{"id":"75d1bd29-ba3a-40ac-9337-5b93c6026042","arxiv_id":"2507.04470","paper_version":1,"verdict":"CONDITIONAL","confidence":"MODERATE","novelty_score":5.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":0,"one_line_summary":"For Hardy-Sobolev inequalities in cones, a sufficiently small first Neumann eigenvalue of the spherical cross-section forces the extremal function to be non-radial.","lead":"This paper studies the best constant in Hardy-Sobolev inequalities on cone-shaped domains and proves that under a spectral condition on the cone's cross-section, the best function is not the obvious radially symmetric one. The result gives a clean criterion for symmetry breaking and yields multiple solutions for the associated Neumann problem.","discovery_kind":"extension","skeptic_critique":{"model":"deepseek-v4-flash","headline":"The symmetry-breaking step never verifies that the test direction h is an admissible tangent variation on the Nehari manifold; Proposition 2 only solves an Euler equation pointwise, so the conclusion that a minimizer is non-radial is not yet supported.","rationale":"The reader's verdict is CONDITIONAL, and my analysis supports that assessment. The most load-bearing unverified premise is that the proposed variation h is actually admissible for the second-variation argument. The paper's Proposition 2 is an explicit pointwise computation of an Euler-type equation for f, and it appears algebraically correct, but admissibility in D_p(ΣD) and tangency to the Nehari manifold are not consequences of that computation alone. The second variation formula is only meaningful for h in the energy space, and the integration by parts over the infinite cone requires decay at 0 and infinity that is not demonstrated. Without these checks, the negative sign of D^2J does not imply that U is not a local minimizer on N. I also note the paper's own statement that the proofs of Proposition 1 and Theorem 1 are omitted; for the subcritical branch this is a real missing support, though not an evident contradiction. These are fillable gaps rather than demonstrated errors, so the appropriate verdict remains CONDITIONAL rather than REJECT. I agree with the reader's identification of the admissibility of h as a fragile premise, and I add the tangency condition as part of the same concern.","tokens_in":9627,"tokens_out":31915,"duration_ms":326582,"concrete_test":"For the Talenti profile U in (2), set α=(1−σ)q/p and f(r)=r^α U'(r). Compute explicitly whether the three radial integrals I1=∫_0^∞ |(r^α U')'|^p r^{n−1}dr, I2=∫_0^∞ |r^{α−1}U'|^p r^{n−1}dr, and I3=∫_0^∞ r^{(σ−1)q}|r^α U'|^q r^{n−1}dr are finite, and check that the boundary terms [r^{n−1}|U'|^{p−2}(r^α U')' r^α U'] vanish at 0 and ∞. If they do, then h∈D_p(ΣD); combined with ∫_D g dS=0 this gives DJ(U;h)=0, so h is tangent to N and the second-variation computation is legitimate. If not, Theorem 3's non-radiality conclusion fails.","verdict_should_be":"UNCHANGED","load_bearing_attack":"The central non-radiality argument in Section 3 uses h = f(r)g(x/|x|), f = r^α U', as a second negative direction for the second variation of J on the Nehari manifold N. To draw the conclusion that U is not a minimizer, h must belong to D_p(ΣD) and must be tangent to N, i.e. DJ(U;h) = 0. Proposition 2 proves only that f satisfies the pointwise Euler equation (9); it does not establish the required integrability of |∇h|^p and of r^{(σ−1)q}|h|^q over the unbounded cone, nor the vanishing of the boundary terms at r=0 and r=∞ that justify the integration by parts leading to the sign D^2J(U;h,h) = (Λ*+λ1)∫|U'|^{p−2}f^2/r^2. Tangency is also not shown, although it follows readily from ∫_D g dS=0; without an explicit verification, the second negative direction might be transverse to N, in which case the constrained second-variation criterion does not apply. The authors also explicitly omit the proofs of Proposition 1 and Theorem 1, saying the Neumann adaptation is straightforward; this compounds the gap, since Theorem 1 is needed for the subcritical branch. None of these omissions is independently fatal, but together they leave the central claim conditional: the algebra of Proposition 2 is consistent, but the variational conclusion requires additional admissibility and boundary checks that the paper does not supply.","agreement_with_reader":"agree"},"referee_report":{"model":"deepseek-v4-flash","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.","tokens_in":9953,"tokens_out":23362,"duration_ms":277557,"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":[{"comment":"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":"Section 2.1, Theorem 1 and Proposition 1"},{"comment":"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":"Section 3, admissibility of h"},{"comment":"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":"Section 3, tangency to the Nehari manifold"},{"comment":"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.","section":"Section 2.2, proof of Theorem 2"}],"minor_comments":[{"comment":"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":"Section 1"},{"comment":"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":"Section 2.1"},{"comment":"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.","section":"Section 3"},{"comment":"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.","section":"Throughout"}],"recommendation":"major_revision","confidential_remarks":"The paper is within the scope of the journal. The explicit computation in Proposition 2 is valuable and suggests the main result is true. However, the current manuscript leaves two load-bearing pillars unproved: attainability in the Neumann setting and the admissibility/tangency of the test direction. Both appear fixable, but the authors should be asked to supply full arguments or very precise reductions, rather than the current sketches. I do not see a circularity problem: the second-variation computation is explicit and parameter-free."