{"id":"200e26a4-7701-4bd7-9977-feac1c6ee1ee","arxiv_id":"2412.09033","paper_version":1,"verdict":"REJECT","confidence":"MODERATE","novelty_score":7.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":0,"one_line_summary":"Every positive finite energy cylindrically symmetric solution of the singular p-Laplace equation is shown to be a scaled translate of one explicit function when 3 <= k <= n-1.","lead":"This paper classifies all cylindrically symmetric positive solutions of a critical p-Laplace equation with a Hardy weight, for k between 3 and n-1. It yields the sharp constant and explicit extremal functions for a family of Hardy-Sobolev-Maz'ya inequalities, answering a 2006 conjecture in this range.","discovery_kind":"extension","skeptic_critique":{"model":"deepseek-v4-flash","headline":"The cutoff error estimate (3.32) has exponent μ=0 for k≤2p, so the ε→0 limit at (3.34)–(3.35) does not close; the proof needs a larger q in Lemma 2.5.","rationale":"The reader identified the same weakest assumption, and my independent reading agrees. The central claim is the classification in Theorem 1.2, and the only place where the proof's estimates must close is the ε→0 limit. The exponent computation is unambiguous: μ=0, so the error term (3.32) is only O(C(R)) and cannot be discarded. Since the monotone-convergence argument cannot replace the missing o(1) bound, the submitted proof is incomplete. I do not see a second, independent fatal objection; the identity and ODE parts are internally consistent, and the best-constant conclusion would follow once the estimate is fixed. The issue is a genuine gap in the written proof rather than a disagreement with external consensus, and it justifies retaining the reader's reject verdict.","tokens_in":17030,"tokens_out":9289,"duration_ms":91654,"concrete_test":"Independently compute μ from (3.31) for q=2(p−1)k/(k−2) and for q_δ=2(p−1)k/(k−2)+δ, using the same Hölder step (3.28)–(3.30). The first computation yields μ=0, confirming the gap; if the second computation yields μ_δ>0 and Lemma 2.5 still supplies (3.29), then the proof is repairable by a parameter change.","verdict_should_be":"UNCHANGED","load_bearing_attack":"With q=2(p−1)k/(k−2) in the k≤2p branch, direct substitution into (3.31) gives q−2p+2=4(p−1)/(k−2), hence μ = k·[4(p−1)/(k−2)] / [2(p−1)k/(k−2)] − 2 = 0. The paper's displayed '0.1' is a typo in the algebra. Consequently the right side of (3.30) is C(R), not o(1), and (3.32) does not tend to 0 as ε→0. In (3.34) one therefore cannot pass from ∫ φ^lζ^m divY ≤ C R^{−τ}+C(R)ε^μ to the ε-independent bound (3.35); the conclusion divY=0 at (3.37) rests on exactly this passage. The gap appears repairable: Lemma 2.5 is stated for every q>1, and taking q_δ=2(p−1)k/(k−2)+δ gives μ_δ=(k−2)−2k(p−1)/q_δ>0, with (3.27) and (3.29) unchanged. But as written, the k≤2p case is not proved. The differential identity (3.11), the boundary vanishing (3.17), and the final ODE step are not implicated in this objection.","agreement_with_reader":"agree"},"referee_report":{"model":"deepseek-v4-flash","summary":"The paper studies the Hardy-Sobolev-Maz'ya inequality (1.1) and its Euler-Lagrange equation (1.2) for the p-Laplace operator. The authors derive a differential identity for a transformed function φ, combine it with a priori C0 and gradient estimates, and aim to classify all positive finite-energy cylindrically symmetric solutions of (1.2) for 3 ≤ k ≤ n−1. From this classification and the symmetrization result of Secchi-Smets-Willem, they claim the best constant and all extremal functions for the inequality. The main theorem is stated as Theorem 1.2, and the proof occupies Section 3.","tokens_in":17315,"tokens_out":8844,"duration_ms":81022,"significance":"If the classification result were fully established, it would confirm the Alvino-Ferone-Trombetti conjecture for k ≥ 3 and extend the p = 2 result of Mancini-Fabbri-Sandeep to the p-Laplace setting. The differential identity in Proposition 3.1 is a nontrivial and interesting contribution, and the final ODE analysis is clean. However, the proof of Theorem 1.2 currently fails as written in the case p ≥ k/2 because a key error term is claimed to decay as ε→0 when, with the chosen parameter q, it does not. Since the gap is local and apparently repairable by a small modification of q, the paper is not without merit.","major_comments":[{"comment":"For the case k ≤ 2p, the authors choose q = 2(p−1)k/(k−2) and compute μ in (3.31), concluding μ > 0. Direct substitution gives q − 2p + 2 = 4(p−1)/(k−2), so k(q−2p+2)/q = 2 and therefore μ = 0, not a positive number. The displayed computation in the manuscript, containing