{"id":"4eb797e2-1c60-4546-8e4d-28b3746feafd","arxiv_id":"1908.02536","paper_version":2,"verdict":"CONDITIONAL","confidence":"HIGH","novelty_score":7.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":0,"one_line_summary":"The authors prove existence of a nontrivial weak solution to a double critical fractional Laplacian equation with a Hardy term, using a new improved Sobolev inequality in weighted Morrey spaces.","lead":"This paper proves that a certain equation with a fractional Laplacian, a Hardy singularity, and two competing critical nonlinearities has at least one nonzero solution on all of R^n. The proof uses a new weighted Morrey space inequality that also simplifies earlier existence proofs.","discovery_kind":"new_method","skeptic_critique":{"model":"deepseek-v4-flash","headline":"Proposition 4.1(3)'s attainment of Lambda(n,s,gamma,alpha) for gamma<0 is asserted, not proved; the mountain-pass threshold c<c* rests on it.","rationale":"The reader's weakest_assumption identifies the same load-bearing concern: the mountain-pass threshold c<c* is established by testing paths along minimizers of S_mu and Lambda, and the attainment of Lambda for negative gamma is only delegated to Remark 4.2 and a citation that does not cover that range. Since Theorem 1.1(I) explicitly advertises gamma<0 as the improvement over earlier work, this omission genuinely affects the central proof. The gap is likely repairable by writing out the analogous concentration argument, so a conditional verdict rather than rejection is appropriate; the secondary boundary case alpha=beta=0 also lacks a demonstrated proof, further supporting conditionality. The verdict is therefore UNCHANGED relative to the reader's CONDITIONAL recommendation.","tokens_in":27033,"tokens_out":46719,"duration_ms":428320,"concrete_test":"Complete the missing proof of Proposition 4.1(3) for gamma<0 by adapting the proof of Proposition 4.1(1): take a minimizing sequence u_k with ||u_k||^2 -> Lambda and integral of |u_k|^{2*_s(alpha)}/|x|^alpha equal to 1; use (1.9)-(1.10) to obtain the Morrey lower bound, rescale to v_k, prove v_k converges weakly to v not identically zero, then split ||v_k||^2 and the weighted L^{2*_s(alpha)} norm via Lemma 2.5 with q=2/2*_s(alpha) to conclude ||v||^2 = Lambda and v is a minimizer. If the split requires a subadditivity step for t goes to t^{2/2*_s(alpha)}, verify it; alternatively, locate a published proof of Lambda attainment covering gamma<0. If neither is available, the gamma<0 portion of Theorem 1.1(I) is unsupported.","verdict_should_be":"UNCHANGED","load_bearing_attack":"Theorem 1.1(I) claims existence for every gamma<gamma_H, including gamma<0, with 0<alpha,beta<2s. The proof of Proposition 5.2 needs minimizers of both variational constants S_mu(n,s,gamma,alpha) and Lambda(n,s,gamma,beta) to build v0 and to obtain the strict mountain-pass bound c<c*. Proposition 4.1(1) proves attainment of S_mu for all gamma<gamma_H, but Proposition 4.1(3) is not proved: Remark 4.2 says only (1)-(2) are proved and that 'the strategy can be applied' to (3), citing [2]. The cited result in [2] covers 0<=gamma<gamma_H, not the negative gamma regime that is the paper's advertised improvement. In the regime where the Lambda-term is the smaller threshold, the strict inequality c<c* has no demonstrated basis; a nonexistent or unattained Lambda(n,s,gamma,beta) would leave the PS level c at or above the critical threshold, and the nonvanishing argument in Theorem 1.1 would not close. This is a proof gap rather than a demonstrated counterexample: the concentration argument used for S_mu in Proposition 4.1(1) is expected to carry over to Lambda by replacing B_alpha with the weighted L^{2*_s(alpha)} norm and using Lemma 2.5 with exponent 2/2*_s(alpha)<1.","agreement_with_reader":"agree"},"referee_report":{"model":"deepseek-v4-flash","summary":"The manuscript studies the existence of nontrivial weak solutions in the homogeneous fractional Sobolev space \\dot{H}^s(\\mathbb{R}^n) to the double critical equation (1.1), which combines a fractional Laplacian with a Hardy potential, a Hardy–Sobolev critical term, and a Choquard-type nonlocal critical term. The main theorem claims existence under either (I) 0<\\alpha,\\beta<2s<n, \\mu\\in(0,n) and \\gamma<\\gamma_H, or (II) \\alpha\\beta=0, 