pith. sign in

arxiv: 2606.27710 · v1 · pith:6CT2XUF5new · submitted 2026-06-26 · 🧮 math.AP

The compactness of Moser-Trudinger functionals with conical metric in the unit ball

Qi Xia , Yufeng Lu This is my paper

Pith reviewed 2026-06-29 04:00 UTC · model grok-4.3

classification 🧮 math.AP
keywords Moser-Trudinger inequalityconical metriccompactnessradial symmetryextremal functionsSobolev embeddingunit ball
0
0 comments X

The pith

The sequence of extremal functions for the weighted Moser-Trudinger inequality with conical singularity converges in C^1 to an extremal for the standard inequality.

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

This paper studies the compactness of maximizers for a family of Moser-Trudinger functionals modified by a radially symmetric weight h_ε that behaves like |x|^{-2ε} near the origin. The authors show that as ε tends to zero, the corresponding extremal functions u_ε converge to a limit u_0 that belongs to C^1 of the closed unit ball. This limit u_0 then serves as an extremal function achieving the supremum in the classical Moser-Trudinger inequality restricted to radial functions with unit Dirichlet energy. The result establishes that the presence of the conical metric does not destroy the attainment of the supremum in the limit case.

Core claim

Zhang proved attainment for the weighted case with factor (1+ε). The paper proves that the sequence of such u_ε converges to u_0 in C^1(¯B), and u_0 attains the sup of ∫_B exp(4π u²) dx over radial u with ∫ |∇u|² ≤1, u≠0.

What carries the argument

The radial symmetry class S combined with the asymptotic condition lim_{x→0} h_ε(x) |x|^{-2ε}=1 on the weight.

If this is right

  • The limit u_0 is continuously differentiable up to the boundary of the ball.
  • u_0 achieves the supremum in the unweighted Moser-Trudinger inequality within the radial class.
  • The functional remains compact under the conical perturbation in the radial setting.
  • No concentration occurs at the origin in the limit.

Where Pith is reading between the lines

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

  • This compactness may extend to cases where the weight approaches other singular metrics.
  • The result implies that the Euler-Lagrange equation for the limit has a regular solution.
  • Similar compactness could hold without radial symmetry if additional assumptions are made.

Load-bearing premise

All candidate functions are restricted to be radially symmetric, and the weight h_ε satisfies the exact limit condition at the origin.

What would settle it

A sequence of radial maximizers u_ε whose gradients blow up or whose limit fails to be C^1 at some point would falsify the claim.

read the original abstract

Let $\mathbb{B}$ be the unit ball in $\mathbb{R}^2$, $W_0^{1,2} \left( \mathbb{B} \right)$ is a standard Sobolev space. Suppose a function $h_{\epsilon}(x)$ is radially symmetric, nonnegative, continuous on $\overline{\mathbb{B}}$ and satifies $\underset{x \rightarrow 0}{\lim} h_{\epsilon}(x) |x|^{- 2 \epsilon} =1 $, with $h_{\epsilon} (x) >0$ on $\overline{\mathbb{B}} \setminus \{0\}$. In \citep{26}, Zhang proved that the supremum in the following inequality can be attained by some function $u_{\epsilon}$, i.e. , \begin{align} \int_{ \mathbb{B} } h_{\epsilon} (x) e^{ 4 \pi \left(1 + \epsilon \right) {u_{\epsilon}}^2 } dx = \underset{u \in W_0^{1,2} \left( \mathbb{B} \right) \cap \mathcal{S} \setminus \{0\} , ~ \int_{ \mathbb{B} } |\nabla u|^2 dx \leq 1}{\sup} \int_{\mathbb{B}} h_{\epsilon} (x) e^{4 \pi (1 + \epsilon) u^2 } dx, \label{eq: 0.1} \end{align} where $4 \pi$ is the best constant in the classical Moser-Trudinger inequality, and $\mathcal{S}$ is the set of radially symmetric functions. In this paper, we consider the compactness of the sequence $\{ u_{\epsilon} \}_{\epsilon} $ and prove that the limit of this sequence is a function $u_0 \in C^1 (\overline{ \mathbb{B}} )$. Moreover, the $u_0$ is an extremal function of the supremum \begin{align*} \underset{u \in W_0^{1,2} \left( \mathbb{B} \right) \cap \mathcal{S} \setminus \{0\} , ~ \int_{ \mathbb{B} } |\nabla u|^2 dx \leq 1}{\sup} \int_{\mathbb{B}} e^{4 \pi u^2 } dx. \end{align*}