expressions such as '2.1 − 2 = 0.1', is algebraically incorrect. Consequently the estimate (3.32) is C(R)ε^0, which does not tend to zero as ε→0. The passage from (3.34) to (3.35), where the ε→0 limit is taken to obtain the ε-independent bound, is therefore unjustified. Since the conclusion divY = 0 at (3.37) rests on this passage, Theorem 1.2 is not proved for p ≥ k/2 as written. This gap appears repairable: Lemma 2.5 is stated for every q > 1, and choosing q_δ = 2(p−1)k/(k−2) + δ gives μ_δ = k(q_δ − 2p + 2)/q_δ − 2 > 0, while (3.27) and (3.29) remain unchanged. The differential identity (3.11), the boundary vanishing (3.17), and the final ODE step are not implicated in this objection.","section":"Section 3, Eqs. (3.30)-(3.32)"},{"comment":"The theorem statement includes a free translation parameter z0 ∈ R^{n−k} in the conclusion u(y,z) = λ^{(n−p)/p} u0(λy, λz + z0), but the proof at the end concludes u(y,z) = λ^{(n−p)/p} u0(λy, λz) with no z0. If 'cylindrically symmetric' is understood with the symmetry axis fixed at the origin, the statement should say so explicitly; otherwise the proof should begin by translating the symmetry axis to the origin and then state the final conclusion with z0 restored. This is not load-bearing for the main argument but should be clarified.","section":"Theorem 1.2 statement vs. proof"}],"minor_comments":[{"comment":"The computation of μ contains corrupted decimal-point notation ('2.1', '4.2', '0.1') that makes the displayed algebra impossible to follow; this should be corrected to the clean fractions and the true value μ = 0 should be acknowledged.","section":"Section 3, Eq. (3.31)"},{"comment":"In the proof of Lemma 2.5, the word 'Anagin' should read 'Again'. Also, the iteration step leading to (2.23) is terse; a few words explaining how α_{i+1} is chosen from the two integrability conclusions would improve readability.","section":"Lemma 2.5 proof"},{"comment":"Reference [22] is cited in the introduction for the symmetrization result, but the statement of Proposition 2.1 as '[22]' could benefit from a page or theorem number to help the reader verify the exact claim.","section":"References"}],"recommendation":"major_revision","confidential_remarks":"The paper contains a serious but local gap in the proof of Theorem 1.2: the error term at (3.32) does not decay for the chosen q in the k ≤ 2p case. The fix appears straightforward — taking q slightly larger than 2(p−1)k/(k−2) restores μ > 0 without changing the rest of the argument. I therefore do not recommend rejection outright, but the current version cannot be accepted. The authors should also clean up the corrupted algebra in (3.31) and align the statement of Theorem 1.2 with the proof regarding the translation parameter z0."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"Here is my take. The paper does something real: it pushes the p=2 classification of Mancini–Fabbri–Sandeep to general p in 1<p<n, for cylindrically symmetric solutions with k≥3, and it comes with a new integral identity for the p-Laplace operator. The identity is derived carefully, the boundary vanishing argument is clean, and once you reach divY=0 the final ODE step is straightforward and gives exactly the expected profile. The paper also states plainly that k=2 remains open, which is honest.\n\nThe soft spot is exactly where the stress-test note lands. In the k≤2p branch the paper chooses q=2(p−1)k/(k−2) in Lemma 2.5. Substituting that q into (3.31) gives μ=0, not the claimed 0.1. The displayed algebra in (3.30) is wrong: the fraction is exactly 2, so μ=0. That is load-bearing, because (3.32) then does not decay as ε→0, and the passage from (3.34) to (3.35) cannot be made. Without that passage you cannot conclude divY=0 at (3.37). The gap appears very repairable—take q to be q0+δ with δ>0, then μδ=(k−2)δ/(q0+δ)>0, and Lemma 2.5 is stated for every q>1—but as written the proof of Theorem 1.2 does not close in the k≤2p case. The differential identity, the boundary term, and the final ODE classification are not implicated in this objection.\n\nOther concerns are minor by comparison. Some limiting and scaling steps in Lemmas 2.4 and 2.5 are a bit quick, and the estimates in (3.20)–(3.23) would benefit from more detail, but nothing else looks as sharp as the μ=0 issue. The citation pattern is honest, the credited results are used appropriately, and there is no circularity.