Minimizers and best constants for a weighted critical Sobolev inequality involving the polyharmonic operator

Jeferson Silva; Jos\'e Francisco de Oliveira

arxiv: 2505.09035 · v2 · submitted 2025-05-14 · 🧮 math.AP

Minimizers and best constants for a weighted critical Sobolev inequality involving the polyharmonic operator

Jos\'e Francisco de Oliveira , Jeferson Silva This is my paper

Pith reviewed 2026-05-22 16:09 UTC · model grok-4.3

classification 🧮 math.AP

keywords Sobolev inequalitypolyharmonic operatorbest constantcritical exponentradial symmetryregularityclassification of solutions

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The pith

The best constant for a weighted critical Sobolev inequality involving the polyharmonic operator is computed explicitly.

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

The paper sets out to determine the precise value of the best constant in a Sobolev-type inequality that uses a polyharmonic operator together with a weight. Knowing the sharp constant matters because it supplies the limiting case for variational problems and yields exact bounds when studying higher-order elliptic equations. The authors reach this value by proving regularity and classification results for solutions of an associated generalized critical polyharmonic equation, but only under radial symmetry.

Core claim

Our main goal is to explicitly compute the best constant for the Sobolev-type inequality involving the polyharmonic operator obtained in a prior reference. To achieve this goal, we also establish both regularity and classification results for a generalized critical polyharmonic equation in the radial setting.

What carries the argument

Classification of radial solutions to the generalized critical polyharmonic equation, which identifies the minimizers and thereby fixes the best constant.

If this is right

The Sobolev inequality holds with the newly computed constant and is attained by explicit radial functions.
The sharp constant supplies the optimal threshold for existence and non-existence results in related polyharmonic equations.
Variational problems involving the same operator and weight can now be analyzed with exact quantitative control.

Where Pith is reading between the lines

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

Removing the radial assumption would extend the result to a wider class of functions and domains.
The same classification technique might apply to other critical exponents or different weight classes.
The explicit constant could serve as a benchmark for numerical approximations of minimizers in non-radial cases.

Load-bearing premise

The regularity and classification results are established only in the radial setting.

What would settle it

A non-radial solution to the critical equation that produces a strictly smaller value for the functional would show the computed constant is not sharp.

read the original abstract

Our main goal is to explicitly compute the best constant for the Sobolev-type inequality involving the polyharmonic operator obtained in (Analysis and Applications 22, pp. 1417-1446, 2024). To achieve this goal, we also establish both regularity and classification results for a generalized critical polyharmonic equation in the radial setting.

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.

Desk Editor's Note private letter to a colleague

The paper gives an explicit best constant for the weighted polyharmonic Sobolev inequality by classifying radial solutions, but leaves open whether that constant is sharp without the radial restriction.

read the letter

The main thing to know is that the authors compute an explicit value for the best constant in the weighted critical Sobolev inequality for the polyharmonic operator, using the radial solutions to the associated Euler-Lagrange equation. They also prove regularity and a classification result for a generalized critical polyharmonic equation, but only when the functions are radial. This extends the existence result from their 2024 paper in Analysis and Applications by turning it into a concrete number. That explicit constant is the clear new output, and the radial classification work looks like careful, self-contained analysis on the reduced ODE or integral equation side. For anyone who needs the sharp value in estimates or further work on radial extremals, this supplies something usable that was not there before. The radial regularity and classification steps appear to be the parts they execute most directly. The soft spot is the gap on global sharpness. The constant is derived from radial minimizers, yet the paper supplies no symmetry-reduction argument such as an adapted moving-plane method or a weighted symmetrization that would show the infimum cannot be smaller for non-radial functions. Without that, it remains possible that the reported number is only the best radial constant rather than the best constant over the whole space. This is not a minor technicality if the title and abstract claim the best constant for the inequality as stated. The concern in the stress-test note holds up on the information given. This work is for specialists in higher-order elliptic inequalities and best-constant problems who already know the 2024 existence result. A reader who wants the explicit number or who studies radial classification will find direct value; broader audiences in geometry or physics will see limited immediate use. The paper has enough concrete content and a verifiable new computation to deserve referee time rather than a desk reject. I would send it to peer review and ask the referees to check the radial classification details and to say whether the global sharpness claim needs qualification or a separate argument.

Referee Report

1 major / 1 minor

Summary. The paper computes the best constant in a weighted critical Sobolev inequality for the polyharmonic operator (building on a 2024 result in Analysis and Applications) by explicitly solving the associated Euler-Lagrange equation under radial symmetry. It additionally proves regularity and classification results for the generalized critical polyharmonic equation, but only within the radial class.