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 / 0 minor

Summary. The manuscript claims that the sequence of radial maximizers {u_ε} for the weighted supercritical Moser-Trudinger functional (with weight h_ε satisfying lim_{x→0} h_ε(x)|x|^{-2ε}=1 and the stated regularity) converges as ε→0 to a limit u_0 ∈ C¹(¯B) that attains the critical supremum sup ∫_B exp(4π u²) dx over W_0^{1,2}(B) ∩ S with ∫_B |∇u|² ≤ 1.

Significance. If correct, the result would establish attainment of the critical Moser-Trudinger supremum in the radial class via a limiting procedure with conical weights, which would be a notable contribution to the analysis of extremals for exponential functionals.

major comments (1)
  1. [Abstract] Abstract, claim following Eq. (0.1): the assertion that the limit u_0 attains the supremum of ∫_B exp(4π u²) dx contradicts the classical non-attainment result for the critical Moser-Trudinger functional on bounded domains. Any such u_0 would solve the Euler-Lagrange equation -Δu = λ u exp(4π u²) with u|∂B=0. This is impossible by the Pohozaev identity, which yields a boundary integral contradiction on the star-shaped domain B. The obstruction persists in the radial class S. This issue is load-bearing for the central compactness-to-attainment claim.

Simulated Author's Rebuttal

1 responses · 0 unresolved

We thank the referee for the careful reading of our manuscript and for identifying the inconsistency in the abstract with the classical non-attainment result for the critical Moser-Trudinger functional. We address the comment point by point below.

read point-by-point responses

  1. Referee: [Abstract] Abstract, claim following Eq. (0.1): the assertion that the limit u_0 attains the supremum of ∫_B exp(4π u²) dx contradicts the classical non-attainment result for the critical Moser-Trudinger functional on bounded domains. Any such u_0 would solve the Euler-Lagrange equation -Δu = λ u exp(4π u²) with u|∂B=0. This is impossible by the Pohozaev identity, which yields a boundary integral contradiction on the star-shaped domain B. The obstruction persists in the radial class S. This issue is load-bearing for the central compactness-to-attainment claim.

    Authors: We agree with the referee that the classical result of non-attainment holds, including within the radial class S, by the Pohozaev identity applied to any solution of the associated Euler-Lagrange equation. Our abstract's assertion that u_0 attains the unweighted supremum is therefore incorrect and leads to the noted contradiction. The compactness result establishing C^1 convergence of the sequence {u_ε} remains valid as a separate statement, but we cannot and do not claim that the limit attains the critical supremum. We will revise the abstract, the statement of the main result, and any related claims to remove the attainment assertion entirely, stating only the C^1 convergence of the maximizers without reference to u_0 being extremal for the unweighted functional. This directly addresses the load-bearing issue raised. revision: yes

Circularity Check

0 steps flagged

No significant circularity detected

full rationale

The paper cites an external result from Zhang (ref 26) for existence of the maximizers u_ε in the weighted supercritical case. The compactness of {u_ε} as ε→0, the C¹ convergence to u_0, and the claim that u_0 attains the unweighted supremum are derived from the functional inequalities, radial symmetry in S, and the given limit condition on h_ε. No steps reduce by definition or construction to the inputs, no self-citations are load-bearing, and the derivation chain is independent of the target claim.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

The paper rests on standard functional-analysis background and a cited existence theorem; no free parameters or invented entities are introduced.

axioms (2)
  • standard math Standard properties of Sobolev space W_0^{1,2}(B) and the classical Moser-Trudinger inequality
    Used to set up the functional and the constraint.
  • domain assumption Existence of each maximizer u_ε from Zhang [26]
    The sequence is defined using this prior existence result.

pith-pipeline@v0.9.1-grok · 5978 in / 1226 out tokens · 80886 ms · 2026-06-29T04:00:21.681803+00:00 · methodology

discussion (0)

Sign in with ORCID, Apple, or X to comment. Anyone can read and Pith papers without signing in.

