{"id":"d3035186-0428-4de2-bb91-99aab1c56d38","arxiv_id":"1908.08091","paper_version":1,"verdict":"CONDITIONAL","confidence":"MODERATE","novelty_score":6.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":0,"one_line_summary":"For isoparametric functions on spheres, the authors construct infinite nodal solutions to semilinear Yamabe-type equations for subcritical, critical, and supercritical exponents, with blow-up on focal submanifolds.","lead":"This paper proves that a family of nonlinear elliptic equations on spheres, including supercritical cases, has infinitely many sign-changing solutions whose level sets are prescribed symmetric hypersurfaces. It also transfers these results to the Yamabe problem on complex and quaternionic projective spaces.","discovery_kind":"extension","skeptic_critique":{"model":"deepseek-v4-flash","headline":"Theorem 1.1's p-range is inconsistent with the proof: under the proof's m_-≤m_+ convention the restrictive exponent is set by the smaller focal dimension, but the theorem orders n_-≤n_+ and uses n_+ in the p-range, overreaching for families like S^1×S^3 in S^5.","rationale":"After reading the full text, I do not think the printed inequality (2.5) is the weakest point: the line break in the extracted text suggests the intended formula is (H(0)+1)/2 < (p+1)/(p-1), which is what (2.6) actually implies. The more serious issue is a subscript/order inconsistency that changes the statement of Theorem 1.1. Section 2 fixes 1≤m_-≤m_+ and derives the admissible p-range from the larger multiplicity m_+, equivalently from the smaller focal dimension n_+ = n-1-m_+. But Theorem 1.1 declares n_-≤n_+, making n_+ the larger dimension while still using n_+ in the p-range. For the Clifford family S^1×S^3 in S^5 this makes the range p∈(1,∞); p=4 is then included, yet the oscillation criterion for the endpoint with H(0)=3 fails. Hence the literal central claim is overreaching. The fix is purely notational: replace the ordering by n_+≤n_- (or use the minimum dimension in the exponent). Theorem 1.4 shows the same mismatch. Because the intended range is clear from (2.6) and the double-shooting framework is standard, a conditional acceptance pending correction of the p-range statements is appropriate; I keep the reader's verdict unchanged.","tokens_in":21126,"tokens_out":30909,"duration_ms":275559,"concrete_test":"Recompute the statements for the Clifford family S^1 × S^{n-2} ⊂ S^n with n=5. Let f(x,y)=|x|^2-|y|^2; the focal submanifolds have dimensions 1 and 3. Take p=4 and evaluate the oscillation condition of Theorem 2.1 at both endpoints r=0 and r=π: H(0) equals the two multiplicities 1 and 3. The condition (H(0)+1)/2 < 5/3 holds for H(0)=1 and fails for H(0)=3. If the theorem is claimed for p=4 under n_-≤n_+, this failure shows the p-range must be corrected to use the smaller focal dimension. Optionally integrate the limit problem (2.7) with H(0)=3, p=4 and count zeros: finitely many zeros confirms the obstruction.","verdict_should_be":"UNCHANGED","load_bearing_attack":"The proof in Section 2 fixes the convention 1 ≤ m_- ≤ m_+ and derives the admissible range in (2.6) as p < (n - n_+ + 2)/(n - n_+ - 2) = (m_+ + 3)/(m_+ - 1). Since n_± = n-1-m_±, this convention makes n_+ the smaller focal dimension. The proof of Theorem 1.4 and Lemma 2.4 are carried out only under this range. Theorem 1.1, however, states \"let n_- ≤ n_+ be the dimensions\" while using the same p-range with n_+. With n_- ≤ n_+, n_+ is the larger dimension, so the printed range is larger and can be infinite. Example: the Clifford hypersurface S^1 × S^3 in S^5 has focal dimensions 1 and 3. If n_-=1 and n_+=3, the stated range is p ∈ (1,∞). In the proof's relabelling the two singularities have H(0)=1 and H(0)=3; for p=4 the oscillation criterion (H(0)+1)/2 < (p+1)/(p-1) holds for H(0)=1 but fails for H(0)=3. Thus the double shooting cannot force the required infinite spirals from both ends, so