Significance. An explicit sharp constant for this polyharmonic inequality would be a useful addition to the literature on weighted Sobolev embeddings and their extremals, particularly if the radial computation can be shown to capture the global infimum. The explicit evaluation of the constant via the radial solution is a concrete strength when the reduction to radial functions is justified.

major comments (1)

[Abstract and main results section] The central claim is that the computed value is the best (sharp) constant for the inequality on the full space. However, regularity and classification results are established only in the radial setting (as stated in the abstract). No symmetry-reduction argument—such as a weighted Schwarz symmetrization or an adaptation of the moving-plane method to the polyharmonic operator with the given weight—is supplied to show that any minimizing sequence can be replaced by a radial one without increasing the Rayleigh quotient. This leaves open the possibility that a non-radial function attains a strictly smaller value, so the reported constant may not be globally sharp.

minor comments (1)

[Introduction] Clarify in the introduction whether the weight function satisfies the precise integrability or decay conditions needed for the polyharmonic operator to be well-defined in the weighted space.

Simulated Author's Rebuttal

1 responses · 0 unresolved

We thank the referee for the careful reading and constructive feedback on our manuscript. We respond to the major comment below, acknowledging the distinction between radial and global settings.

read point-by-point responses

Referee: [Abstract and main results section] The central claim is that the computed value is the best (sharp) constant for the inequality on the full space. However, regularity and classification results are established only in the radial setting (as stated in the abstract). No symmetry-reduction argument—such as a weighted Schwarz symmetrization or an adaptation of the moving-plane method to the polyharmonic operator with the given weight—is supplied to show that any minimizing sequence can be replaced by a radial one without increasing the Rayleigh quotient. This leaves open the possibility that a non-radial function attains a strictly smaller value, so the reported constant may not be globally sharp.

Authors: We agree that the manuscript establishes regularity and classification results only within the radial class, as explicitly stated in the abstract, and does not supply a symmetry-reduction argument (such as weighted Schwarz symmetrization or a moving-plane adaptation for the polyharmonic operator) to justify that the infimum over all functions coincides with the radial infimum. The explicit computation of the constant proceeds by solving the Euler-Lagrange equation under radial symmetry, building on the inequality established in the 2024 Analysis and Applications paper. In the revised manuscript we will qualify the main claims to state that the reported value is the best constant in the radial setting. We will also add a brief discussion noting that extension to the global case would require a separate symmetry argument, which is left for future work if it cannot be obtained by adapting existing techniques for higher-order operators. revision: yes

Circularity Check

0 steps flagged

No significant circularity; derivation is self-contained via new radial analysis

full rationale

The paper cites a 2024 result solely for the existence of the underlying weighted critical Sobolev inequality and then derives the best constant through fresh regularity and classification theorems established in this manuscript for the generalized critical polyharmonic equation under radial symmetry. These classification results are obtained by direct analysis of the Euler-Lagrange equation in the radial class and are not defined in terms of the target constant or fitted to prior outputs by construction. The explicit evaluation of the constant follows from substituting the classified radial extremal into the Rayleigh quotient. While the absence of a symmetry-breaking or rearrangement argument leaves open whether the radial value is globally minimal, this is a completeness gap rather than a reduction of the claimed result to self-definition, self-citation chains, or renamed inputs. The self-citation is limited to the base inequality and carries no load-bearing role in the constant computation itself.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

Paper relies on standard Sobolev embedding theory and polyharmonic operator properties from prior literature; no new free parameters or invented entities are indicated in the abstract.

axioms (1)

standard math Standard functional-analytic properties of the polyharmonic operator and weighted Sobolev spaces hold in the radial setting.
Invoked implicitly to obtain the inequality and its best constant.

pith-pipeline@v0.9.0 · 5575 in / 1032 out tokens · 30212 ms · 2026-05-22T16:09:24.325473+00:00 · methodology

discussion (0)

Lean theorems connected to this paper

Citations machine-checked in the Pith Canon. Every link opens the source theorem in the public Lean library.

IndisputableMonolith/Cost/FunctionalEquation washburn_uniqueness_aczel unclear

?

unclear
Relation between the paper passage and the cited Recognition theorem.