Reference graph

Works this paper leans on

27 extracted references

  1. [1]

    A Trudinger–Moser inequality for a conical metric in the unit ball

    Yang Y, Zhu X. A Trudinger–Moser inequality for a conical metric in the unit ball. Archiv der Mathematik, 2019, 112(5): 531-545

    2019

  2. [2]

    A nonlinear Dirichlet problem on the unit ball and its applications

    Ni W. A nonlinear Dirichlet problem on the unit ball and its applications. Indiana University Mathematics Journal, 1982, 31(6): 801-807

    1982

  3. [3]

    Compactness of Extremals for Trudinger-Moser Functionals on the Unit Ball inR 2

    Shan W, Li X. Compactness of Extremals for Trudinger-Moser Functionals on the Unit Ball inR 2. Acta Mathematica Sinica, English Series, 2024, 40(11): 2840-2854

    2024

  4. [4]

    Extremal functions for singular Trudinger–Moser inequalities in the entire Euclidean space

    Li X, Yang Y. Extremal functions for singular Trudinger–Moser inequalities in the entire Euclidean space. Journal of Differential Equations, 2018, 264(8): 4901-4943. 18

    2018

  5. [5]

    On the existence of an extremal function for an inequality of J

    Carleson L. On the existence of an extremal function for an inequality of J. Moser. Bull. Sci. Math, 1986, 110(2): 113-127

    1986

  6. [6]

    Non-radial Maximizers For Functionals With Exponential Non- linearity inR 2

    Calanchi M, Terraneo E. Non-radial Maximizers For Functionals With Exponential Non- linearity inR 2. Advanced Nonlinear Studies, 2005, 5(3): 337-350

    2005

  7. [7]

    Symmetry of extremal functions in Moser-Trudinger inequal- ities and a Hnon type problem in dimension two

    Bonheure D, Serra E, Tarallo M. Symmetry of extremal functions in Moser-Trudinger inequal- ities and a Hnon type problem in dimension two. Advances in Differential Equations, 2008, 13: 105-138

    2008

  8. [8]

    Proceedings of the American Mathematical Society, 2016, 144(8): 3369-3380

    deFigueiredoD.G., do ´O, J.M.B., dosSantos, E.M.Trudinger-Moserinequalitiesinvolvingfast growth and weights with strong vanishing at zero. Proceedings of the American Mathematical Society, 2016, 144(8): 3369-3380

    2016

  9. [9]

    Regularity for a more general class of quasilinear elliptic equations

    Tolksdorf P. Regularity for a more general class of quasilinear elliptic equations. Journal of Differential equations, 1984, 51(1): 126-150

    1984

  10. [10]

    A sharp form of Moser–Trudinger inequality in high dimension

    Yang Y. A sharp form of Moser–Trudinger inequality in high dimension. Journal of Functional Analysis, 2006, 239(1): 100-126

    2006

  11. [11]

    Singular solutions of the p-Laplace equation

    Kichenassamy S, V´ eron L. Singular solutions of the p-Laplace equation. Mathematische An- nalen, 1986, 275(4): 599-615

    1986

  12. [12]

    Moser-Trudinger inequality on compact Riemannian manifolds of dimension two

    Li Y. Moser-Trudinger inequality on compact Riemannian manifolds of dimension two. Journal of Partial Differential Equations, 2001, 14(2): 163-192

    2001

  13. [13]

    Extremal functions for Trudinger–Moser inequalities of Adimurthi–Druet type in dimension two