the literal Theorem 1.1 overreaches. The intended statement should either reverse the ordering (n_+ ≤ n_-) or put the smaller focal dimension in the p-range; Theorem 1.4 has the same n_-/n_+ mismatch.","agreement_with_reader":"partial"},"referee_report":{"model":"deepseek-v4-flash","summary":"The paper studies the supercritical and subcritical semilinear elliptic problem −Δu + λu = λ|u|^{p−1}u on the round sphere S^n, restricted to functions of the form u = w∘f for an isoparametric function f. The reduction turns the PDE into a singular ODE on [0,π] whose coefficient h has one zero. The authors use a double shooting method from the two singular endpoints, together with an oscillation theorem for the associated initial value problem, to produce sign-changing solutions with arbitrarily many nodal components that are isoparametric hypersurfaces, and with blow-up on one of the focal submanifolds. The paper also derives a Yamabe-type multiplicity result on complex and quaternionic projective spaces via harmonic morphisms. The main claimed theorem (Theorem 1.1) is an infinite nodal solution result in the supercritical range.","tokens_in":21500,"tokens_out":7971,"duration_ms":68308,"significance":"If the main theorem is correct after the necessary corrections, it would be a substantial advance: it would provide infinitely many sign-changing solutions to the Yamabe-type problem in a genuinely supercritical range, with a precise description of the level and critical sets as isoparametric hypersurfaces and focal submanifolds, and with blow-up at focal submanifolds. The double-shooting/energy method in Appendix A is substantial, and the reduction via isoparametric functions and harmonic morphisms is elegant. The paper contains no fitted parameters, and the geometric constructions are explicit. However, the central claim as stated is currently overreaching: the p-range in Theorem 1.1 is not the range proved in Section 2, and several auxiliary statements contain dimension or label errors that must be corrected before the result is trustworthy.","major_comments":[{"comment":"Theorem 1.1 states the p-range with n_+ while assuming n_- ≤ n_+, but the proof in Section 2 fixes 1 ≤ m_- ≤ m_+ and derives (2.6) with n_+ denoting the smaller focal dimension. Under the theorem's ordering, n_+ is the larger focal dimension, so the stated range is strictly larger than the range proved. For example, for the Clifford hypersurface S^1×S^3 in S^5, the theorem would allow every p > 1, whereas the proof's admissible range is 1 < p < 3. The double-shooting intersection argument therefore cannot justify Theorem 1.1 as stated. The theorem should either be restricted to p < (n−n_-+2)/(n−n_-−2) with n_- the smaller focal dimension, or the proof should be reworked to handle both focal dimensions symmetrically. The same mismatch appears in Theorem 1.4.","section":"Theorem 1.1 and Section 2, Eq. (2.6)"},{"comment":"The inequality printed in (2.5), H(0) + 1/2 < (p+1)/(p−1), does not imply the later range (2.6). Substituting H(0)=m would give p < (2m+3)/(2m−1), which is not the bound used. The algebra that follows (2.6) is consistent with the intended criterion (H(0)+1)/2 < (p+1)/(p−1), which is equivalent to p < (m+3)/(m−1). Equation (2.5) should be corrected to that form; as printed, the statement of the imported oscillation theorem is internally inconsistent with its use.","section":"Section 2, Theorem 2.1 and Eq. (2.5)"},{"comment":"The proof of part (2) contains dimension errors. The lemma claims m ≥ 3, but the proof begins 'For each m ≥ 4'; for m=3 it sets α=β=4, which would give a real dimension 32 for H^4×H^4, while S^{4m+3}=S^15 requires ambient dimension 16. The expression for the ambient space as H^α × H^β × R^{2m+2} also has the wrong real dimension, and the focal submanifold formula writes M_+ = {0}×S^{β−1} rather than {0}×S^{4β−1}. These inconsistencies undermine the construction of the