0\\le \\alpha,\\beta<2s<n, \\mu\\in(0,n) and 0\\le\\gamma<\\gamma_H. The proof introduces weighted Morrey-space embeddings and an improved Sobolev inequality, uses them to prove attainment of the variational constants S_\\mu(n,s,\\gamma,\\alpha) and \\Lambda(n,s,\\gamma,\\alpha), and then applies the mountain-pass lemma with a threshold c^* to produce a Palais-Smale sequence whose rescaled weak limit is shown to be a nontrivial solution.","tokens_in":27310,"tokens_out":15214,"duration_ms":154344,"significance":"If the proof is completed, the result is a genuine advance: it removes the restriction \\mu\\in(n-2s,n) appearing in [41], it allows negative \\gamma when both \\alpha and \\beta are positive, and it offers a direct argument that avoids the extension-space and truncation machinery used in [2]. The improved Sobolev inequality in weighted Morrey spaces, Proposition 1.3 and Corollary 1.4, is a useful independent contribution. The manuscript is also transparent about the external origin of the inequalities it uses, and I found no circular dependence on the existence statement. However, a load-bearing part of the attainment argument for \\Lambda is only asserted, not proved, so the current version is not yet acceptable.","major_comments":[{"comment":"The attainment of \\Lambda(n,s,\\gamma,\\alpha) for 0<\\alpha<2s and of \\Lambda(n,s,\\gamma,0) is stated as Proposition 4.1(3)-(4), but Remark 4.2 explicitly says that only items (1)-(2) are proved and that the strategy 'can be applied' to (3)-(4). This is not a proof. Proposition 5.2 then uses item (3) to construct the minimizer V_{\\gamma,\\beta} and to establish the strict mountain-pass inequality c<c^*, and Theorem 1.1(I) depends on this in the negative-\\gamma regime, which is exactly the advertised improvement over [2]. Since [2] covers only 0\\le\\gamma<\\gamma_H, the negative-\\gamma case is not covered by the cited result. The missing proof should be supplied; the concentration argument used for Proposition 4.1(1) appears transferable by replacing B_\\alpha with the weighted norm \\int_{\\mathbb{R}^n}|u|^{2^*_s(\\alpha)}/|x|^\\alpha\\,dx and using the subadditivity exponent 2/2^*_s(\\alpha)<1, but this must be written out.","section":"Section 4, Proposition 4.1(3)-(4) and Remark 4.2"},{"comment":"The statement of Proposition 5.3 covers the two cases \\alpha=0<\\beta<2s and \\beta=0<\\alpha<2s, but its proof says that items (2) and (4) of Proposition 4.1 are used, which are the \\alpha=0 items. For the case \\alpha=0<\\beta, the required constant is \\Lambda(n,s,\\gamma,\\beta), i.e., item (3), not item (4); for the case \\beta=0<\\alpha, one needs items (1) and (4). The proof therefore does not match the statement and must be corrected, with the needed attainment results identified precisely.","section":"Section 5, Proposition 5.3"},{"comment":"The final paragraph of the proof of Theorem 1.1 dismisses the case \\alpha=\\beta=0 by saying that the Nehari manifold method in [41] gives a nontrivial weak solution for 0\\le\\gamma<\\gamma_H. However, the hypotheses of [41] as summarized in the introduction are 0<\\beta<2s, \\mu\\in(n-2s,n), and 0<\\gamma<\\gamma_H; these do not directly cover \\alpha=\\beta=0 with \\mu\\in(0,n) and \\gamma=0 or more generally \\gamma\\in[0,\\gamma_H). This part of Theorem 1.1 therefore needs either a proof adapted to the present parameter range or a precise reference whose assumptions match the claim.","section":"Theorem 1.1(II), case \\alpha=\\beta=0"}],"minor_comments":[{"comment":"The Hardy term in several displayed formulas is missing the division slash; it should read \\int_{\\mathbb{R}^n} u^2/|x|^{2s}\\,dx rather than \\int_{\\mathbb{R}^n} u^2|x|^{2s}\\,dx.","section":"Equations (2.3) and surrounding text"},{"comment":"The notation L^{p,\\frac{n-2s}{2}p+pr}(\\mathbb{R}^n,|y|^{-pr}) is ambiguous; it should be written as L^{p,\\frac{n-2s}{2}p+pr}(\\mathbb{R}^n,|y|^{-pr}) is ambiguous; it should be written as L^{p,((n-2s)/2)p+pr}(\\mathbb{R}^n,|y|^{-pr}) to make the Morrey exponent clear.","section":"Equations (0.2), (1.9), and Section 3"},{"comment":"There is a stray parenthesis in the domain: it should read u\\in\\dot{H}^s(\\mathbb{R}^n)\\setminus\\{0\\}, not u\\in\\dot{H}^s(\\mathbb{R}^n))\\setminus\\{0\\}.","section":"Equation (1.12)"},{"comment":"The