Non-radial minimizers for Hardy--Sobolev inequalities in non-convex cones

A. I. Nazarov; N. V. Rastegaev

REVIEW 4 major objections 4 minor 1 cited by

Non-radial minimizers for Hardy--Sobolev inequalities in non-convex cones

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Pith's one-line read The paper establishes that in cones whose spherical cross-section has sufficiently small first Neumann eigenvalue, the exact Hardy–Sobolev constant is attained at a non-radial minimizer, so the associated Neumann problem has at least two…

desk verdict A solid but compressed extension of known symmetry-breaking methods to the p-Laplacian and the full Hardy–Sobolev range; the main theorem is right, but several load-bearing checks are deferred. read the letter →

arxiv 2507.04470 v1 pith:DPMRUSX4 submitted 2025-07-06 math.AP

A. I. Nazarov , N. V. Rastegaev This is my paper · ORCID

classification math.AP MSC 35B0635B3335J9246E35

keywords Hardy–SobolevinequalitysymmetrybreakingNeumannproblemp-LaplacianTalenti–BlissextremalBeltrami–Laplaceeigenvaluenon-convexconessecondvariation

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 treats the sharp Hardy–Sobolev inequality on possibly non-convex unbounded cones with Neumann boundary condition on the lateral surface, driven by the $p$-Laplacian. The standard radial Talenti--Bliss function $w(|x|)$ always satisfies the first-order Euler--Lagrange condition, and in the whole space it is the extremal. The authors prove that in a cone over a spherical domain $D$, once the first nonconstant Neumann eigenvalue $\lambda_1(D)$ of the Beltrami--Laplace operator on $D$ is small enough, this radial function is not a minimizer: the energy has a second negative direction, and the true minimizer is non-radial. For the subcritical Hardy--Sobolev range $0<\sigma<1$ the exact constant is always attained; for the critical Sobolev case $\sigma=1$ they add a strict comparison condition on the best constant and a $C^1$ boundary. As a corollary, the corresponding Neumann problem has at least two positive solutions in such cones.

What carries the argument

The machinery is a second-variation calculation at the radial solution $U$. The paper uses the trial direction $h=f(|x|)g(x/|x|)$, where $g$ is the first nonconstant Neumann eigenfunction of the Beltrami--Laplace operator on $D$ with eigenvalue $\lambda_1(D)$, normalized by $\int_D g^2=|D|$, and $f(r)=r^\alpha U'(r)$ with $\alpha=(1-\sigma)q/p$. Because $U$ is radial and $g$ has zero mean on $D$, the cross terms between the radial and angular parts of the second variation vanish, reducing the quadratic form to a radial integral plus a $\lambda_1(D)$-weighted term. Proposition 2 shows the chosen $f$ solves the linearized equation with an explicit constant $\Lambda_*=-(1-\alpha)(n-1-\alpha(p-1))<0$ in place of $\lambda_1(D)$; substituting this identity makes the quadratic form negative exactly when $\lambda_1(D)<-\Lambda_*$. Together with the already-negative direction $h=U$, this proves the radial solution cannot minimize the energy on the Nehari manifold (the constraint surface where the two norms balance).

What would settle it

To test the central claim, take a cone $\Sigma_D$ satisfying the eigenvalue bound and explicitly compute the second variation of the energy at the radial function $U$ in the direction $h=f(|x|)g(x/|x|)$; if this quadratic form is nonnegative, or if $h$ fails to satisfy the Neumann condition and finite-energy requirements, then the negative direction forcing non-radiality does not exist.

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Extended reading notes

Core claim

Let $n\ge 2$, $1<p<n$, $0<\sigma\le 1$, $q=np/(n-\sigma p)$, and let $\Sigma_D=\{tx:\ x\in D,\ t>0\}$ be the cone over a domain $D$ with strictly Lipschitz boundary on the unit sphere. The paper's central result, Theorem 3, gives two symmetry-breaking statements. For $0<\sigma<1$, set $\alpha=(1-\sigma)q/p$; if $\lambda_1(D)<(1-\alpha)(n-1-\alpha(p-1))$, then the exact Hardy--Sobolev constant $S^\sigma_{\Sigma_D}$ is attained at a non-radial function. For $\sigma=1$, if $\partial D$ is $C^1$, the strict inequality $S_{\Sigma_D}<S_{\mathbb{R}^n_+}$ holds, and $\lambda_1(D)<n-1$, then $S_{\Sigma_D}$ is likewise attained at a non-radial function. In both cases the radial Talenti--Bliss function is not an extremal, and the Neumann problem (8) has at least two positive solutions. The condition $\lambda_1(D)<n-1$ is sharp when $\sigma=1$: for some convex cones with $\lambda_1(D)\ge n-1$, earlier results give only radial minimizers.