Theorem 1.4: Suppose that (1.23) holds. Then the best constant S^{-1/2}(α,m) is given by S^{-1/2}=P^{-1/2}[2Γ(α+1)/Γ((α+1)/2)Γ((α+1)/2)]^{m/(α+1)} where P=∏_{h=-m}^{m-1}(α+1+2h).
IndisputableMonolith/Foundation/DimensionForcing alexander_duality_circle_linking unclear

?

unclear
Relation between the paper passage and the cited Recognition theorem.

Theorem 1.2 (classification): positive (2m-1)-th nonsingular solutions with odd-order zero initial conditions are exactly w_ε(r)=P^{(α-2m+1)/(4m)}(ε/(ε²+r²))^{(α-2m+1)/2}.

What do these tags mean?

matches: The paper's claim is directly supported by a theorem in the formal canon.
supports: The theorem supports part of the paper's argument, but the paper may add assumptions or extra steps.
extends: The paper goes beyond the formal theorem; the theorem is a base layer rather than the whole result.
uses: The paper appears to rely on the theorem as machinery.
contradicts: The paper's claim conflicts with a theorem or certificate in the canon.
unclear: Pith found a possible connection, but the passage is too broad, indirect, or ambiguous to say the theorem truly supports the claim.

Reference graph

Works this paper leans on

33 extracted references · 33 canonical work pages

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Sharp Sobolev and Adams-Trudinger-Moser embeddings on weighted Sobolev spaces and their applications

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Trudinger-Moser embeddings on weighted Sobolev spaces on unbounded domains

J.M. do ´O, G. Lu, and R. Ponciano. “Trudinger-Moser embeddings on weighted Sobolev spaces on unbounded domains”. In:Discrete Contin. Dyn. Syst.45 (2025), pp. 557– 584. REFERENCES 35

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A sharp Adams-type inequality for weighted Sobolev spaces

J.M. do ´O, A. Macedo, and J.F. de Oliveira. “A sharp Adams-type inequality for weighted Sobolev spaces”. In:Q. J. Math.71 (2020), pp. 517–538

work page 2020
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Concentration-compactness and extremal problems for a weighted Trudinger-Moser inequality

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On a class of quasilinear elliptic problems with critical exponential growth on the whole space

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On a sharp inequality of Adimurthi-Druet type and extremal functions

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Trudinger-Moser type inequalities for weighted Sobolev spaces involving fractional dimensions

J.F. de Oliveira and J.M. do ´O. “Trudinger-Moser type inequalities for weighted Sobolev spaces involving fractional dimensions”. In:Proc. Amer. Math. Soc.142 (2014), pp. 2813–2828

work page 2014
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Hardy-Sobolev type inequality and su- percritical extremal problem

J.F. de Oliveira, J.M. do ´O, and P. Ubilla. “Hardy-Sobolev type inequality and su- percritical extremal problem”. In:Discrete Contin. Dyn. Syst.39 (2019), pp. 3345– 3364

work page 2019
[26]

On a Sobolev-type inequality and its minimizers

J.F. de Oliveira and J.N. Silva. “On a Sobolev-type inequality and its minimizers”. In:Anal. Appl.22 (2024), pp. 1417–1446

work page 2024
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Schoen.Variational theory for the total scalar curvature functional for Riemann- ian metrics and related topics

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[28]

The Brezis-Nirenberg result for the fractional Lapla- cian

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[29]

Existence of solitary waves in higher dimensions

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Best constant in Sobolev inequality

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Classification of solutions of higher order conformally invariant equations

J. Wei and X. Xu. “Classification of solutions of higher order conformally invariant equations”. In:Math. Ann.313 (1999), pp. 207–228. (J.F. de Oliveira) Department of Mathematics Federal University of Piau´ı 64049-550 Teresina, PI, Brazil Email address:jfoliveira@ufpi.edu.br REFERENCES 36 (J.N. Silva) Department of Mathematics Federal University of the D...

work page 1999

[1] [1]

On a weighted Trudinger-Moser inequality inR N

E. Abreu and L.G. Fernandes. “On a weighted Trudinger-Moser inequality inR N”. In: J. Differ. Equations269 (2020), pp. 3089–3118

work page 2020

[2] [2]

W., Holmes, C

J.H. Andrade and J. Wei. “Classification for positive singular solutions to critical sixth order equations”. arXiv:2210.04376.url:https://doi.org/10.48550/arXiv.2210. 04376

work page doi:10.48550/arxiv.2210

[3] [3]

Andrews, R

G.E. Andrews, R. Askey, and R. Roy.Special functions. Paperback ed. Vol. 71. Encycl. Math. Appl. Cambridge: Cambridge University Press, 2000

work page 2000

[4] [4]