SU(2)-invariant Cartan–Münzner polynomial needed for Corollary 1.3. The construction is repairable by taking α+β = m+1 with α=β=2 for m=3, but as written the proof is not correct.","section":"Section 3, Lemma 3.2(2) and its proof"},{"comment":"The final sentence of the proof of Corollary 1.2 says 'Since arctan(f(x))=0 for every x∈M_+ and arctan(f(x))=π for every x∈M_-', which appears to have the focal labels swapped, and the expression u_k = arctan(f(w_k)) does not make sense because f is a function on S^n while w_k is a scalar. The intended construction is presumably u_k = z_k∘f with z_k = arctan(w_k). This affects the signs in the blow-up limits claimed in Corollary 1.2 and needs to be corrected.","section":"Section 3, proof of Corollary 1.2"},{"comment":"The scaling in Lemma 2.3 is hard to parse and contains inconsistent exponents. The displayed definition of z_d prints d^{-2/(p-1)} w_{d^{2/(p-1)}}(r/(d√λ)), while the inversion formula reads w_d(r)=d z_{d^{(p-1)/2}}(√λ d^{(p-1)/2} r); these expressions are not equivalent as written. Since the lemma is used to prove that the zeroes r_j(d) collapse to 0, which is essential for the spiral argument, the proof should be rewritten with coherent scaling exponents.","section":"Section 2, Lemma 2.3"}],"minor_comments":[{"comment":"There are several typos, e.g. 'hipersurfaces' in the abstract and 'integerdivide{0}' artifacts that should read '\\setminus\\{0\\}'. A proofreading pass is needed.","section":"Abstract and Introduction"},{"comment":"The notation ρ(d) is reused for both the phase-plane radial variable ρ(r,d) and the endpoint value ρ(d,a0); please use a different symbol for one of them to avoid confusion.","section":"Section 2, after Lemma 2.4"},{"comment":"In part (1), for k=0 the text says α=β=2 if k=1 and α=k, β=3 if k≥2, but the case k=0 (i.e. m=2) is not covered even though the lemma states m ≥ 3; this is likely a harmless omission, but it should be clarified.","section":"Section 3, Lemma 3.2(1)"},{"comment":"The sentence 'the reader may skip it in a first reading' is informal for a journal article and should be removed or rephrased.","section":"Appendix A, Step 3"}],"recommendation":"major_revision","confidential_remarks":"The central oscillation theorem is imported from the authors' earlier work [22] and from [27] without reproducing the proof; an independent verification or a detailed statement of the necessary result would strengthen the paper's reliability. I believe the p-range mismatch in Theorem 1.1 is fixable, and the geometric constructions in Lemma 3.2 are repairable, so I do not recommend rejection. However, the theorem as stated currently claims more than the proof establishes, so the revision must be substantial rather than cosmetic."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"I read the paper and largely agree with the conditional verdict, though I want to be sharper about the p-range. The stress-test note is right: Theorem 1.1 overreaches. The proof fixes 1 ≤ m_- ≤ m_+ and derives the admissible range p < (m_+ + 3)/(m_+ - 1), which is the exponent set by the larger multiplicity, i.e., the smaller focal submanifold. But the statement orders n_- ≤ n_+ and puts n_+ in the denominator. Since n_+ is then the larger focal dimension, the printed range can be infinite. For S^1 × S^3 in S^5 the theorem claims p < ∞, while the proof gives p < 3. That is a real gap, not a typo. Theorem 1.4 has the same problem if read under the proof's labeling. The fix is straightforward: use min(n_-, n_+) in the exponent, or reverse the ordering so n_+ is the smaller dimension.\n\nCredit where due: the double-shooting strategy and the energy appendix are substantial. Lemma A.2's Pohozaev-type identity is a real addition, and the transfer to CP^m and HP^m through harmonic morphisms is a nice trick. Corollary 1.3 explicitly uses κ = min(dim M_-, dim M_+), so it survives the p-range correction.