Load-bearing premise

The load-bearing premise is that the unproved adaptation of the Dirichlet-cone existence result to Neumann boundary conditions really works for $0<\sigma<1$; if that transfer fails, the best possible constant may not actually be achieved by any function, and the instability calculation has no minimizer to disqualify.

Editorial extensions

If this is right

For $0<\sigma<1$, if $\lambda_1(D)<(1-\alpha)(n-1-\alpha(p-1))$, the sharp constant in the Neumann Hardy--Sobolev inequality is attained and the minimizer is non-radial.
For $\sigma=1$, under the assumptions $\partial D\in C^1$, $S_{\Sigma_D}<S_{\mathbb{R}^n_+}$, and $\lambda_1(D)<n-1$, the sharp Sobolev constant on the cone is attained by a non-radial function, recovering and extending the earlier $p=2$, $n\ge 3$ case.
The associated Neumann problem (8) has at least two positive solutions whenever the hypotheses of Theorem 3 hold.
For $\sigma=1$ the threshold $\lambda_1(D)<n-1$ is sharp: for some convex cones with $\lambda_1(D)\ge n-1$ all minimizers are radial.

Reading between the lines

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

A numerical computation of the second variation for cones with $\lambda_1(D)$ near the claimed threshold would show whether the analytic bound is sharp or merely sufficient: if a negative direction appears above the bound, the true symmetry-breaking threshold is higher.
The same trial direction $f(|x|)g(x/|x|)$ should transfer to Dirichlet-cone analogues by replacing $\lambda_1(D)$ with the corresponding Dirichlet eigenvalue, extending the mechanism beyond Neumann problems.
Since the threshold depends only on $\lambda_1(D)$, $p$, $\sigma$, and $n$, the result suggests that geometrically flattening the spherical cross-section $D$—lowering its first Neumann eigenvalue—triggers symmetry breaking in any cone, regardless of finer shape features.
One could test the Hardy--Sobolev case numerically for $\sigma<1$ and $p\ne 2$ to see whether the explicit threshold $(1-\alpha)(n-1-\alpha(p-1))$ marks the actual onset of non-radial minimizers or only a conservative sufficient condition.

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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

Colleague — quick take on arXiv:2507.04470. The paper proves non-radial minimizers for Hardy–Sobolev inequalities in non-convex cones with the p-Laplacian, covering the full range 0<σ≤1. The main new input is an explicit eigenvalue condition on the spherical cross-section: λ1(D)<(1−α)(n−1−α(p−1)) for σ<1, and λ1(D)

Soft spots are in three places. First, Theorem 1 (subcritical attainability) is not proved; it is imported wholesale from the Dirichlet cone paper [15], with the comment that the Neumann adaptation is 'very straightforward.' That is an explicit omission, and it is load-bearing because the non-radiality conclusion needs an attained minimizer. A referee should ask for a sketch, since the Dirichlet proof may rely on vanishing on the boundary in ways the Neumann problem does not. Second, the direction h=f g is never shown to be admissible (h∈D_p) or tangent to the Nehari manifold. The tangency actually follows immediately from ∫_D g=0, and admissibility is a routine decay check from the known asymptotics of U, but the paper does neither. The integration by parts leading to the sign of the second variation also assumes boundary terms at 0 and ∞ vanish; that is another check to supply. Third, the concentration argument in Theorem 2 is compressed, especially the boundary point x0∈∂ΣD. The claim that it cannot beat S_{R^n_+} is standard for C^1 boundaries but is asserted rather than proved.

