Minimization problem associated with an improved Hardy-Sobolev type inequality

Megumi Sano

REVIEW 2 major objections 5 minor 26 references

Minimization problem associated with an improved Hardy-Sobolev type inequality

T0 review · 2 major / 5 minor · reviewed 2026-08-14 · deepseek-v4-flash

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Pith's one-line read This paper proves that for an improved Hardy–Sobolev minimization problem with a boundary-singular weight, non-radial minimizers appear for a parameter interval near $a=1$, while none exist for smaller $a$.

desk verdict The main symmetry-breaking result is likely correct, but the stated constant A in Theorem 1(iii) is miscalculated and the theorem as written is not fully established. read the letter →

arxiv 1908.03915 v3 pith:SO2IFGTQ submitted 2019-08-11 math.AP

Megumi Sano This is my paper · ORCID

classification math.AP MSC 35A2335J2035A08

keywords Hardy-Sobolevinequalityminimizationproblemnon-radialminimizersymmetrybreakingcriticalexponentSchwarzsymmetrizationimprovedinfinite-dimensionalSobolev

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 the infimum $I_a$ of the Rayleigh quotient $\int_{B_R}|\nabla u|^p\,dx\,/\,(\int_{B_R}|u|^{p^*(s)}V_a\,dx)^{p/p^*(s)}$, where $V_a(x)=|x|^{-s}(1-a(|x|/R)^{(N-p)/(p-1)})^{-\beta}$ is a Hardy weight with an additional boundary singularity when $a=1$. The main claim is that for $0

What carries the argument

The central object is the radial change of variables $r^{-(N-p)/(p-1)}-R^{-(N-p)/(p-1)}=t^{-(N-p)/(p-1)}-T^{-(N-p)/(p-1)}$, which maps radial functions on $B_R$ with the weight $V_a$ to radial functions on $B_T$ with the classical weight $|y|^{-s}$. This transformation shows that $I_{a,\mathrm{rad}}=C_{N,p,s}$ for every $a\in[0,1]$, so the whole question is whether any non-radial competitor can beat the radial constant. For small $a$, rearrangement inequalities force $I_a=I_{a,\mathrm{rad}}$; for $a$ close to $1$, a comparison of Schwarz symmetrizations, $V_a^\#(x)<V_1(x)$, together with Lemma 3 (if $I_a<C_{N,p,s}$, then $I_a$ is attained by a non-radial function) yields the existence of the threshold $a_*>A$ and the symmetry breaking.

What would settle it

Compute, numerically or analytically, the Schwarz symmetrization $V_a^\#$ of $V_a$ on $B_R$ for $0<s<p$ and $a\in(A,A+\varepsilon)$, and check whether the inequality $V_a^\#(x)<V_1(x)$ holds everywhere; alternatively, evaluate $I_a$ directly for $a$ slightly above $A$ and compare with $C_{N,p,s}$—if $I_a=C_{N,p,s}$ for such $a$, then the asserted threshold satisfies $a_*\le A$, contradicting Theorem 1(iii).

Watch

Extended reading notes

Core claim

For the critical exponent $p^*(s)=p(N-s)/(N-p)$ with $0<s<p$, the minimization problem on $B_R$ with potential $V_a$ behaves differently from its radial restriction. A radial transformation identifies $I_{a,\mathrm{rad}}$ with the classical Hardy-Sobolev constant $C_{N,p,s}$, independent of $a$, but the full problem satisfies $I_a<C_{N,p,s}$ for $a$ close to $1$ and $I_a=C_{N,p,s}$ for $a\in[0,a_*]$. Once $I_a<C_{N,p,s}$, concentration-compactness plus a Brezis-Lieb splitting argument forces the infimum to be attained, and the attainment cannot be radial. Hence the Euler-Lagrange equation $-\mathrm{div}(|\nabla u|^{p-2}\nabla u)=bV_a(x)|u|^{p^*(s)-2}u$ on $B_R$ with zero boundary condition has non-radial ground states for a full interval of parameters. The paper also determines the endpoints: for $s=0$, $I_a=C_{N,p,0}(1-a)^{(N-1)p/N}$ and is not attained for $a\in[0,1)$; for $s=p$, $I_a=I_{a,\mathrm{rad}}=C_{N,p,p}$ and is not attained for any $a\in[0,1]$.

