{"id":"914acf71-5081-4d96-b75c-7dc3fbfac036","arxiv_id":"2506.01652","paper_version":2,"verdict":"REJECT","confidence":"LOW","novelty_score":6.0,"correctness_risk":"high","formal_verification":"none","parameter_count":2,"one_line_summary":"The paper claims existence of multi-peak solutions with O(μ^{-1/2}) peaks for a slightly supercritical Hénon-type problem in the 3D unit ball, but the reduced system on which the proof depends has no positive solution as stated.","lead":"This paper claims to construct solutions with many sharp peaks near the boundary of a 3D ball for a supercritical equation. The proof appears to contain a scaling error that breaks the main reduction step.","discovery_kind":"extension","skeptic_critique":{"model":"deepseek-v4-flash","headline":"System (2.21) has no positive root: L(σ)>0 for every σ>0, so the degree argument in Section 4.2 cannot produce the required (λ*,σ*), and the proof of Theorem 1.1 fails.","rationale":"The reader's weakest_assumption identifies exactly the load-bearing failure: the reduced system (2.21) has no positive solution because L(σ)>0 for all σ>0. This is verifiable by a one-line computation and is internal to the paper, so it does not rely on specialist judgment. The proof's final degree argument in Section 4.2 is anchored to the nonexistent root (λ*,σ*), and Lemma 4.3's leading-order formula for ∂λJ makes the obstruction quantitative: the first reduced equation cannot be satisfied. The secondary issue in (2.6), where the bubble scaling ε^{1/2} is inconsistent with the equation -ΔU=K(r)U^5, is also real and independently supports rejection, but the nonexistence of the root of (2.21) is the most load-bearing because it breaks the central existence mechanism. No machine-checked verification or reproducible computational evidence is present to offset these internal inconsistencies. The paper should remain rejected; the reader's verdict and confidence are appropriate, and no adjustment is needed.","tokens_in":19622,"tokens_out":5860,"duration_ms":60080,"concrete_test":"Evaluate L(1) using (2.19): each term 1/(|j|π)-1/sqrt(j^2π^2+1) is positive and the sum converges, so L(1)>0; indeed L(σ)>0 for all σ>0. An independent re-derivation of Lemma 2.1 and Lemma 4.3 should confirm that ∂λJ is proportional to L(σ) in the leading order; if so, ∂λJ cannot vanish for σ in the positive range (2.4), and the degree argument in Section 4.2 has no zero. Separately, test the scaling identity by plugging Ui from (2.6) into -ΔUi and K(r)Ui^5; the two sides differ by ε^4, showing the ansatz does not satisfy the stated bubble equation.","verdict_should_be":"UNCHANGED","load_bearing_attack":"The proof of Theorem 1.1 depends on the reduced system (2.21), L(σ)=0 and K'(1)A1λ=A2L'(σ). With L(σ) defined in (2.19) as Σ_{j∈Z\\{0}}(1/|jπ| - 1/sqrt((jπ)^2+σ^2)), every summand is positive for σ>0, so L(σ)>0 for all σ>0; the only zero is L(0)=0. Hence (2.21) has no positive root (λ*,σ*), contradicting the assertion after (2.21) that such a unique root exists. This is not a minor gap: the admissible parameter intervals in (2.4) are centered at (λ*,σ*), Lemma 4.3 gives ∂λJ(λ,σ,μ)=λ^{-2}A2L(σ)+O(μ^{1/2}|ln μ|), so ∂λJ cannot vanish for any positive σ in the admissible range. The degree argument in Section 4.2 therefore searches for a zero of a vector field whose first component is strictly positive, up to the error term, so no critical point of the reduced energy can be found by the stated construction. A second internal inconsistency reinforces this: the ansatz (2.6) sets Ui=K^{-1/4}(r)ε^{1/2}Φ((x-rm_i)/ε), but a direct computation gives -ΔUi=K^{-1/4}(r)ε^{-3/2}Φ^5 and K(r)Ui^5=K^{-1/4}(r)ε^{5/2}Φ^5, so the claimed identity -ΔUi=K(r)Ui^5 fails by a factor ε^4. The correct blow-up scaling would use ε^{-1/2}. Both issues are internal inconsistencies, not disagreements with external consensus, and they invalidate the central construction.","agreement_with_reader":"agree"},"referee_report":{"model":"deepseek-v4-flash","summary":"The paper claims existence of positive multi-peak solutions to -Δu = K(x) u^{5+μ} in the unit ball B ⊂ R^3 with