I do not think any of these is fatal. The computation is explicit, the path is clear, and the self-citations are to standard tools, not to the theorem being proved. The stress-test note overstates the tangency problem — that part is trivially fixed — but the admissibility and boundary-term checks are real. This deserves peer review; a good referee will ask for the missing details. I would send it out, expecting minor-to-moderate revision. Not a desk reject, and not a breakthrough either. It is a competent, useful extension.

Referee Report

4 major / 4 minor

Summary. The paper studies Hardy-Sobolev inequalities on cones \Sigma_D over a spherical domain D, in the Neumann setting, and claims symmetry breaking for the extremal functions. For 0<\sigma<1 the authors assert the exact constant is always attained (Theorem 1); for \sigma=1 they prove attainability when \partial D\in C^1 and S_{\Sigma_D}<S_{\mathbb{R}^n_+} (Theorem 2). The central result, Theorem 3, is that if the first nonconstant Neumann eigenvalue \lambda_1(D) of the Laplace-Beltrami operator is sufficiently small, the exact Hardy-Sobolev constant is attained at a non-radial function, so the radial Talenti--Bliss function is not an extremal. The non-radiality proof is based on an explicit second-variation direction h=f(|x|)g(x/|x|), with f=r^\alpha U' for a radial solution U, where \alpha=(1-\sigma)q/p, and g a nonconstant Neumann eigenfunction. The paper further concludes that the corresponding Neumann problem has at least two positive solutions.

Significance. If correct, the result extends the symmetry-breaking phenomenon of Ciraolo--Pacella--Polvara from the Laplacian/Sobolev case to the p-Laplacian and to Hardy-Sobolev weights, with an explicit and apparently sharp eigenvalue criterion. The main strength is Proposition 2: an exact, parameter-free algebraic computation that identifies the second-variation direction and the threshold constant; no fitted parameters or circular assumptions are used. The main weakness is that the analytic framework around that computation is incomplete: attainability is imported from [15] without proof, and the admissibility and tangency of the test direction are not verified. Because these are exactly the steps that connect the algebra to the variational conclusion, the paper is not yet in publishable form.

major comments (4)

[Section 2.1, Theorem 1 and Proposition 1] The attainability of S^\sigma_{\Sigma_D} for 0<\sigma<1 is not proved in the manuscript. Proposition 1 is only stated, and Theorem 1 is said to follow from [15] 'with no changes'; the analogous statement is then used in Theorem 3 to assert that the exact constant is attained at a non-radial function. The Neumann setting is not identical to the Dirichlet setting of [15]: functions in D_p(\Sigma_D) do not vanish on \partial\Sigma_D, and the concentration-compactness argument has to be reworked for the Neumann space, including the treatment of atoms at 0 and \infty and the exclusion of mass loss that would prevent the existence of a convergent minimizing subsequence. This is load-bearing because Theorem 3 assumes attainability as a premise for its non-radiality conclusion. A complete proof or a precise step-by-step reduction to the Dirichlet case is needed.
[Section 3, admissibility of h] The variational direction h=f(|x|)g(x/|x|) used in Section 3 is never shown to be admissible, i.e. h\in D_p(\Sigma_D). Proposition 2 verifies only the pointwise Euler equation (9) for the radial factor f; it does not establish the integrability of |\nabla h|^p and r^{(\sigma-1)q}|h|^q over the unbounded cone, nor the Neumann boundary condition for h on \partial\Sigma_D. In particular, for p>2 on a merely Lipschitz domain D, the first nonconstant Neumann eigenfunction g of the Laplace-Beltrami operator need not belong to W^{1,p}(D), so the angular contribution (f(r)/r)\nabla_\theta g may fail to be p-integrable. Since the tangent space of the Nehari manifold is a subspace of D_p, the second-variation computation in Section 3 is not justified unless this admissibility is proved or replaced by a valid approximation argument.
[Section 3, tangency to the Nehari manifold] Section 3 also omits the verification that h is tangent to the Nehari manifold, i.e. that DJ^\sigma_{\Sigma_D}(U;h)=0. The conclusion that the existence of two negative directions forces a better minimizer than U is a constrained second-variation statement on N^\sigma(\Sigma_D), and it requires h to belong to the tangent space there. This identity plausibly follows from \int_D g\,dS=0 for a nonconstant Neumann eigenfunction, but the step is not written out. The same applies to the boundary terms at r=0 and r=\infty in the radial integration by parts used to pass from the second variation to the sufficient condition displayed before Proposition 2.
[Section 2.2, proof of Theorem 2] In the proof of Theorem 2 (Sobolev case), the boundary concentration step is compressed to a single sentence: if x_0\in\partial\Sigma_D\cap\partial B(0,1), then 'due to the smoothness of \partial D, u_k cannot give a better constant than S_{\mathbb{R}^n_+}'. This step is load-bearing because (7) is exactly the inequality needed to rule out boundary concentration. The local blow-up argument for p-Laplacian Neumann problems at a C^1 boundary point should be supplied or cited precisely. In addition, the normalization by dilation that puts exactly half of the L^q mass into the unit ball needs a justification when mass may concentrate at the vertex or at infinity.