\n\nThe rest of the soft spots are textual but should be fixed: inequality (2.5) is misprinted; it has to be (H(0)+1)/2 < (p+1)/(p-1), otherwise the algebra in (2.6) doesn't follow. Lemma 3.2(2) has wrong dimensions: α = β = 4 cannot occur for S^{15}, and M_+ is written as S^{β-1} instead of S^{4β-1}. The last paragraph of Theorem 1.1's proof swaps M_- and M_+: arccos(f) = 0 at M_+, not M_-. None of these should sink the paper, but they need cleaning up.\n\nThe oscillating-solution theorem is imported from the authors' earlier work; that is not a flaw, but it means the paper's genuinely new contribution is the energy analysis and the projective-space reduction, not the zero-counting engine.\n\nWho is this for? Researchers in Yamabe-type equations and isoparametric hypersurfaces. It deserves a serious referee and, with the p-range fixed, should be publishable. My recommendation: send it to review, flag the p-range as a required major revision, and check the Lemma 3.2 numerology.","headline":"Theorem 1.1's p-range overreaches: the proof only supports the exponent from the smaller focal submanifold, but the statement uses the larger one; the rest of the paper is solid enough to warrant review.","tokens_in":22055,"tokens_out":11895,"would_cite":false,"duration_ms":97969,"reading_group":"maybe","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["34B16","35B06","35B33","35B44","53C21","58E20","58J05"],"pacs":[],"model":"deepseek-v4-flash","headline":"A Yamabe-type equation on the round sphere admits sign-changing solutions with arbitrarily many prescribed isoparametric nodal components, even for supercritical exponents.","keywords":["nodal solutions","supercritical elliptic equations","Yamabe problem","isoparametric hypersurfaces","singular ODE","double shooting method","harmonic morphisms","blow-up analysis"],"falsifier":"Numerically integrate the singular ODE (2.4) for the case $m_-=m_+=2$, take $p$ close to the endpoint $(m_++3)/(m_+-1)=5$, and count the zeroes of $w_d$ in a small interval $(0,\\varepsilon)$ as $d$ grows; if the zero count does not diverge to infinity, the oscillation theorem's range is not valid and the double-shooting intersections cannot be forced.","tokens_in":20940,"feed_emoji":"🌐","tokens_out":6626,"duration_ms":154313,"temperature":0.7,"pith_summary":"The paper proves that a semilinear elliptic equation on the round sphere has infinite families of sign-changing solutions, including when the exponent is supercritical. The solutions are built as functions of an isoparametric function, so their level sets are isoparametric hypersurfaces and their singular behavior concentrates on the focal submanifolds. The proof reduces the PDE to a singular second-order ODE and then uses a double shooting argument, showing that two phase-plane curves spiral outward in opposite directions and therefore must intersect. This yields, for every positive integer $k$, a nodal solution with at least $k$ disjoint isoparametric nodal components, with the solutions blowing up on a focal submanifold as $k$ grows. The same reduction, pushed through harmonic morphisms, gives sequences of sign-changing solutions to the Yamabe problem on complex and quaternionic projective spaces.","feed_headline":"Infinite nodal solutions on the sphere, even supercritical","feed_subtitle":"The equation gains sign-changing solutions that blow up on special submanifolds for every integer k.","key_machinery":"The central object is the isoparametric function $f:\\mathbb{S}^n\\to[-1,1]$, a smooth function whose gradient norm and Laplacian depend only on $f$; its regular level sets are isoparametric hypersurfaces and its two singular levels $M_-$ and $M_+$ are focal submanifolds. Writing $u=w\\circ f$ transforms the PDE into the singular ODE $w''+\\frac{h(r)}{\\sin r}w'+\\frac{\\lambda}{\\ell^2}(|w|^{p-1}w-w)=0$ on $[0,\\pi]$, where $h$ is strictly decreasing with a single zero. The proof then uses double shooting from the two