Load-bearing premise

The proof of the quantitative lower bound $a_*>A$ assumes that the Schwarz symmetrization of $V_a$ satisfies $V_a^\#(x)<V_1(x)$ on $B_R$ for $a\in[A,A+\varepsilon]$; if this pointwise comparison fails, the stated lower bound is not justified, although the existence of some threshold still follows from continuity.

Editorial extensions

If this is right

For $0<s<p$ and $a\in(a_*,1)$, a bounded domain admits a minimizer at the critical Hardy-Sobolev exponent, whereas the classical $a=0$ case has no minimizer.
The ground states of the associated Euler-Lagrange equation break radial symmetry for $a$ near $1$; this is a concrete non-radial phenomenon driven by a boundary-singular potential.
For $s=0$, the infimum equals $C_{N,p,0}(1-a)^{(N-1)p/N}$ and is never attained for $a\in[0,1)$; for $s=p$, the infimum is the Hardy constant $C_{N,p,p}$ and is never attained.
The radial infimum $I_{a,\mathrm{rad}}$ is independent of $a$ and equals $C_{N,p,s}$, so the radial problem is equivalent to the classical Hardy-Sobolev problem on $\mathbb{R}^N$ through the radial transformation.
As a by-product of the transformation viewpoint, the classical Sobolev inequality in dimension $m$ tends, as $m\to\infty$, to the Hardy inequality in fixed dimension $N$, giving an infinite-dimensional form of the Sobolev inequality.

Reading between the lines

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

Editorial inference: The threshold $a_*$ should depend continuously on $N$, $p$, $s$, and $R$, and could be probed numerically by testing whether $I_a<C_{N,p,s}$ for $a$ just above $A$; such a computation would also test the quantitative lower bound $a_*>A$.
Editorial inference: The mechanism suggests that other potentials with a boundary singularity of similar order, not necessarily of the exact form $V_a$, will also force symmetry breaking once their Schwarz symmetrization dips below the radial comparison function.
Editorial inference: The open question posed in the appendix about non-radial compactness of a complementary embedding, if answered affirmatively, would establish a kind of 'non-radial Strauss compactness,' the opposite of the classical radial compactness of Strauss.
Editorial inference: One natural testable extension is to replace the ball $B_R$ by a general bounded Lipschitz domain, as the author notes in Remark 4; the same strategy should give non-radial minimizers there as well.

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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,

Read Sano's paper on improved Hardy-Sobolev minimizers. The main qualitative result is credible and interesting: for 0

The soft spot is the quantitative part of Theorem 1(iii). The proof that a* > A hinges on the identity V_A(R)=V_1(R_1), where R_1 is the critical radius of V_1. The paper evaluates V_1(R_1) as R^{-s} t0^{-s}(1-t0)^{-β} with t0=s(p-1)/(p(N-1)). But direct substitution gives R_1 = t0^{(p-1)/(N-p)}R, so V_1(R_1)=R^{-s} t0^{-s(p-1)/(N-p)}(1-t0)^{-β}. The exponent on t0 is wrong unless N=2p-1. Consequently the stated formula for A is not the value for which V_A(R)=V_1(R_1). The rearrangement comparison V_a^# < V_1 derived from it is therefore unsupported, and the theorem's claim a* > A is not established as written.

That said, the qualitative existence of some threshold a*<1 and of non-radial minimizers for a close to 1 is robust: it follows from I_1=0, Lemma 2, and Lemma 3, and does not need the precise A. The proof of non-attainability below the threshold also survives with an appropriately redefined a*. So the main phenomenon is likely salvageable, but the manuscript as posted contains a real error in a stated constant.

Other issues: the proof of Theorem 1(i) is omitted entirely ('we omit the proof'), and the rearrangement comparison step in the final paragraph is sketched rather than demonstrated. These are fixable but need to be addressed.