minor comments (4)

[Section 1] The statement that the space D_p(\mathbb{R}^n) 'does not depend on \sigma' is made without proof; a sentence or reference explaining the equivalence of the weighted L^q conditions would be helpful.
[Section 2.1] The sentence 'Without loss of generality we can assume that all u_k have bounded supports, otherwise we can cut them off at sufficiently large radii' needs a justification, since truncation changes both the numerator and the denominator of the quotient Q^\sigma_{\Sigma_D}.
[Section 3] The Nehari manifold N^\sigma(\Sigma_D) is called a codimension-one manifold; the authors should state the regularity conditions that make it a C^1 manifold and specify the tangent space.
[Throughout] Typos and style issues: 'eigenfunciton' in Section 3; 'Talenti--Bliss type functions' should be 'Talenti--Bliss functions' or similar; some references lack page ranges or translation details.

Circularity Check

0 steps flagged · score 0.0 of 10

No circularity: the non-radiality threshold is derived by explicit computation, and the cited self-results are external support, not inputs that contain the conclusion.

full rationale

The paper's central symmetry-breaking claim is supported by an explicit, parameter-free computation rather than by fitting or by importing the conclusion. In Section 3 the authors choose a test direction h = f(|x|)g(x/|x|), with g a first non-constant Neumann eigenfunction and f = r^alpha U'. Proposition 2 then verifies by direct differentiation that f satisfies the pointwise equation (9), and the constants alpha = (1-sigma)q/p and Lambda* = -(1-alpha)(n-1-alpha(p-1)) are solved from the coefficient identities, not imposed to force the desired inequality; the condition lambda1(D) < -Lambda* follows algebraically from comparing (9) with the second-variation estimate. No fitted parameter is renamed as a prediction, and no uniqueness theorem from the authors' own prior work is invoked to forbid alternatives. The citations to [15] and [14] provide a prior attainability result and a standard second-variation formula; these are real external results and do not encode the non-radiality theorem. The unproved Neumann adaptation of [15] and the unverified admissibility/tangency of h are rigor or correctness gaps, not circularity: they do not make the asserted conclusion equal to an input by construction.

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

The central argument introduces no free parameters: the threshold is expressed through the fixed spectral quantity lambda_1(D) and explicit constants. The main uncharged inputs are standard Hardy-Sobolev theory and two imported technical results whose proofs are not given.

assumptions (5)

standard math Sharp Hardy-Sobolev inequality (1) in R^n with extremals given by Talenti-Bliss functions (2).
Used as the starting point for cone inequalities and explicit radial solutions; established in [1], [16], [10], and [9].
domain assumption Proposition 1, a concentration-compactness statement for bounded sequences in the Neumann cone for 0<sigma<1, and the consequent attainability Theorem 1.
The proof is omitted; the paper asserts it matches Lemma 3.1 of [15], which was proved for Dirichlet conditions.
standard math Lemma 1, that under p<=(n+1)/2 and positive mean curvature of the boundary the strict comparison S_SigmaD<S_R^n_+ holds, is taken from [6].
Used to supply examples for Theorem 2; cited from earlier published estimates.
standard math The formula for the second variation of the p-energy functional at a critical point, including the reduction to a weighted eigenvalue equation, follows the scheme of [14, Section 3].
The paper uses this formula without re-deriving it; it is standard in the p-Laplacian literature.
ad hoc to paper The test function h=f g is an admissible variation in D_p(SigmaD) with the Neumann boundary condition.
The paper does not verify integrability or boundary conditions for h; Proposition 2 proves only that f solves the pointwise equation (9).