zero Dirichlet boundary condition, for μ > 0 sufficiently small and radial K satisfying K(1) > 0 and K'(1) > 0. The construction places k = ⌊μ^{-1/2}⌋ peaks on a regular polygon of radius r = 1 - σ μ^{1/2}, uses a reflected bubble ansatz W = Σ_i(U_i - U_i^*), solves a constrained nonlinear problem by a finite-dimensional reduction, and then finds (λ,σ) by a degree argument based on the reduced system L(σ) = 0 and K'(1)A_1 λ = A_2 L'(σ). The theorem states the number of strict local maxima is of order μ^{-1/2} as μ→0.","tokens_in":20066,"tokens_out":15603,"duration_ms":161604,"significance":"If the theorem were established, it would extend Liu-Peng's higher-dimensional supercritical result to dimension three, and it would provide boundary-peak solutions for the Hénon-type equation with a count of peaks growing as μ^{-1/2}. The reduction framework follows earlier work by Wei-Yan and Hao-Chen-Zhang, and the paper contains a substantial amount of technical estimation. However, the two load-bearing issues described below are internal inconsistencies in the construction itself; as written, the proof does not establish the theorem. The significance of the intended result is therefore conditional on a successful repair of the reduced system and the ansatz scaling.","major_comments":[{"comment":"The reduced system (2.21) has no positive solution. With L(σ) defined in (2.19) as Σ_{j∈Z\\{0}}(1/|jπ| - 1/√((jπ)^2+σ^2)), every summand is strictly positive for every σ > 0 because (jπ)^2+σ^2 > (jπ)^2. Hence L(σ) > 0 for all σ > 0, with the only zero at σ = 0. The assertion after (2.21) that there is a unique positive root (λ*,σ*) is therefore false, and the admissible rectangle in (2.4) is empty. This is not a cosmetic issue: Lemma 4.3 gives ∂_λ J = λ^{-2} A_2 L(σ) + O(μ^{1/2}|ln μ|), so the first component of the reduced gradient cannot vanish for any positive σ in the claimed range. The degree argument in §4.2, which searches for a zero of this vector field in a rectangle around (λ*,σ*), consequently has no object to find. The proof of Theorem 1.1 fails at this point.","section":"§2.3, Eqs. (2.19)–(2.21) and §4.2"},{"comment":"The ansatz scaling in (2.6) is inconsistent with the claimed identity -ΔU_i = K(r) U_i^5. Writing Φ((x-rm_i)/ε), if U_i = K^{-1/4}(r) ε^{1/2} Φ, then a direct computation gives -ΔU_i = K^{-1/4}(r) ε^{-3/2} Φ^5, while K(r)U_i^5 = K^{-1/4}(r) ε^{5/2} Φ^5; these differ by a factor ε^4. The correct bubble scaling for the critical equation in R^3 is ε^{-1/2}. The inconsistency is not confined to (2.6): Lemma 2.2 and the subsequent estimates use U_i = λ^{1/2} K^{-1/4}(r) Φ(0) / (μ^{1/2} d_i), which is effectively ε^{-1/2} scaling, and Remark 1 also writes ε^{-1/2}. Because the error estimates in Lemmas 2.2 and 2.3 and the energy estimates in Lemma 4.3 are derived from the asserted amplitude, the manuscript's computations do not form a coherent proof of the ansatz's leading-order behavior. If (2.6) is meant to be a typo, the change to ε^{-1/2} must be propagated through all subsequent estimates, which is a substantial revision.","section":"§2.1, Eq. (2.6); §2.4, Lemma 2.2"}],"minor_comments":[{"comment":"The last sentence of the proof of Lemma 2.3 says 'Combining all these estimates above, we obtain the assertion of Lemma 4.3'; this should refer to Lemma 2.3.","section":"Lemma 2.3, proof"},{"comment":"The derivative formula L'(σ) = Σ_{j∈Z} σ/(σ^2+(jπ)^2)^{3/2} includes the j = 0 term, which is not the derivative of any term in L(σ) as defined; the sum should be over j ∈ Z\\{0}, or L(σ) should be defined consistently. For σ > 0 this does not change positivity, but it is a mathematical mismatch in a definition used in the main system.","section":"Eq. (2.19)"},{"comment":"Remark 1 states that the solution has the form u(x) = max_i ε^{-1/2} Φ((x-rm_i)/ε) + O(1), which is incompatible with the ε^{1/2} amplitude written in Eq. (2.6). This reinforces the need to correct the ansatz scaling.","section":"Remark 1 and Eq. (2.6)"}],"recommendation":"reject","confidential_remarks":"The two major comments are concrete and independent of any external consensus: the reduced system (2.21) as defined has no positive root, and the ansatz in (2.6) does not satisfy the leading-order equation. Both are central to the finite-dimensional reduction, so I do not see how the paper can be accepted without a major rewrite of the ansatz and the reduced system. I would not rule out that a corrected argument is possible, but the present manuscript does not provide it."