Who is this for? People working on sharp constants, symmetry breaking in elliptic equations, or Hardy-Sobolev inequalities. It deserves a serious referee; the core idea is sound and the error is localized. I would send it to review, with a strong request to correct the constant and fill the gaps.

Best,

Referee Report

2 major / 5 minor

Summary. The paper studies the minimization problem I_a = inf (∫|∇u|^p)/(∫|u|^{p*(s)} V_a(x) dx)^{p/p*(s)} on W_0^{1,p}(B_R), where V_a(x)=|x|^{-s}(1-a(|x|/R)^{(N-p)/(p-1)})^{-β}. For radial functions, Ioku's transformation identifies I_{a,rad} with the classical Hardy-Sobolev constant C_{N,p,s}, independent of a. The paper derives this and related transformations, gives an infinite-dimensional form of the Sobolev inequality, and then analyzes I_a for all functions. The main theorem states: for s=p, I_a is the Hardy constant and is not attained; for s=0, I_a = C_{N,p,0}(1-a)^{(N-1)p/N} and is not attained; for 0<s<p, there exists a*∈(A,1), with an explicit A, such that I_a<I_{a,rad} and I_a is attained by a non-radial function for a∈(a*,1), while I_a=I_{a,rad} and is not attained for a∈[0,a*). The proof uses rearrangement, Ekeland's principle, Boccardo-Murat compactness, and concentration-compactness arguments. The qualitative symmetry-breaking phenomenon for a close to 1 is the central new claim.

Significance. If the main theorem is correct, the paper provides a new and interesting example of symmetry breaking for the Hardy-Sobolev critical exponent on bounded domains, in contrast to the classical case a=0. The proof strategy is sound in outline: the concentration level is identified as I_{a,rad}=C_{N,p,s}, and the strict inequality I_a<I_{a,rad} is used to exclude vanishing and to produce a non-radial minimizer. The transformation viewpoint in Section 2 is a useful contribution, and Proposition 4 gives a concrete comparison of two nonlinear scalings. The paper is careful with standard tools (Brezis-Lieb, Boccardo-Murat, Ekeland) and states several auxiliary results with proofs. However, the quantitative part of Theorem 1(iii), namely the explicit lower bound a*>A, rests on an erroneous algebraic identity and an unproved rearrangement comparison; this part needs correction.

major comments (2)

[§3, proof of Theorem 1(iii), final paragraph] The displayed identity V_1(R_1)=R^{-s}(s(p-1)/(p(N-1)))^{-s}(1-s(p-1)/(p(N-1)))^{-β} uses the wrong exponent. Since V_1(r)=r^{-s}(1-(r/R)^{(N-p)/(p-1)})^{-β}, substituting R_1=(s(p-1)/(p(N-1)))^{(p-1)/(N-p)}R gives V_1(R_1)=R^{-s}(s(p-1)/(p(N-1)))^{-s(p-1)/(N-p)}(1-s(p-1)/(p(N-1)))^{-β}, not the expression with exponent -s in the first factor. Consequently the value of A stated in Theorem 1(iii) is not the one that makes V_A(R)=V_1(R_1); the correct equality gives A=1-(s(p-1)/(p(N-1)))^{s(p-1)/(\beta(N-p))}(1-s(p-1)/(p(N-1))), and the two expressions agree only when N=2p-1. Since the proof of the lower bound a*>A relies on this identity, Theorem 1(iii) as stated is not established. The qualitative existence of some a*<1 and of non-radial minimizers for a close to 1 still follows from I_1=0, Lemma 2, and Lemma 3, so the main phenomenon is likely salvageable, but the quantitative statement and its proof must be corrected.
[§3, proof of Theorem 1(iii), final paragraph] The claim V_a^#(x)<V_1(x) for a∈[A,A+ε] is asserted without a rigorous derivation, and the two monotonicity facts given do not by themselves imply it. From V_A^#(\tilde R)=V_1(R_1), V_A^# decreasing on B_R\setminus B_{\tilde R}, V_1 increasing on B_R\setminus B_{R_1}, and \tilde R<R_1, one only obtains a comparison at different points. At x=\tilde R, one has V_1(\tilde R)<V_1(R_1)=V_A^#(\tilde R), so the desired pointwise inequality can fail on (\tilde R,R_1) unless an additional quantitative rearrangement estimate is supplied. A complete proof of the rearrangement comparison is needed before the bound a*>A can be accepted; this is load-bearing for the stated lower bound, not merely a presentation issue.