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","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)<n−1 for σ=1, with α as defined. The σ=1 condition being independent of p is new for general p, and the authors note it is sharp in view of the convex cone results. Proposition 2, the computational core, is correct: the radial derivative f=r^α U' solves the pointwise equation with Λ*, and the sign argument works.\n\nSoft 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.\n\nI 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.","headline":"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.","tokens_in":10506,"tokens_out":7743,"would_cite":true,"duration_ms":80670,"reading_group":"maybe","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["35B06","35B33","35J92","46E35"],"pacs":[],"model":"deepseek-v4-flash","headline":"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…","keywords":["Hardy–Sobolev inequality","symmetry breaking","Neumann problem","p-Laplacian","Talenti–Bliss extremal","Beltrami–Laplace eigenvalue","non-convex cones","second variation"],"falsifier":"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.","tokens_in":9426,"feed_emoji":"🔺","tokens_out":21617,"duration_ms":195482,"temperature":0.7,"pith_summary":"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.","feed_headline":"Small Neumann eigenvalue forces non-radial Hardy–Sobolev minimizers","feed_subtitle":"Below a sharp eigenvalue threshold, the radial Talenti–Bliss extremal loses to a non-radial one.","key_machinery":"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).","core_discovery":"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.","pith_inferences":["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."],"forward_implications":["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."],"supporting_citations":[{"why":"This is one of the two independent proofs of the sharp Sobolev constant whose radial extremal is the trial function under test.","marker":"[1]"},{"why":"This is the earlier symmetry-breaking result for the Laplacian case that the present theorem extends, and its eigenfunction-direction idea is reused.","marker":"[4]"},{"why":"This supplies the estimates behind Lemma 1 that yield the strict comparison of best constants needed for attainability.","marker":"[6]"},{"why":"This establishes the exact Hardy–Sobolev constant and its radial extremals in the subcritical case.","marker":"[9]"},{"why":"This gives the concentration–compactness statement used in the Appendix and the convex-cone result showing the critical threshold is sharp.","marker":"[13]"},{"why":"This provides the second-variation formulas for the p-Laplacian energy used to compute the instability.","marker":"[14]"},{"why":"This gives the Dirichlet-cone attainability proof that is adapted, with a note that the Neumann case is straightforward, to prove Theorem 1.","marker":"[15]"},{"why":"This is the second independent proof of the sharp Sobolev constant, and its explicit extremal is the function whose radial symmetry is broken.","marker":"[16]"}],"fun_headline_variants":["Low Neumann eigenvalue breaks radial symmetry in Hardy–Sobolev","Eigenvalue threshold forces non-radial extremals in cones","Radial minimizer fails below critical Neumann eigenvalue","Symmetry breaking in Hardy–Sobolev: eigenvalue condition rules","Sharp eigenvalue condition yields non-radial minimizers"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"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.","fun_headline_variants_meta":{"raw":{"variants":["Low Neumann eigenvalue breaks radial symmetry in Hardy–Sobolev","Eigenvalue threshold forces non-radial extremals in cones","Radial minimizer fails below critical Neumann eigenvalue","Symmetry breaking in Hardy–Sobolev: eigenvalue condition rules","Sharp eigenvalue condition yields non-radial minimizers"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000198,"raw_usage":{"total_tokens":1384,"prompt_tokens":975,"completion_tokens":409,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":591,"completion_tokens_details":{"reasoning_tokens":326}},"tokens_in":591,"tokens_out":409,"duration_ms":5175,"temperature":1.0,"reasoning_tokens":326,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-06T19:47:21.532875+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"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.","supporting_citations":[{"cited_title":"Journal of Differ- ential Geometry 11 (1976), N4, pp.573-598","cited_arxiv_id":null,"evidence_quote":"This is one of the two independent proofs of the sharp Sobolev constant whose radial extremal is the trial function under test."},{"cited_title":"and Polvara, C","cited_arxiv_id":null,"evidence_quote":"This is the earlier symmetry-breaking result for the Laplacian case that the present theorem extends, and its eigenfunction-direction idea is reused."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"This supplies the estimates behind Lemma 1 that yield the strict comparison of best constants needed for attainability."},{"cited_title":"and Yuan, C., Multiple solutions for quasi-linear PDEs involv- ing the critical Sobolev and Hardy exponents","cited_arxiv_id":null,"evidence_quote":"This establishes the exact Hardy–Sobolev constant and its radial extremals in the subcritical case."},{"cited_title":"L., Pacella, F","cited_arxiv_id":null,"evidence_quote":"This gives the concentration–compactness statement used in the Appendix and the convex-cone result showing the critical threshold is sharp."},{"cited_title":"I., On solutions of the Dirichlet problem for an equation involving the p-Laplacian in a spherical layer","cited_arxiv_id":null,"evidence_quote":"This provides the second-variation formulas for the p-Laplacian energy used to compute the instability."},{"cited_title":"I., Hardy-Sobolev inequalities in a cone","cited_arxiv_id":null,"evidence_quote":"This gives the Dirichlet-cone attainability proof that is adapted, with a note that the Neumann case is straightforward, to prove Theorem 1."},{"cited_title":"Annali di Matematica pura ed Applicata 110 (1976), pp.353-372","cited_arxiv_id":null,"evidence_quote":"This is the second independent proof of the sharp Sobolev constant, and its explicit extremal is the function whose radial symmetry is broken."}],"review_version":1}