An integral inequality

G. Bliss. “An integral inequality”. In:J. London Math. Soc.5 (1930), pp. 40–46

work page 1930

[5] [5]

Positive solutions of nonlinear elliptic equations involving critical Sobolev exponents

H. Br´ ezis and L. Nirenberg. “Positive solutions of nonlinear elliptic equations involving critical Sobolev exponents”. In:Commun. Pure Appl. Math.36 (1983), pp. 437–477

work page 1983

[6] [6]

Asymptotic symmetry and local behavior of semilinear equations with critical Sobolev growth

L. Caffarelli, B. Gidas, and J. Spruck. “Asymptotic symmetry and local behavior of semilinear equations with critical Sobolev growth”. In:Comm. Pure Appl. Math.42 (1989), pp. 271–289

work page 1989

[7] [7]

Classification of solutions of some nonlinear elliptic equations

W. Chen and C. Li. “Classification of solutions of some nonlinear elliptic equations”. In:Duke Math. J.63 (1991), pp. 615–622

work page 1991

[8] [8]

Quasilinear elliptic equations with critical exponents

P. Cl´ ement, D.G. de Figueiredo, and E. Mitidieri. “Quasilinear elliptic equations with critical exponents”. In:Topol. Methods Nonlinear Anal.7 (1996), pp. 133–170

work page 1996

[9] [9]

Classification and non-degeneracy of positive radial solutions for a weighted fourth-order equation and its application

S. Deng and X. Tian. “Classification and non-degeneracy of positive radial solutions for a weighted fourth-order equation and its application”. In:Nonlinear Anal.240 (2024), p. 113468

work page 2024

[10] [10]

Critical exponents, critical dimensions and the biharmonic operator

D.E. Edmunds, D. Fortunato, and E. Jannelli. “Critical exponents, critical dimensions and the biharmonic operator”. In:Arch. Rational Mech. Anal.112 (1990), pp. 269– 289

work page 1990

[11] [11]

Nonoscillation and eventual disconjugacy

U. Elias. “Nonoscillation and eventual disconjugacy”. In:Proc. Am. Math. Soc.66 (1977), pp. 269–275

work page 1977

[12] [12]

Classification of positive singular solutions to a nonlinear biharmonic equation with critical exponent

R.L. Frank and T. K¨ onig. “Classification of positive singular solutions to a nonlinear biharmonic equation with critical exponent”. In:Anal. PDE12 (2019), pp. 1101–111

work page 2019

[13] [13]

Gazzola, H.C

F. Gazzola, H.C. Grunau, and G. Sweers.Polyharmonic Boundary Value Problems: Positivity Preserving and Nonlinear Higher Order Elliptic Equations in Bounded Do- mains. Springer-Verlag, 1991

work page 1991

[14] [14]

Classification to the positive radial solutions with weighted biharmonic equation

X. Huang and L. Wang. “Classification to the positive radial solutions with weighted biharmonic equation”. In:Discrete Contin. Dyn. Syst.40 (2020), pp. 4821–4837

work page 2020

[15] [15]

D´ ecomposition orthogonale d’un espace hilbertien selon deux cˆ ones mutuellement polaires

J.J. Moreau. “D´ ecomposition orthogonale d’un espace hilbertien selon deux cˆ ones mutuellement polaires”. In:Comptes rendus hebdomadaires des s´ eances de l’Acad´ emie des sciences255 (1962), pp. 238–240

work page 1962

[16] [16]

Sharp higher order Adams’ inequality with exact growth condition on weighted Sobolev spaces

J.M. do ´O, G. Lu, and R. Ponciano. “Sharp higher order Adams’ inequality with exact growth condition on weighted Sobolev spaces”. In:J. Geom. Anal.34.53 (2024)

work page 2024

[17] [17]

Sharp Sobolev and Adams-Trudinger-Moser embeddings on weighted Sobolev spaces and their applications

J.M. do ´O, G. Lu, and R. Ponciano. “Sharp Sobolev and Adams-Trudinger-Moser embeddings on weighted Sobolev spaces and their applications”. In:Forum Math.36 (2024), pp. 1279–1320

work page 2024

[18] [18]

Trudinger-Moser embeddings on weighted Sobolev spaces on unbounded domains

J.M. do ´O, G. Lu, and R. Ponciano. “Trudinger-Moser embeddings on weighted Sobolev spaces on unbounded domains”. In:Discrete Contin. Dyn. Syst.45 (2025), pp. 557– 584. REFERENCES 35

work page 2025

[19] [19]