minor comments (5)

[§3, proof of Theorem 1(iii)] In the non-attainment argument for a<a*, the author assumes a nonnegative minimizer u without explanation; one should first replace a minimizer by its absolute value, which is possible because |∇|u||=|∇u| almost everywhere and the weight is positive.
[Proposition 3] The condition 'f /nequivalence0' is a rendering error; it should read f\not\equiv 0.
[§4, proof of Proposition 4] In the proof of Proposition 4, the notation B_R appears in the change of variables although the domain is B_1; this should be clarified.
[§3, final paragraph] The definition of \tilde R is introduced only inside the proof of Theorem 1(iii); stating it before the rearrangement argument would improve readability.
[Remark 2] In the displayed definition of u_λ, the condition 't∈[0,\tilde R]' appears to be a typo; the variable in the support condition should be r.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity; the derivation is self-contained.

full rationale

The central claim (Theorem 1(iii)) is derived from standard external results and from Ioku's transformation, which the paper re-derives in Section 2. The threshold a* follows from Lemma 2 (continuity/monotonicity) and Proposition 2 (I_1=0), not from assuming the conclusion. Lemma 3 is proved via Ekeland's principle and external results (Brezis-Lieb, Boccardo-Murat). The quantitative bound a* > A is obtained by an explicit pointwise comparison of Schwarz symmetrizations of the potentials, based on the shape of V_a; the constant A is defined independently by a geometric condition, not fitted to the conclusion. While a reviewer suspects a computational error in the evaluation of V_1(R_1) that determines A, this is a correctness issue, not circularity. Self-citations to [12], [16], [19] provide techniques and analogies but are not load-bearing; the paper contains no fitted parameters, no uniqueness arguments imported from the author's prior work, and no 'prediction' that is equal to an input by construction.

Assumptions & free parameters 0 free parameters · 7 assumptions · 0 invented entities

The central theorem rests on standard sharp inequalities, rearrangement theory, and two variational convergence lemmas, plus one highlighted comparison of rearranged potentials that is not fully proved. No free parameters are fitted and no new entities are postulated.

assumptions (7)

standard math Sharp Hardy-Sobolev inequality on R^N: C_{N,p,s} is the best constant, attained only when T=infinity and 0<=s<p.
Used throughout as the benchmark value for I_{a,rad}; cited to [2], [14] and [8].
standard math The transformation (5) preserves the L^p norm of the gradient and maps the Hardy-Sobolev weight |y|^{-s} to V_a(x) for radial functions (equalities (7),(8)).
Rederived in Section 2 after Ioku [12].
standard math Polya-Szego and Hardy-Littlewood rearrangement inequalities are valid for u in W^{1,p}_0(B_R).
Used to prove I_a=I_{a,rad} for a<=s(p-1)/(p(N-1)).
standard math Brezis-Lieb lemma applies to the weighted measure V_a dx.
The weight is integrable since 0<=s<p<N; used in Lemma 3.
standard math Boccardo-Murat almost everywhere gradient convergence theorem (Lemma 5) applies to the sequence of approximate solutions.
Used to pass to a.e. convergence of gradients in Lemma 3.
ad hoc to paper The V_a^#(x)<V_1(x) comparison for a in [A, A+epsilon] holds.
Asserted without a full derivation; needed to prove a*>A.
standard math Regularity and strong maximum principle for the p-Laplacian equation (3) hold for minimizers.
Used in the non-attainment part for a<a*; cited to [9],[15].