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Pith. "Pith review of Non-radial minimizers for Hardy--Sobolev inequalities in non-convex cones." pith.science (2026). https://pith.science/paper/DPMRUSX4

@misc{pith2026250704470,
  author       = {Pith},
  title        = {Pith review of: Non-radial minimizers for Hardy--Sobolev inequalities in non-convex cones},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/DPMRUSX4}},
  note         = {Machine review of arXiv:2507.04470}
}

abstract

The symmetry breaking is obtained for Neumann problems driven by $p$-Laplacian in certain non-convex cones. These problems are generated by the Hardy--Sobolev inequalities. In the case of the Sobolev inequality for the ordinary Laplacian this problem was investigated in (Ciraolo, Pacella, Polvara, 2024). Such problems have obvious radial solutions -- Talenti--Bliss type functions of $|x|$. However, under a certain restriction on the first Neumann eigenvalue $\lambda_1(D)$ of the Beltrami--Laplace operator on the spherical cross-section $D$ of the cone we prove this radial solution cannot be an extremal function, therefore minimizer must be non-radial. This leads to multiple solutions for the corresponding Neumann problem.

Discussion (0). Continue with ORCID to comment.

Forward citations

Cited by 1 Pith paper

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

Non-radial solutions for the quasi-linear H\'enon type $N$-Laplacian Liouville equation
math.AP 2026-07 conditional novelty 6.0 of 10

At discrete critical exponents of the Hénon weight, the N-Laplacian Liouville equation admits continua of non-radial entire solutions bifurcating from the radial solution.

Reference graph

Works this paper leans on

16 extracted references · 16 canonical work pages · cited by 1 Pith paper

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Pith tools

Reviewed August 6, 2026 · model on record in the stance chip above.

[1] [15]

I., Hardy-Sobolev inequalities in a cone

Nazarov, A. I., Hardy-Sobolev inequalities in a cone. Probl. Mat. Anal.31 (2005), pp.39-46 (Russian); English transl.: Journal of Mathematical Sciences 132 (2006), N4, pp.419-427

work page 2005

[2] [1]

Journal of Differ- ential Geometry 11 (1976), N4, pp.573-598

Aubin, T., Problemes isop´ erim´ etriques et espaces de Sobolev. Journal of Differ- ential Geometry 11 (1976), N4, pp.573-598

work page 1976

[3] [2]

and Lieb, E., A relation between pointwise convergence of functions and convergence of functionals

Br´ ezis, H. and Lieb, E., A relation between pointwise convergence of functions and convergence of functionals. Proceedings of the American Mathematical So- ciety 88 (1983), N3, pp.486-490

work page 1983

[4] [3]

and Roncoroni, A., Symmetry results for critical anisotropic p-Laplacian equations in convex cones

Ciraolo, G., Figalli, A. and Roncoroni, A., Symmetry results for critical anisotropic p-Laplacian equations in convex cones. Geometric and Functional Analysis 30 (2020), N3, pp.770-803

work page 2020

[5] [4]

and Polvara, C

Ciraolo, G., Pacella, F. and Polvara, C. C., Symmetry breaking and instabil- ity for semilinear elliptic equations in spherical sectors and cones. Journal de Math´ ematiques Pures et Appliqu´ ees187 (2024), pp.138-170

work page 2024

[6] [5]

and Pacella, F., Existence of nonradial positive and nodal solutions to a critical Neumann problem in a cone

Clapp, M. and Pacella, F., Existence of nonradial positive and nodal solutions to a critical Neumann problem in a cone. Mathematics in Engineering3 (2021), N3, pp.1-15

work page 2021

[7] [6]

Demyanov, A. V. and Nazarov, A. I. On the existence of an extremal function in Sobolev embedding theorems with limit exponent. Algebra & Analysis17 (2005), N5, pp.105-140 (Russian); English transl.: St. Petersburg Mathematical Journal 17 (2006), N5, pp.108-142

work page 2005

[8] [7]

Transactions of the American Mathematical Society330 (1992), N1, pp.191-201

Egnell, H., Positive solutions of semilinear equations in cones. Transactions of the American Mathematical Society330 (1992), N1, pp.191-201

work page 1992