\n\nWho this is for: people working on sharp functional inequalities, Hardy–Sobolev–Maz'ya constants, and classification of positive solutions to quasilinear elliptic equations. A serious referee should see it because the method is promising and the fix looks straightforward. My own verdict is not “accept as is”—the central claim needs a repaired cutoff argument—but this is a desk-reject only if the editors want to avoid the extra round. I would send it to peer review and ask the authors to fix the q choice and re-check the algebra in (3.30). I would not cite it in my own work until that repair is on the page.","headline":"A genuinely new classification and identity, but the k≤2p branch of the proof has a μ=0 cutoff error that must be fixed before the main theorem is established.","tokens_in":17847,"tokens_out":4475,"would_cite":false,"duration_ms":47469,"reading_group":"maybe","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["35J92","35B33","35B06"],"pacs":[],"model":"deepseek-v4-flash","headline":"This paper proves that every positive finite-energy cylindrically symmetric solution of the critical p-Laplace Hardy-Sobolev-Maz'ya equation is an explicit rescaling and translation of one profile, which yields the best constant and…","keywords":["Hardy-Sobolev-Maz'ya inequality","p-Laplace equation","best constant","extremal functions","cylindrical symmetry","differential identity","classification of solutions","critical weighted Sobolev inequality"],"falsifier":"Evaluate $\\mu$ in (3.31) for $k\\le 2p$ with $q=2(p-1)k/(k-2)$: substituting gives $\\mu=0$, so the error term (3.32) is of order one as $\\varepsilon\\to 0$; that direct computation would falsify the claim that this term vanishes in that parameter range.","tokens_in":16827,"feed_emoji":"📐","tokens_out":9412,"duration_ms":86960,"temperature":0.7,"pith_summary":"This paper aims to prove that the positive extremals of a class of Hardy-Sobolev-Maz'ya inequalities have a completely explicit shape. For $s=1$, $n\\ge 4$, and $3\\le k\\le n-1$, the authors show that every positive finite-energy cylindrically symmetric solution of the associated $p$-Laplace equation must be a dilation and a $z$-translation of the single profile $u_0$ in (1.3). Combined with the cylindrically symmetric minimization result of Secchi-Smets-Willem, the classification gives the best constant and the extremal functions for inequality (1.1). If the proof is correct, the paper settles the Alvino-Ferone-Trombetti conjecture for $k\\ge 3$ and leaves the two-dimensional singular set case $k=2$ open.","feed_headline":"Explicit form found for all symmetric Hardy-Sobolev-Maz'ya extremals","feed_subtitle":"New identity classifies the positive solutions for k≥3, settling the conjecture and giving the best constant.","key_machinery":"The carrying mechanism is the differential identity in Proposition 3.1, a $p$-Laplace generalization of the Garofalo-Vassilev identity. After changing variables $u=\\kappa\\,\\varphi^{-(n-p)/p}$, the identity writes $\\mathrm{div}\\,Y$ as the sum of three nonnegative squared terms: a traceless second-derivative term, a deviation term for $\\Delta_p\\varphi-\\frac{2(p-1)}{p}|\\nabla\\varphi|^p/\\varphi$, and a mixed radial-tangential term. Integration over cylindrical shells with cutoffs, together with the $C^0$ decay estimate and the gradient estimates in Lemmas 2.4 and 2.5, shows the total integral is smaller than any positive constant. Hence $\\mathrm{div}\\,Y=0$, and the nonnegativity of the three terms forces each to vanish, yielding the two relations (3.38) and (3.39) that integrate to the explicit profile.","core_discovery":"The central claim is Theorem 1.2: for $s=1$, $n\\ge 4$, and $3\\le k\\le n-1$, any cylindrically symmetric solution $u(y,z)$ of (1.2) has the form $u(y,z)=\\lambda^{(n-p)/p}u_0(\\lambda y,\\lambda z+z_0)$, where $u_0(x)=C_{p,n,k}[(1+|y|)^2+|z|^2]^{-(n-p)/(2(p-1))}$. The proof establishes an integral identity with a nonnegative right-hand side, integrates it against carefully chosen cutoffs, and uses the decay and gradient estimates to force the right-hand side to vanish. That forces the solution to satisfy two differential equations that integrate directly to the explicit profile. As a consequence, the best constant in (1.1) for $s=1$ and $k\\ge 3$ is attained by these explicit functions, resolving the conjecture in [1] for that range.","pith_inferences":["A reader checking (3.31) for $k\\le 2p$ with the stated $q=2(p-1)k/(k-2)$ finds $\\mu=0$, not $\\mu>0$; the printed computation of $\\mu$ in the $k\\le 2p$ case does not close the limit $\\varepsilon\\to 0$ unless another choice of $q$ or an additional estimate is supplied.","If the classification can be repaired for $p\\ge k/2$, the same identity method would yield the sharp constant by evaluating the Rayleigh quotient at $u_0$, giving a closed-form expression for $S_{p,1}$.","The differential-identity