A sharp Adams-type inequality for weighted Sobolev spaces

J.M. do ´O, A. Macedo, and J.F. de Oliveira. “A sharp Adams-type inequality for weighted Sobolev spaces”. In:Q. J. Math.71 (2020), pp. 517–538

work page 2020

[20] [20]

Concentration-compactness and extremal problems for a weighted Trudinger-Moser inequality

J.M. do ´O and J .F. de Oliveira. “Concentration-compactness and extremal problems for a weighted Trudinger-Moser inequality”. In:Commun. Contemp. Math.19 (2017), p. 1650003

work page 2017

[21] [21]

On a class of quasilinear elliptic problems with critical exponential growth on the whole space

J.F. de Oliveira. “On a class of quasilinear elliptic problems with critical exponential growth on the whole space”. In:Topol. Methods Nonlinear Anal.49 (2017), pp. 529– 550

work page 2017

[22] [22]

Equivalence of critical and subcritical sharp Trudinger- Moser inequalities in fractional dimensions and extremal functions

J.F. de Oliveira and J.M. do ´O. “Equivalence of critical and subcritical sharp Trudinger- Moser inequalities in fractional dimensions and extremal functions”. In:Rev. Mat. Iberoam.39 (2023), pp. 1073–1096

work page 2023

[23] [23]

On a sharp inequality of Adimurthi-Druet type and extremal functions

J.F. de Oliveira and J.M. do ´O. “On a sharp inequality of Adimurthi-Druet type and extremal functions”. In:Calc. Var.62.162 (2023)

work page 2023

[24] [24]

Trudinger-Moser type inequalities for weighted Sobolev spaces involving fractional dimensions

J.F. de Oliveira and J.M. do ´O. “Trudinger-Moser type inequalities for weighted Sobolev spaces involving fractional dimensions”. In:Proc. Amer. Math. Soc.142 (2014), pp. 2813–2828

work page 2014

[25] [25]

Hardy-Sobolev type inequality and su- percritical extremal problem

J.F. de Oliveira, J.M. do ´O, and P. Ubilla. “Hardy-Sobolev type inequality and su- percritical extremal problem”. In:Discrete Contin. Dyn. Syst.39 (2019), pp. 3345– 3364

work page 2019

[26] [26]

On a Sobolev-type inequality and its minimizers

J.F. de Oliveira and J.N. Silva. “On a Sobolev-type inequality and its minimizers”. In:Anal. Appl.22 (2024), pp. 1417–1446

work page 2024

[27] [27]

Schoen.Variational theory for the total scalar curvature functional for Riemann- ian metrics and related topics

R.M. Schoen.Variational theory for the total scalar curvature functional for Riemann- ian metrics and related topics. Topics in calculus of variations, Lect. Notes Math. 1365, 120-154. 1989

work page 1989

[28] [28]

The Brezis-Nirenberg result for the fractional Lapla- cian

R. Servadei and E. Valdinoci. “The Brezis-Nirenberg result for the fractional Lapla- cian”. In:Trans. Am. Math. Soc.367 (2015), pp. 67–102

work page 2015

[29] [29]

Existence of solitary waves in higher dimensions

W.A. Strauss. “Existence of solitary waves in higher dimensions”. In:Commun.Math. Phys.55 (1977), pp. 149–162

work page 1977

[30] [30]

The best Sobolev constant

C.A. Swanson. “The best Sobolev constant”. In:Appl. Anal.47 (1992), pp. 227–239

work page 1992

[31] [31]

Best constant in Sobolev inequality

G. Talenti. “Best constant in Sobolev inequality”. In:Ann. Mat. Pura Appl.110 (1976), pp. 353–372

work page 1976

[32] [32]

Walter.Ordinary differential equations

W. Walter.Ordinary differential equations. Transl. from the German by Russell Thomp- son. English. Vol. 182. Grad. Texts Math. New York NY: Springer-Verlag, 1998.isbn: 0-387-98459-3

work page 1998

[33] [33]

Classification of solutions of higher order conformally invariant equations

J. Wei and X. Xu. “Classification of solutions of higher order conformally invariant equations”. In:Math. Ann.313 (1999), pp. 207–228. (J.F. de Oliveira) Department of Mathematics Federal University of Piau´ı 64049-550 Teresina, PI, Brazil Email address:jfoliveira@ufpi.edu.br REFERENCES 36 (J.N. Silva) Department of Mathematics Federal University of the D...

work page 1999