{"id":"33511d72-bab5-4866-b961-3b7f87bd7ca2","arxiv_id":"1908.03915","paper_version":3,"verdict":"CONDITIONAL","confidence":"MODERATE","novelty_score":6.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":0,"one_line_summary":"For an improved Hardy-Sobolev inequality on a ball, minimizers exist and are non-radial when the boundary-singular weight parameter a is close to 1, while for small a the infimum equals the classical constant and is not attained.","lead":"This paper studies a minimization problem for an improved Hardy-Sobolev inequality on a ball, with a singular weight that can also blow up at the boundary. It proves that for a range of the parameter a, the minimizer exists and is non-radial, in contrast to the classical Hardy-Sobolev case.","discovery_kind":"extension","skeptic_critique":{"model":"deepseek-v4-flash","headline":"The constant A in Theorem 1(iii) is miscalculated: the proof's identity V_A(R)=V_1(R_1) uses the wrong exponent for R_1, so the stated lower bound a* > A is unsupported.","rationale":"The reader's weakest assumption identified the unproved rearrangement comparison at the end of Theorem 1(iii) as the soft spot. My read agrees this is load-bearing, and it is worse than a missing justification: the comparison begins from an identity V_A(R)=V_1(R_1) that is algebraically false for the A stated in the theorem. The derivation of A in the final proof evaluates V_1(R_1) as if R_1=t0 R, but the paper's own definition has R_1=t0^{(p-1)/(N-p)}R. This changes the exponent in A. The numerical example makes the failure unambiguous. The qualitative existence of non-radial minimizers for a close to 1 does not depend on the exact value of A and is supported by I_1=0, Lemma 2, and Lemma 3, so I would not reject the whole phenomenon; I would require a corrected constant and a valid proof of the lower bound before accepting Theorem 1(iii) as stated.","tokens_in":19629,"tokens_out":20229,"duration_ms":213527,"concrete_test":"Take N=5, p=2, s=1, R=1. With the paper's definitions, alpha=3, beta=7/3, t0=1/8, R_1=0.5. Direct substitution gives V_1(R_1)=0.5^{-1}(1-0.5^3)^{-7/3}=2(7/8)^{-7/3}\\approx 2.73, whereas for the stated A=1-(1/8)^{3/7}(7/8)\\approx 0.641, V_A(1)=(1-A)^{-7/3}\\approx 10.9. The identity V_A(R)=V_1(R_1) used to define A therefore fails. Re-run the last paragraph with the corrected identity; the lower bound must use A_correct = 1 - t0^{s(p-1)/(\\beta(N-p))}(1-t0), which for this case is \\approx 0.344. If the claimed a*>A_paper cannot be derived without this identity, Theorem 1(iii)'s quantitative statement must be revised.","verdict_should_be":"CONDITIONAL","load_bearing_attack":"In the final paragraph of the proof of Theorem 1(iii), the author defines R_a := ((s(p-1))/(a p(N-1)))^{(p-1)/(N-p)} R and then writes V_1(R_1) = R^{-s}(s(p-1)/(p(N-1)))^{-s}(1 - s(p-1)/(p(N-1)))^{-\\beta}. This is not the value of V_1 at the displayed R_1. Since V_1(r)=r^{-s}(1-(r/R)^{(N-p)/(p-1)})^{-\\beta}, substituting r=R_1 gives exponent -s(p-1)/(N-p), not -s, in the first factor. Consequently the equality V_A(R)=V_1(R_1), which fixes A, is false in general. Solving the correct equality gives A_correct = 1 - t0^{s(p-1)/(\\beta(N-p))}(1-t0), where t0=s(p-1)/(p(N-1)), whereas the paper uses exponent s/\\beta. The two agree only when N=2p-1. For N=5, p=2, s=1, the stated A \\approx 0.641 gives V_A(R) \\approx 10.9 R^{-1} while V_1(R_1) \\approx 2.73 R^{-1}, a factor of four. Since the rearrangement comparison and the lower bound a*>A rest on this identity, Theorem 1(iii) as stated is not established. The qualitative existence of some threshold a*<1 and of non-radial minimizers for a close to 1 follows from I_1=0, Lemma 2, and Lemma 3, so the main symmetry-breaking phenomenon is probably salvageable once A is corrected.","agreement_with_reader":"partial"},"referee_report":{"model":"deepseek-v4-flash","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.","tokens_in":19989,"tokens_out":12373,"duration_ms":124373,"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":[{"comment":"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.","section":"§3, proof of Theorem 1(iii), final paragraph"},{"comment":"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.","section":"§3, proof of Theorem 1(iii), final paragraph"}],"minor_comments":[{"comment":"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.","section":"§3, proof of Theorem 1(iii)"},{"comment":"The condition 'f /nequivalence0' is a rendering error; it should read f\\not\\equiv 0.","section":"Proposition 3"},{"comment":"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.","section":"§4, proof of Proposition 4"},{"comment":"The definition of \\tilde R is introduced only inside the proof of Theorem 1(iii); stating it before the rearrangement argument would improve readability.","section":"§3, final paragraph"},{"comment":"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.","section":"Remark 2"}],"recommendation":"major_revision","confidential_remarks":null},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"Colleague,\n\nRead Sano's paper on improved Hardy-Sobolev minimizers. The main qualitative result is credible and interesting: for 0<s<p, the improved potential V_a on a ball produces non-radial minimizers for a close to 1, contrasting with the classical a=0 non-existence. The proof via continuity of I_a, the fact I_1=0, and the concentration lemma is coherent, and the non-attainability argument below the threshold is standard. Section 2's transformation viewpoint is a nice summary, and the infinite-dimensional form of Sobolev is a pleasing observation, though peripheral.\n\nThe 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.\n\nThat 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.