strategy is tied to the cylindrical symmetry and to the weight $|y|^{-1}$; extending it to other weights $s$ or to the case $k=2$ would require new pointwise estimates rather than only integral ones."],"forward_implications":["The best constant $S_{p,1}$ in (1.1) for $s=1$ and $k\\ge 3$ is attained by the explicit profile $u_0$ under dilations and $z$-translations, making the extremal value effectively explicit.","The Alvino-Ferone-Trombetti conjecture is true for $k\\ge 3$: the candidate $u_0$ is the only extremal profile up to the stated transformations.","Every positive finite-energy cylindrically symmetric solution of (1.2) is the explicit one, so the classification within this symmetry class is complete.","The case $k=2$ remains open because the a priori estimates used in the proof are not available for that singular set."],"supporting_citations":[{"why":"supplies the cylindrically symmetric minimization result that lets the classification of symmetric solutions imply extremals for (1.1).","marker":"[22]"},{"why":"supplies the sharp $C^0$ decay estimate used throughout the integral estimates.","marker":"[15]"},{"why":"provides the gradient estimate used to prove Lemma 2.4 for $k>2p$.","marker":"[27]"},{"why":"provides the interior regularity and gradient estimates used in Lemma 2.5 for $p\\ge k/2$.","marker":"[2]"},{"why":"supplies the differential identity that the paper generalizes to the $p$-Laplace setting.","marker":"[16]"},{"why":"supplies the $p=2$ classification and the integration-by-parts strategy that the proof of Theorem 1.2 follows.","marker":"[19]"},{"why":"states the Hardy-Sobolev-Maz'ya inequality and its Euler-Lagrange equation (1.2).","marker":"[4]"},{"why":"states the conjecture that the paper confirms for $k\\ge 3$.","marker":"[1]"}],"fun_headline_variants":["Explicit extremals for Hardy-Sobolev-Maz'ya found","Symmetric extremals classified, best constant attained","Integral identity unlocks HSM extremal profile","All symmetric solutions explicit for k≥3 range","Hardy-Sobolev-Maz'ya extremals: form fully determined"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"The whole argument rests on one limiting step: the inner-cutoff error term must shrink to zero as $\\varepsilon\\to 0$, which requires the exponent $\\mu$ in (3.31) to be positive in every parameter range used.","fun_headline_variants_meta":{"raw":{"variants":["Explicit extremals for Hardy-Sobolev-Maz'ya found","Symmetric extremals classified, best constant attained","Integral identity unlocks HSM extremal profile","All symmetric solutions explicit for k≥3 range","Hardy-Sobolev-Maz'ya extremals: form fully determined"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000139,"raw_usage":{"total_tokens":1103,"prompt_tokens":840,"completion_tokens":263,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":456,"completion_tokens_details":{"reasoning_tokens":179}},"tokens_in":456,"tokens_out":263,"duration_ms":3641,"temperature":1.0,"reasoning_tokens":179,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-11T17:23:56.853480+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"Evaluate $\\mu$ in (3.31) for $k\\le 2p$ with $q=2(p-1)k/(k-2)$: substituting gives $\\mu=0$, so the error term (3.32) is of order one as $\\varepsilon\\to 0$; that direct computation would falsify the claim that this term vanishes in that parameter range.","supporting_citations":[{"cited_title":"Secchi, D","cited_arxiv_id":null,"evidence_quote":"supplies the cylindrically symmetric minimization result that lets the classification of symmetric solutions imply extremals for (1.1)."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"supplies the sharp $C^0$ decay estimate used throughout the integral estimates."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"provides the gradient estimate used to prove Lemma 2.4 for $k>2p$."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"provides the interior regularity and gradient estimates used in Lemma 2.5 for $p\\ge k/2$."},{"cited_title":"Garofalo, D","cited_arxiv_id":null,"evidence_quote":"supplies the differential identity that the paper generalizes to the $p$-Laplace setting."},{"cited_title":"Mancini, I","cited_arxiv_id":null,"evidence_quote":"supplies the $p=2$ classification and the integration-by-parts strategy that the proof of Theorem 1.2 follows."},{"cited_title":"Badiale, G","cited_arxiv_id":null,"evidence_quote":"states the Hardy-Sobolev-Maz'ya inequality and its Euler-Lagrange equation (1.2)."},{"cited_title":"Alvino, V","cited_arxiv_id":null,"evidence_quote":"states the conjecture that the paper confirms for $k\\ge 3$."}],"review_version":1}