^γ as a test function. That requires an approximation argument across the critical set. It is not automatic, especially for p<2. Since Theorem 3.1 is used to generate the contradiction in every rigidity theorem, this is load-bearing. A referee should demand a proof or a precise lemma.

Second, Theorem II: the proof sets q=(n+2)/(n-2) and claims q3. So the proof does not cover n=2,3. The theorem statement says all n. Either the theorem needs a low-dimension argument or a restriction to n≥4. This is a mathematical error, not just a presentation issue.

Also, the Cheng-Yau gradient estimate is explicitly deferred for n=2; minor.

Who this is for: geometric analysts working on rigidity and quasilinear PDEs. It deserves a serious referee, but the referee's job is real: verify the weak-regularity extension and fix the n=2,3 case. I would not cite it as-is.

Referee Report

3 major / 4 minor

Summary. The paper studies positive weak solutions of the critical p-Laplace equation -Δ_p u = u^{np/(n-p)-1} and the quasilinear Liouville equation -Δ_n u = e^{nu} on complete, noncompact Riemannian manifolds with nonnegative Ricci curvature. The authors introduce a P-function construction, prove a nonlinear Kato inequality and a Bochner-type identity for the p-Laplacian, and use them to derive integral growth estimates of Karp type and a Cheng-Yau type gradient estimate. From these they conclude, under either a range of p, a finite-energy/decay condition, or a pointwise polynomial upper bound, that the manifold is isometric to Euclidean space and the solution is an Aubin-Talenti bubble; an analogous rigidity statement is given for the n-Laplace Liouville equation. The main theorems are Theorem I (parts A, B, C) and Theorem II.

Significance. If the technical gaps noted below are closed, this is a substantial contribution: it gives the first classification of critical p-Laplace solutions on manifolds with nonnegative Ricci curvature under broad analytic conditions, extends the semilinear results of Catino-Monticelli and Ciraolo-Farina-Polvara to the quasilinear setting, and answers part of Problem 2 in the introduction. The algebraic core is explicit and verifiable: Lemma 2.1 and Lemma 2.4 contain concrete identities and inequalities, and the conditions in Theorems I and II are stated with explicit exponents, so the predictions are falsifiable. A particular strength is that the argument is not a reduction to previously fitted parameters; the central contradiction comes from the growth dichotomy in Theorem 3.1. The main reservations are about the weak-solution justification of the distributional Bochner machinery and one incorrect linear-algebra bound in the gradient estimate; both appear repairable.

major comments (3)

[§3, Theorem 3.1 (Eqs. (3.3)-(3.7))] The proof of Theorem 3.1 is the engine of the paper, but the passage from the pointwise Bochner identity (2.15) and Kato inequality (2.4), both proved only for w ∈ C^3 with ∇w ≠ 0, to the distributional inequality used for weak solutions is not justified. The text asserts after citing [1,22,51,52] that div(w^{1-n}E(W)) ≥ w^{1-n}trE^2 “holds in the sense of distribution” and then tests it against f^{-α}η^γ. However, f is only C^{0,α} across the critical set Z, ∇f is not classically defined on Z, and f^{-α}η^γ is not a C∞_0 test function; moreover, for p<2 the available regularity |∇u|^{p-2}∇²u ∈ L²_loc does not by itself control the boundary terms that arise when integrating by parts on M\Z. Since Theorem 3.1 supplies the growth contradiction in Theorems 5.1–5.3 and Theorem II, an approximation or density argument (or a directly weak formulation of (2.15)) must be supplied before the main claims are established.
[§4, Lemma 4.3, Eq. (4.14)] The eigenvalue assertion preceding (4.14) is incorrect as stated: A_φ = Id + (p-2)|∇φ|^{-2}∇φ⊗∇φ has eigenvalues {1, p-1}, so for p<2 it is bounded below by p-1 and above by 1, not “below by 1 and above by p−1”. Consequently, the factor (p−1)/(tp) multiplying ∫|∇(η|∇φ|^t)|² in (4.14) is not a valid consequence when p>2, where the lower bound is 1. The inequality should use min{1,p−1} (or an equivalent p-dependent constant) in that factor. This affects the proof of Theorem 4.1, which Theorem 5.3 invokes for p up to n²/(3n−2); for n≥6 this range includes p>2, so the error occurs in a parameter range actually used in the paper.
[§5.4, proof of Theorem II (Eqs. (5.3)-(5.4))] The final contradiction in Theorem II is not written correctly. Theorem 3.2 is applied with G(s)=F(2s) to obtain (5.4), a limsup with denominator R²G(R) of an integral over B_R. The next display, however, asserts a limsup with denominator R²F(R) over B_{R/2} and declares it infinite; (5.3) only bounds ∫_{B_{R/2}} f w^{3-n-2/n} by C R²F(R), while doubling the radius gives the bound C R²G(R), not C R²F(R). The argument can be repaired by keeping G in the final step: bound ∫_{B_R} f^{1-α}... by C R²G(R) and contradict (5.4). As it stands, the displayed implication does not follow from the preceding lines.

minor comments (4)