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Cite this review

Pith. "Pith review of Minimization problem associated with an improved Hardy-Sobolev type inequality." pith.science (2026). https://pith.science/paper/SO2IFGTQ

@misc{pith2026190803915,
  author       = {Pith},
  title        = {Pith review of: Minimization problem associated with an improved Hardy-Sobolev type inequality},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/SO2IFGTQ}},
  note         = {Machine review of arXiv:1908.03915}
}

abstract

We consider the existence and the non-existence of a minimizer of the following minimization problems associated with an improved Hardy-Sobolev type inequality introduced by Ioku. $$ I_a := \inf_{u \in W_0^{1,p}(B_R ) \setminus \{ 0\} } \frac{\int_{B_R} |\nabla u |^{p} \,dx}{\left( \int_{B_R} |u|^{p^*(s)} V_a(x) \,dx \right)^{\frac{p}{p^*(s)}}}, \,\,\text{where}\,\, V_a (x) =\frac{1}{|x|^s \left( 1- a \,\left( \frac{|x|}{R} \right)^{\frac{N-p}{p-1}} \right)^\beta} \ge \frac{1}{|x|^s}. $$ Only for radial functions, the minimization problem $I_a$ is equivalent to it associated with the classical Hardy-Sobolev inequality on $\mathbb{R}^N$ via a transformation. First, we summarize various transformations including that transformation and give a viewpoint of such transformations. As an application of this viewpoint, we derive {\it an infinite dimensional form} of the classical Sobolev inequality in some sense. Next, without the transformation, we investigate the minimization problems $I_a$ on balls $B_R$. In contrast to the classical results for $a=0$, we show the existence of non-radial minimizers for the Hardy-Sobolev critical exponent $p^* (s)=\frac{p (N-s)}{N-p}$ on bounded domains. Finally, we give remarks of a different structure between two nonlinear scalings which are equivalent to the usual scaling only for radial functions under some transformations.

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Pith tools

Reviewed August 14, 2026 · model on record in the stance chip above.

[1] [1]

Alvino, A., A limit case of the Sobolev inequality in Lorentz spaces , Rend. Accad. Sci. Fis. Mat. Napoli (4) 44 (1977), 105-112 (1978)

work page 1977

[2] [2]

Diﬀerential Geometry 11 (1976), no

Aubin, T., Probl` emes isop´ erim´ etriques et espaces de Sobolev, J. Diﬀerential Geometry 11 (1976), no. 4, 573-598

work page 1976

[3] [3]

A., The heat equation with a singular potential , Trans

Baras, P ., Goldstein, J. A., The heat equation with a singular potential , Trans. Amer. Math. Soc., 284 (1984), 121-139

work page 1984

[4] [4]

Beckner, W., Pearson, M., On sharp Sobolev embedding and the loga- rithmic Sobolev inequalitiy, Bull

W. Beckner, W., Pearson, M., On sharp Sobolev embedding and the loga- rithmic Sobolev inequalitiy, Bull. London Math. Soc., 30 (1998), 80-84. 22

work page 1998

[5] [5]

Boccardo, L., Murat, F., Almost everywhere convergence of the gradients of solutions to elliptic and parabolic equations , Nonlinear Anal. TMA. 19 (1992), 581-597

work page 1992

[6] [6]

Brezis, E

H. Brezis, E. Lieb, A relation between pointwise convergence of functions and convergence of functionals , Proc. Amer. Math. Soc., 88 (1983), 486- 490

work page 1983

[7] [7]

L.,Blow-up solutions of some nonlinear elliptic prob- lems, Rev

Brezis, H., V´ azquez, J. L.,Blow-up solutions of some nonlinear elliptic prob- lems, Rev. Mat. Univ. Complut. Madrid 10 (1997), No. 2, 443-469

work page 1997

[8] [8]

39 (2013), no

Cassani, D., Ruf, B., Tarsi, C., Optimal Sobolev type inequalities in Lorentz spaces, Potential Anal. 39 (2013), no. 3, 265-285

work page 2013