\n\nOther 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.\n\nWho 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.\n\nBest,","headline":"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.","tokens_in":20541,"tokens_out":3503,"would_cite":true,"duration_ms":32552,"reading_group":"maybe","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["35A23","35J20","35A08"],"pacs":[],"model":"deepseek-v4-flash","headline":"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 $a$.","keywords":["Hardy-Sobolev inequality","minimization problem","non-radial minimizer","symmetry breaking","critical exponent","Schwarz symmetrization","improved Hardy-Sobolev inequality","infinite-dimensional Sobolev inequality"],"falsifier":"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).","tokens_in":19371,"feed_emoji":"🎯","tokens_out":5412,"duration_ms":49569,"temperature":0.7,"pith_summary":"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 $0<s<p$ there is a threshold $a_*\\in(A,1)$ such that for $a\\in(a_*,1)$ the infimum is strictly below the radial level $C_{N,p,s}$ and is attained by a non-radial function, while for $a\\in[0,a_*)$ no minimizer exists. This matters because it shows that a bounded-domain Hardy-Sobolev problem at the critical exponent can have ground states once the potential develops a boundary singularity, and those ground states break radial symmetry. For $s=0$ the infimum is never attained, and for $s=p$ the infimum equals the Hardy constant and is never attained.","feed_headline":"Near-boundary potential makes Hardy-Sobolev minimizers non-radial","feed_subtitle":"For 0<s<p and a close to 1 the infimum drops below the radial constant, so a ball admits a symmetry-breaking ground state.","key_machinery":"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.","core_discovery":"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 $a\\in[0,1]$.","pith_inferences":["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."],"forward_implications":["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."],"supporting_citations":[{"why":"Introduces the improved Hardy-Sobolev inequality for radial functions and the transformation (5) that identifies $I_{a,\\mathrm{rad}}$ with the classical constant.","marker":"[12]"},{"why":"Provides the explicit family of minimizers $W_\\lambda$ and the value of the classical Hardy-Sobolev constant used throughout.","marker":"[8]"},{"why":"Supplies the almost-everywhere convergence of gradients for solutions of elliptic equations, used to apply the Brezis-Lieb lemma in the concentration argument.","marker":"[5]"},{"why":"Gives the Brezis-Lieb relation between pointwise convergence and convergence of functionals, the key tool for splitting the $L^p$ norm in Lemma 3.","marker":"[6]"},{"why":"Shows non-radial ground states for the Hénon equation; its argument is adapted to prove the $s=0$ case and to frame the symmetry-breaking phenomenon.","marker":"[20]"},{"why":"The author's earlier analogue for a generalized critical Hardy inequality, which provides the template for proving existence of a non-radial minimizer and for the non-compact sequence construction.","marker":"[16]"}],"fun_headline_variants":["Symmetry breaking in Hardy-Sobolev minimizers","Non-radial minimizers for improved Hardy-Sobolev inequality","Potential near boundary forces non-radial ground states","Hardy-Sobolev infimum drops below radial constant","Bounded domain admits non-radial Hardy-Sobolev minimizers"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"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.","fun_headline_variants_meta":{"raw":{"variants":["Symmetry breaking in Hardy-Sobolev minimizers","Non-radial minimizers for improved Hardy-Sobolev inequality","Potential near boundary forces non-radial ground states","Hardy-Sobolev infimum drops below radial constant","Bounded domain admits non-radial Hardy-Sobolev minimizers"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000322,"raw_usage":{"total_tokens":1937,"prompt_tokens":1196,"completion_tokens":741,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":812,"completion_tokens_details":{"reasoning_tokens":656}},"tokens_in":812,"tokens_out":741,"duration_ms":7580,"temperature":1.0,"reasoning_tokens":656,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-14T13:59:32.286103+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"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).","supporting_citations":[{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Introduces the improved Hardy-Sobolev inequality for radial functions and the transformation (5) that identifies $I_{a,\\mathrm{rad}}$ with the classical constant."},{"cited_title":"39 (2013), no","cited_arxiv_id":null,"evidence_quote":"Provides the explicit family of minimizers $W_\\lambda$ and the value of the classical Hardy-Sobolev constant used throughout."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Supplies the almost-everywhere convergence of gradients for solutions of elliptic equations, used to apply the Brezis-Lieb lemma in the concentration argument."},{"cited_title":"Brezis, E","cited_arxiv_id":null,"evidence_quote":"Gives the Brezis-Lieb relation between pointwise convergence and convergence of functionals, the key tool for splitting the $L^p$ norm in Lemma 3."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Shows non-radial ground states for the Hénon equation; its argument is adapted to prove the $s=0$ case and to frame the symmetry-breaking phenomenon."},{"cited_title":"Differential Equations 267 (2019), no","cited_arxiv_id":null,"evidence_quote":"The author's earlier analogue for a generalized critical Hardy inequality, which provides the template for proving existence of a non-radial minimizer and for the non-compact sequence construction."}],"review_version":1}