[Definition 1.2] In the definition of weak solution to the n-Laplace Liouville equation, the integrand should be |∇u|^{n-2}⟨∇u,∇φ⟩, not |∇u|^{p-2}⟨∇u,∇φ⟩.
[Corollary 1.2] The phrase “Euclidean plane R^n” should read “Euclidean space R^n”, since the corollary covers n = 3, 4, 5.
[Eq. (3.6)] The notation “(n−1)α+” appears to mean the positive part α_+, but this is never defined; please define it explicitly or rewrite the inequality with α_+ = max{α,0}.
[Theorem 3.2] The proof of Theorem 3.2 is dismissed with “the rest of the proof is similar”; given that Theorem II depends on this statement, a brief indication of the changed exponents and the new constant α<1 would improve readability.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: the rigidity argument is derived from an algebraic Kato inequality and a Bochner-type identity, then combined with external Euclidean classification and Tashiro's rigidity theorem.

full rationale

The paper's central derivation chain is self-contained rather than circular. The authors introduce w and f by explicit formulas involving u and |\nabla w|, then compute Δ_p w = f, derive the pointwise Bochner-type identity div(w^{1-n}E(W)) = w^{1-n}tr E^2 + w^{1-n}Ric(W,W) in Lemma 2.4, and use the nonlinear Kato inequality (2.4) to obtain the integral lower bounds in Theorem 3.1 and Theorem 3.2. No parameter is fitted to the data that is later 'predicted', and no conclusion is assumed as an input. The self-citation to He, Sun and Wang [31] occurs in Remark 2.2 as a parenthetical attribution of one inequality in (2.3), but the proof of that inequality is actually reproduced in the proof of Lemma 2.1 using a local orthonormal frame and the self-duality of EA; it is not load-bearing in the sense of an unverified imported result. The rigidity step after f is concluded constant uses Tashiro's theorem [50] for the isometry conclusion and the external Euclidean classification results [10,21,47,53] to identify the solution as an Aubin-Talenti bubble; these are independent external facts rather than assumptions smuggled from the present paper. The Cheng-Yau gradient estimate in Theorem 4.1 is presented with a proof via Moser iteration, not merely cited as an ansatz. A possible analytic concern is that the pointwise Kato/Bochner computations are carried out away from the critical set and then extended to weak solutions by regularity citations; this is a regularity or approximation issue, not a circular reduction. Since no equation is used to define its own conclusion and no fitted or renamed input is presented as a new result, the circularity score is 0.

Assumptions & free parameters 0 free parameters · 6 assumptions · 0 invented entities

No free parameters or invented entities. The paper relies on standard geometric analysis background and explicitly cited external classification theorems.

assumptions (6)

standard math Bishop-Gromov volume comparison and Laplacian comparison for manifolds with nonnegative Ricci curvature.
Used in Lemma 3.4, Lemma 3.6 and Theorem 5.2 to convert volume integrals into polynomial bounds in R.
standard math Regularity theory for weak solutions to p-Laplace type equations: C^{1,α}_loc and C∞ away from the critical set, with critical set of measure zero.
Invoked in Theorem 3.1 proof to justify pointwise and distributional computations.
standard math Weak comparison principle for p-superharmonic functions.
Used in Lemma 3.6 to get the lower bound w ≤ C r^{p/(p-1)}.
standard math Euclidean classification: any positive entire solution to -Δ_p u = u^{p_S-1} in R^n is an Aubin-Talenti bubble.
Final step of Theorem I, citing [10,21,47,53].
standard math Tashiro's theorem: a complete Riemannian manifold admitting a nontrivial homothetic vector field is Euclidean.
Used to conclude M^n is isometric to R^n from ∇W = c g, citing [50].
domain assumption Assumption that M^n is complete, connected, noncompact with nonnegative Ricci curvature.
The entire paper studies this setting; the integral divergence identities use Ric(W,W)≥0.

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Pith. "Pith review of Critical quasilinear equations on Riemannian manifolds." pith.science (2026). https://pith.science/paper/TFFZ432N

@misc{pith2026250208495,
  author       = {Pith},
  title        = {Pith review of: Critical quasilinear equations on Riemannian manifolds},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/TFFZ432N}},
  note         = {Machine review of arXiv:2502.08495}
}

abstract

In this paper, we investigate critical quasilinear elliptic partial differential equations on a complete Riemannian manifold with nonnegative Ricci curvature. By exploiting a new and sharp nonlinear Kato inequality and establishing some Cheng-Yau type gradient estimates for positive solutions, we classify positive solutions to the critical $p$-Laplace equation and show rigidity concerning the ambient manifold. Our results extend and improve some previous conclusions in the literature. Similar results are obtained for solutions to the quasilinear Liouville equation involving the $n$-Laplace operator, where $n$ corresponds to the dimension of the ambient manifold.

Discussion (0). Continue with ORCID to comment.

Forward citations

Cited by 3 Pith papers

Reviewed papers in the Pith corpus that reference this work. Sorted by Pith novelty score. Full citation record

Optimal Rigidity and Classification Results for the $k$-Hessian Equation of Lane--Emden Type
math.AP 2026-08 accept novelty 8.0 of 10

For sigma_k(-D^2u)=u^p in R^n, the paper proves all nonnegative entire solutions vanish for the previously open exponent range, and identifies the critical exponent as the sharp Liouville threshold.
Classification results for bounded positive solutions to the critical $p$-Laplace equation
math.AP 2025-10 conditional novelty 7.0 of 10

Bounded positive local weak solutions of the critical p-Laplace equation in R^n, for all 1<p<n, are classified as p-bubbles via new optimal integral estimates.
Liouville theorem of the subcritical biharmonic equation on complete manifolds
math.AP 2025-08 unverdicted novelty 5.0 of 10

For n>=5 and 1<alpha<(n+4)/(n-4), the equation Delta^2 u = u^alpha admits no positive C^4 solution on complete noncompact manifolds with nonnegative Ricci curvature.

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