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"Colleague \n\nThe paper is a legitimate attack on a real open problem: the 3D case of Liu-Peng's many-peaked supercritical construction. That's the good news. The bad news is that the proof as written has two load-bearing internal errors, and the central argument collapses.\n\nWhat's good: the statement is new and would be significant if true. The paper correctly identifies the gap left by Liu-Peng (n≥4) and adopts a sensible ansatz from Hao-Chen-Zhang's critical case. The reduction framework is detailed and the estimates are carefully laid out; the author is clearly in command of the method.\n\nWhere it fails. First, the reduced system (2.21) is L(σ)=0 with L(σ) defined in (2.19) as Σ_{j∈Z\\{0}} (1/|jπ| − 1/√((jπ)^2+σ^2)). For every σ>0 each term is positive, so L(σ)>0 for all positive σ; the only zero is σ=0. The paper asserts, after (2.21), that this system has a unique positive root (λ*,σ*), but that is false. Consequently the admissible intervals (2.4) are empty, and the degree argument in Section 4.2 has no point to search for. This is fatal.\n\nSecond, the ansatz (2.6) sets U_i = K^{-1/4}(r) ε^{1/2} Φ((x-rm_i)/ε). A direct calculation gives −ΔU_i = K^{-1/4}(r) ε^{-3/2} Φ^5, while K(r)U_i^5 = K^{-1/4}(r) ε^{5/2} Φ^5; the claimed identity −ΔU_i=K(r)U_i^5 fails by a factor ε^4. The correct bubble amplitude in R^3 is ε^{-1/2}. The estimates later in the paper (e.g., Lemma 2.2) implicitly use the ε^{-1/2} scaling, so the written ansatz is inconsistent with the rest of the proof. This is not a typo that can be brushed aside; it changes all the power counting.\n\nNeither issue is a matter of disagreement with a consensus; they are internal inconsistencies in the manuscript. So the proof of Theorem 1.1 does not go through.\n\nIs the approach salvageable? Possibly. The reduction scaffolding is standard, and the errors are specific and identifiable. If the author can find the correct reduced system (maybe a different definition of L) and fix the ansatz scaling, the paper could be reshaped. But the present version does not establish the claimed result.\n\nRecommendation: I would not cite this version. It deserves a serious referee only in the sense that a referee could quickly identify these issues; but the result is not proven. I'd send it back for major revision with a clear explanation of these two problems.\n\nBest,","headline":"A serious attempt at a real open problem, but the proof has two fatal internal inconsistencies: the reduced system has no positive root and the ansatz scaling is off by a factor of epsilon^4.","tokens_in":20589,"tokens_out":7897,"would_cite":false,"duration_ms":67806,"reading_group":"no","serious_thinker":"no","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["35J25","35J61"],"pacs":[],"model":"deepseek-v4-flash","headline":"A slightly supercritical Dirichlet problem in the 3D unit ball admits positive solutions whose number of peaks grows like $\\mu^{-1/2}$.","keywords":["peak solutions","supercritical equation","reduction method","multi-peak solutions","Hénon equation","boundary concentration","slightly supercritical exponent"],"falsifier":"Evaluate $L(\\sigma)=\\sum_{j\\in\\mathbb{Z}\\setminus\\{0\\}}\\bigl(1/|j\\pi|-1/\\sqrt{(j\\pi)^2+\\sigma^2}\\bigr)$ for a specific positive value such as $\\sigma=1$: every term is positive, so the sum is positive, and the same holds for every $\\sigma>0$. That calculation makes the equation $L(\\sigma)=0$ impossible to satisfy, so the positive root $(\\lambda^*,\\sigma^*)$ asserted in Section 2.3 and used in Section 4.2 does not exist.","tokens_in":19397,"feed_emoji":"🧮","tokens_out":9563,"duration_ms":89539,"temperature":0.7,"pith_summary":"This