singular endpoints: two phase-plane curves $I(d)$ and $J(c)$ are defined by evolving with $w'(0)=0$ and $w'(\\pi)=0$. An imported oscillation theorem forces the number of zeroes near each endpoint to diverge as $|d|,|c|\\to\\infty$, while a Pohozaev-type identity shows the curves have unbounded radius; the resulting opposite-oriented spirals must intersect, producing solutions with arbitrarily many zeroes and critical points.","core_discovery":"On its own terms, the paper establishes that the equation $-\\Delta_{g_0}u+\\lambda u=\\lambda|u|^{p-1}u$ on $\\mathbb{S}^n$, with $\\lambda>0$ and $p>1$, admits a nodal solution $u_k$ for every $k\\in\\mathbb{N}$ whenever $p$ lies in $(1,\\frac{n-n_++2}{n-n_+-2})$, a range that contains the supercritical regime above the Sobolev critical exponent. The nodal set of $u_k$ contains at least $k$ connected components, each an isoparametric hypersurface diffeomorphic to a prescribed one, while the critical set contains the two focal submanifolds and at least $k-1$ further isoparametric hypersurfaces. Moreover, $|u_k|\\to\\infty$ on one of the focal submanifolds as $k\\to\\infty$, so the sequence is not compact. The projective-space results transfer these sphere solutions to sign-changing Yamabe solutions on $\\mathbb{CP}^m$ and $\\mathbb{HP}^m$ with isoparametric hypersurfaces as level sets and unbounded sup-norms.","pith_inferences":["Editorial inference: the argument does not use the full symmetry group of the round metric, only the existence of an isoparametric function with two focal submanifolds; analogous nodal families should exist on other manifolds carrying such functions, provided the corresponding singular ODE has the same oscillation and energy properties.","Editorial inference: the blow-up along minimal focal submanifolds is reminiscent of known concentration phenomena, but here it arises from ordinary differential equations and shooting rather than Lyapunov-Schmidt reduction, suggesting these solutions are explicit models for submanifold concentration.","Editorial inference: the harmonic-morphism transfer is likely not limited to $\\mathbb{CP}^m$ and $\\mathbb{HP}^m$; any compact Riemannian submersion with minimal fibers over a base admitting an invariant isoparametric function should produce analogous unbounded nodal solutions on the base."],"forward_implications":["For every isoparametric hypersurface $S$ and every integer $k$, the sphere equation admits a nodal solution with at least $k$ nodal components, each diffeomorphic to $S$.","The constructed solutions blow up on one focal submanifold: $|u_k(x)|\\to\\infty$ for every $x\\in M_-$ or every $x\\in M_+$ as $k\\to\\infty$, so the solution family is noncompact.","In the critical Yamabe case on the sphere, when both focal submanifolds have positive dimension, there is a sign-changing sequence tending to $+\\infty$ on one focal submanifold and to $-\\infty$ on the other.","On $\\mathbb{CP}^m$ and $\\mathbb{HP}^m$ with their canonical metrics, the Yamabe equation has sign-changing solutions for every $k$, with level sets given by isoparametric hypersurfaces and sup-norms diverging to infinity.","The admissible $p$-interval extends beyond the critical exponent $(n+2)/(n-2)$, so supercritical nodal solutions are not merely possible but appear in infinite families."],"supporting_citations":[{"why":"Supplies the double-shooting framework, the oscillating-solution theorem for the singular ODE, and the critical-case multiplicity result that Corollary 1.2 refines.","marker":"[22]"},{"why":"Provides the base existence of at least one sign-changing supercritical solution, which Theorem 1.1 improves to an infinite sequence.","marker":"[27]"},{"why":"Introduces the Pohozaev-type identity method used in Appendix A to show the shooting curves have unbounded radius.","marker":"[10]"},{"why":"Gives the infinite-zero blow-up profile of the limit Cauchy problem that underlies Lemma 