    Yang Y. Extremal functions for Trudinger–Moser inequalities of Adimurthi–Druet type in dimension two. Journal of Differential Equations, 2015, 258(9): 3161-3193

    2015

  14. [14]

    Blow-up analysis in dimension 2 and a sharp form of Trudinger–Moser inequality

    Adimurthi, Druet O. Blow-up analysis in dimension 2 and a sharp form of Trudinger–Moser inequality. Communications in Partial Differential Equations, 2004, 29: 295–322

    2004

  15. [15]

    An improved Trudinger–Moser inequality and its extremal functions involvingLp-norm inR 2

    Li X. An improved Trudinger–Moser inequality and its extremal functions involvingLp-norm inR 2. Turkish Journal of Mathematics, 2020, 44(4): 1092-1114

    2020

  16. [16]

    An improved singular Trudinger–Moser inequality in unit ball

    Yuan A, Zhu X. An improved singular Trudinger–Moser inequality in unit ball. Journal of Mathematical Analysis and Applications, 2016, 435(1): 244-252

    2016

  17. [17]

    Improved Moser–Trudinger inequality of Tintarev type in dimension n and the existence of its extremal functions

    Nguyen V. Improved Moser–Trudinger inequality of Tintarev type in dimension n and the existence of its extremal functions. Annals of Global Analysis and Geometry, 2018, 54(2): 237-256. 19

    2018

  18. [18]

    Compactness of extremals for critical singular Trudinger-Moser functionals

    Wang Y, Yang Y. Compactness of extremals for critical singular Trudinger-Moser functionals. Journal of Mathematical Analysis and Applications, 2021, 496(2): 124841

    2021

  19. [19]

    Blow-up analysis concerning singular Trudinger–Moser inequalities in dimen- sion two

    Yang Y, Zhu X. Blow-up analysis concerning singular Trudinger–Moser inequalities in dimen- sion two. Journal of Functional Analysis, 2017, 272(8): 3347-3374

    2017

  20. [20]

    A sharp form of an inequality by N

    Moser J. A sharp form of an inequality by N. Trudinger. Indiana University Mathematics Journal, 1971, 20(11): 1077-1092

    1971

  21. [21]

    On imbeddings into Orlicz spaces and some applications

    Trudinger N. On imbeddings into Orlicz spaces and some applications. Journal of Mathematics and Mechanics, 1967, 17(5): 473-483

    1967

  22. [22]

    Blow-up analysis for solutions of−∆u=V e u in dimension two

    Li Y, Shafrir I. Blow-up analysis for solutions of−∆u=V e u in dimension two. Indiana University Mathematics Journal, 1994, 43(4): 1255-1270

    1994

  23. [23]

    Blow-up analysis for Liouville type equation in self-dual gauge field theories

    Ohtsuka H, Suzuki T. Blow-up analysis for Liouville type equation in self-dual gauge field theories. Communications in Contemporary Mathematics, 2005, 7(02): 177-205

    2005

  24. [24]

    The concentration-compactness principle in the calculus of variations

    Lions P L. The concentration-compactness principle in the calculus of variations. The limit case, part 1. Revista matem´ atica iberoamericana, 1985, 1(1): 145-201

    1985

  25. [25]

    The concentration-compactness principle in the calculus of variations

    Lions P L. The concentration-compactness principle in the calculus of variations. The limit case, part 2. Revista matem´ atica iberoamericana, 1985, 1(2): 45-121

    1985

  26. [26]

    Extremals for a Trudinger-Moser inequality with a vanishing weight in the unit disk

    Zhang M. Extremals for a Trudinger-Moser inequality with a vanishing weight in the unit disk. Analysis Mathematica, 2020, 46(3): 639-654

    2020

  27. [27]

    Extremal functions for the singular Moser-Trudinger inequality in 2 dimen- sions

    Csató G, Roy P. Extremal functions for the singular Moser-Trudinger inequality in 2 dimen- sions. Calculus of Variations and Partial Differential Equations, 2015, 54(2): 2341-2366. 20

    2015