paper aims to extend to dimension three the known construction of multi-peak positive solutions for slightly supercritical equations of Hénon type. Its main theorem states that, if the radial coefficient $K$ satisfies $K(1)>0$ and $K'(1)>0$, then for every sufficiently small $\\mu>0$ the problem $-\\Delta u=K(x)u^{5+\\mu}$ in the unit ball $\\mathbf{B}\\subset\\mathbb{R}^3$, with $u=0$ on $\\partial\\mathbf{B}$, has a positive solution whose number of strict local maxima is of order $\\mu^{-1/2}$. The peaks all approach the boundary as $\\mu\\to0$, so the solutions are strongly non-radial and become spikier as the exponent approaches the critical value $6$. The result would fill the missing three-dimensional case after the previously known $n\\geq4$ construction, and it would produce a family of solutions whose peak count is not fixed but grows with the small parameter.","feed_headline":"3D supercritical equation gets μ^{-1/2} peaks","feed_subtitle":"For small μ>0, positive solutions concentrate near the boundary with peak count growing like μ^{-1/2}.","key_machinery":"The load-bearing object is the $k$-peak ansatz $W=\\sum_{i=0}^{k-1}(U_i-U_i^*)$, built from the standard bubble $\\Phi(x)=3^{1/4}(1+|x|^2)^{-1/2}$ and its scaled, reflected copies; the reflected terms $U_i^*$ make the ansatz vanish exactly on the boundary. Around this ansatz the paper runs a Lyapunov-Schmidt reduction: it solves the constrained linear problem in a symmetric subspace $H_k$ and then uses energy estimates and degree theory to choose the parameters so that the multipliers $c_1,c_2$ vanish. The pivot is the reduced system $L(\\sigma)=0$ and $K'(1)A_1\\lambda=A_2L'(\\sigma)$, where $L(\\sigma)=\\sum_{j\\in\\mathbb{Z}\\setminus\\{0\\}}(1/|j\\pi|-1/\\sqrt{(j\\pi)^2+\\sigma^2})$; the claimed positive root $(\\lambda^*,\\sigma^*)$ of this system is what provides the parameter rectangle around which the degree argument is centered.","core_discovery":"The paper claims that the finite-dimensional reduction method can be made to work in $\\mathbb{R}^3$ by building an ansatz with $k=\\lfloor\\mu^{-1/2}\\rfloor$ bubbles centered at the vertices $r m_i$ of a regular $k$-gon near the boundary, together with reflected terms $U_i^*$ so that the whole ansatz $W=\\sum_{i=0}^{k-1}(U_i-U_i^*)$ vanishes on $\\partial\\mathbf{B}$. The reduction leads to a two-parameter system in $(\\lambda,\\sigma)$, equation (2.21), which the paper asserts has a unique positive root $(\\lambda^*,\\sigma^*)$; around that root a degree argument is used to make the Lagrange multipliers vanish. At the resulting parameter choice the corrected function $u=W+\\varphi$ solves the original equation, and the correction $\\varphi$ is controlled in weighted norms so that the $k$ peaks survive as strict local maxima. Because $k=\\lfloor\\mu^{-1/2}\\rfloor$, the number of peaks tends to infinity as $\\mu\\to0$.","pith_inferences":["If, as its definition suggests, $L(\\sigma)>0$ for every $\\sigma>0$, then the reduced system (2.21) has no positive root, and the degree argument in Section 4.2 would have no zero to surround; this is a direct check a reader can perform.","The normalization of the ansatz bubbles in (2.6) appears inconsistent with the equation $-\\Delta U_i=K(r)U_i^5$ and with the scaling used in Lemmas 2.2 and 2.3, because the prefactor $K^{-1/4}(r)\\varepsilon^{1/2}$ does not match the powers of $\\varepsilon$ and $\\mu$ in the nonlinearity.","A repaired construction would need either a reduced system with a genuine positive root or an additional parameter able to cancel the positive lattice sum $L(\\sigma)$; numerically evaluating $L$ at a few values of $\\sigma$ would test this directly.","If the theorem survives a corrected argument, the method suggests that the boundary-peaked dihedral pattern is stable under small perturbations of $K$ near $r=1$ that keep $K'(1)>0$."],"forward_implications":["For every sufficiently small $\\mu>0$, the equation $-\\Delta u=K(x)u^{5+\\mu}$ in the unit ball would have a positive non-radial solution with peak count of order $\\mu^{-1/2}$, all peaks