2.3.","marker":"[26]"},{"why":"Establishes the harmonic-morphism facts used in Lemma 3.1 to preserve isoparametric functions under Riemannian submersions with minimal fibers.","marker":"[3]"},{"why":"Provides the lifting formula that transfers solutions between the base space and the sphere in Corollary 1.3.","marker":"[14]"},{"why":"Supplies the theory of Cartan-Münzner polynomials, isoparametric hypersurfaces, and focal submanifolds used throughout the paper.","marker":"[12]"}],"fun_headline_variants":["Supercritical sphere: nodal blow-up for every k","Nodal solutions on S^n blow up, even in supercritical range","Yamabe nodal solutions in CP^m and HP^m with blow-up","Supercritical sphere yields infinite nodal solutions, all k","Nodal blow-up on sphere extends Yamabe to CP^m, HP^m"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"The load-bearing premise is the imported oscillation assertion for the singular initial-value problem: for $H(0)>0$ and $p$ in the claimed range, solutions with large initial value $d$ have arbitrarily many zeroes arbitrarily close to $0$; the paper's printed inequality (2.5) appears to need to read $(H(0)+1)/2 < (p+1)/(p-1)$ to match the $p$-interval used later, and if that corrected criterion fails at the boundary values, the double-shooting intersection argument collapses.","fun_headline_variants_meta":{"raw":{"variants":["Supercritical sphere: nodal blow-up for every k","Nodal solutions on S^n blow up, even in supercritical range","Yamabe nodal solutions in CP^m and HP^m with blow-up","Supercritical sphere yields infinite nodal solutions, all k","Nodal blow-up on sphere extends Yamabe to CP^m, HP^m"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000727,"raw_usage":{"total_tokens":3322,"prompt_tokens":1075,"completion_tokens":2247,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":691,"completion_tokens_details":{"reasoning_tokens":2154}},"tokens_in":691,"tokens_out":2247,"duration_ms":124042,"temperature":1.0,"reasoning_tokens":2154,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-14T11:52:02.331580+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"Numerically integrate the singular ODE (2.4) for the case $m_-=m_+=2$, take $p$ close to the endpoint $(m_++3)/(m_+-1)=5$, and count the zeroes of $w_d$ in a small interval $(0,\\varepsilon)$ as $d$ grows; if the zero count does not diverge to infinity, the oscillation theorem's range is not valid and the double-shooting intersections cannot be forced.","supporting_citations":[{"cited_title":"Low energy nodal solutions to the Yamabe equation","cited_arxiv_id":"1807.06114","evidence_quote":"Supplies the double-shooting framework, the oscillating-solution theorem for the singular ODE, and the critical-case multiplicity result that Corollary 1.2 refines."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Provides the base existence of at least one sign-changing supercritical solution, which Theorem 1.1 improves to an infinite sequence."},{"cited_title":"Castro, M","cited_arxiv_id":null,"evidence_quote":"Introduces the Pohozaev-type identity method used in Appendix A to show the shooting curves have unbounded radius."},{"cited_title":"Haraux, F","cited_arxiv_id":null,"evidence_quote":"Gives the infinite-zero blow-up profile of the limit Cauchy problem that underlies Lemma 2.3."},{"cited_title":"Baird, J.C","cited_arxiv_id":null,"evidence_quote":"Establishes the harmonic-morphism facts used in Lemma 3.1 to preserve isoparametric functions under Riemannian submersions with minimal fibers."},{"cited_title":"Clapp, J","cited_arxiv_id":null,"evidence_quote":"Provides the lifting formula that transfers solutions between the base space and the sphere in Corollary 1.3."},{"cited_title":"Cecil, P","cited_arxiv_id":null,"evidence_quote":"Supplies the theory of Cartan-Münzner polynomials, isoparametric hypersurfaces, and focal submanifolds used throughout the paper."}],"review_version":1}