lying near $\\partial\\mathbf{B}$.","As $\\mu\\to0$, the number of peaks diverges, so the construction would give positive solutions with arbitrarily many boundary peaks accumulating at the critical exponent.","For $K(x)=|x|^\\alpha$, the result would give the same conclusion for the Hénon equation, a concrete family of coefficients satisfying the hypotheses.","The symmetry of the ansatz would produce solutions with the dihedral symmetry of a regular $k$-gon, making the non-radial character of the solutions explicit.","The peak count $\\mu^{-1/2}$ identifies the correct bubble number in dimension three, differing from the order $\\mu^{-1/(n-1)}$ used in the earlier $n\\geq4$ construction."],"supporting_citations":[{"why":"supplies the n≥4 multi-peak result for the slightly supercritical problem that this paper extends to dimension three.","marker":"[16]"},{"why":"supplies the ansatz idea and the basic estimates (A.1)-(A.7) that the three-dimensional construction adapts.","marker":"[12]"},{"why":"classifies the kernel of the linearized operator, which is used to prove the a priori estimate in Proposition 3.2.","marker":"[1]"},{"why":"gives the critical-exponent boundary-peak constructions whose ansatz structure is adapted to the supercritical case.","marker":"[22]"}],"fun_headline_variants":["3D supercritical equation admits μ^{-1/2} peaks","Peak count μ^{-1/2} in 3D supercritical case","Supercritical 3D: many peaks near boundary","3D slightly supercritical: μ^{-1/2} peaks exist"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"The construction depends on the reduced system (2.21) having a positive solution $(\\lambda^*,\\sigma^*)$; if the sum $L(\\sigma)$ appearing there is positive for every positive $\\sigma$, as its definition suggests, no such solution exists and the degree argument in Section 4.2 has no zero to work with.","fun_headline_variants_meta":{"raw":{"variants":["3D supercritical equation admits μ^{-1/2} peaks","Peak count μ^{-1/2} in 3D supercritical case","Supercritical 3D: many peaks near boundary","3D slightly supercritical: μ^{-1/2} peaks exist"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000208,"raw_usage":{"total_tokens":1407,"prompt_tokens":951,"completion_tokens":456,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":567,"completion_tokens_details":{"reasoning_tokens":380}},"tokens_in":567,"tokens_out":456,"duration_ms":4813,"temperature":1.0,"reasoning_tokens":380,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-07T11:41:20.859183+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"Evaluate $L(\\sigma)=\\sum_{j\\in\\mathbb{Z}\\setminus\\{0\\}}\\bigl(1/|j\\pi|-1/\\sqrt{(j\\pi)^2+\\sigma^2}\\bigr)$ for a specific positive value such as $\\sigma=1$: every term is positive, so the sum is positive, and the same holds for every $\\sigma>0$. That calculation makes the equation $L(\\sigma)=0$ impossible to satisfy, so the positive root $(\\lambda^*,\\sigma^*)$ asserted in Section 2.3 and used in Section 4.2 does not exist.","supporting_citations":[{"cited_title":"Solutions with large number of peaks for the supercritical Hénon equation.Pacific J","cited_arxiv_id":null,"evidence_quote":"supplies the n≥4 multi-peak result for the slightly supercritical problem that this paper extends to dimension three."},{"cited_title":"Infinitely many spike solutions for the Hénon equation with critical growth.J","cited_arxiv_id":null,"evidence_quote":"supplies the ansatz idea and the basic estimates (A.1)-(A.7) that the three-dimensional construction adapts."},{"cited_title":"A Sobolev inequality with re- mainder term and critical equations on domains with topology for the polyharmonic operator","cited_arxiv_id":null,"evidence_quote":"classifies the kernel of the linearized operator, which is used to prove the a priori estimate in Proposition 3.2."},{"cited_title":"Infinitely many nonradial solutions for the Hénon equation with critical growth.Rev","cited_arxiv_id":null,"evidence_quote":"gives the critical-exponent boundary-peak constructions whose ansatz structure is adapted to the supercritical case."}],"review_version":1}