coefficient in the second term of c^* should be parenthesized as (2s-\\beta)/(2(n-\\beta)), and the exponent as (n-\\beta)/(2s-\\beta), to avoid misreading.","section":"Proposition 5.2 and Theorem 1.1, definition of c^*"},{"comment":"The symbol 2^\\#_\\mu(\\alpha) is easily confused with a superscripted hash or with 2 times a number; introducing a separate symbol such as q_\\mu(\\alpha) for the Choquard exponent would improve readability.","section":"Notation throughout"},{"comment":"Theorem 5.5 states a p-Laplace analogue without proof. If it is intended as a mathematical result, a proof or a precise reference is needed; otherwise it should be labeled as a conjecture or an extension remark.","section":"Section 5, Theorem 5.5"}],"recommendation":"major_revision","confidential_remarks":"I do not recommend rejection because the missing attainment proof for \\Lambda appears repairable by the paper's own concentration-compactness-style argument, and the central variational strategy is coherent. The authors should be asked to supply Proposition 4.1(3)-(4) in full, to correct Proposition 5.3, and either to prove or to properly reference the \\alpha=\\beta=0 case with matching hypotheses. The relation to [2] and [41] should also be rechecked so that the claimed parameter ranges are justified in every branch of Theorem 1.1."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"Quick take: this is a plausible paper with one real proof gap. The new double-critical combination (Hardy-Sobolev plus weighted Choquard) is genuinely new, and the improved Sobolev inequality in weighted Morrey spaces is a useful tool in its own right. But Theorem 1.1 overclaims: the attainment of Λ for γ<0 is not proved, and the α=β=0 case is outsourced to a citation that does not cover the stated parameters.\n\nWhat's good: The mountain pass structure is standard but cleanly executed. The minimizer argument for S_μ in Proposition 4.1(1) is direct and convincing, using the new inequality to rule out vanishing at infinity. The strict inequality c<c* in Proposition 5.2 is fine—the uniqueness of the one-variable maximum rules out equality. Proposition 1.3 seems correct; the Sawyer-Wheeden proof is a bit compressed but the logic checks out. If the paper were restricted to parameter regimes where the required minimizers are actually proved to exist, I'd be comfortable with the main theorem.\n\nWhere it's soft: Proposition 4.1(3) and (4) are just asserted in Remark 4.2, and the cited [2] does not handle γ<0. That is load-bearing because Proposition 5.2 needs a Λ-minimizer V_{γ,β} for the same γ. The gap is probably fixable—the same concentration argument that works for S_μ should adapt to Λ—but as written the paper does not provide the proof. Less severe: Proposition 5.3 is a one-line sketch, and the α=β=0 case in Theorem 1.1(II) points to [41], whose parameter ranges (μ∈(n-2s,n), γ>0) do not include the claimed range. That part of the theorem is unsupported.\n\nVerdict: send to peer review, but the referee should insist on a complete proof of Proposition 4.1(3)-(4) (or a revised statement that only claims γ≥0 for the Λ threshold) and a proper treatment of the α=β=0 case. The paper is worth engaging with; the main idea is sound, and the improved inequalities are independently valuable.","headline":"A promising new double-critical fractional problem with a genuinely useful weighted Morrey inequality, but the main theorem overreaches because the Λ-minimizers for γ<0 are asserted, not proved.","tokens_in":27878,"tokens_out":8598,"would_cite":true,"duration_ms":87828,"reading_group":"maybe","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["35A01","35A23","35B33","35R11","35R70"],"pacs":[],"model":"deepseek-v4-flash","headline":"The paper proves that the doubly critical fractional Laplacian equation with a Hardy term admits at least one nontrivial weak solution in the parameter ranges of Theorem 1.1.","keywords":["fractional Laplacian","Hardy term","double critical exponents","nontrivial weak solution","weighted Morrey space","improved Sobolev inequality","mountain pass lemma","Riesz potential nonlinearity"],"falsifier":"Compute the Hardy–Sobolev ratio $\\Lambda(n,s,\\gamma,\\alpha)$ for a concrete negative $\\gamma$ (for example $n=3$, $s=1/2$, $\\alpha=1$, $\\gamma=-1$). If a minimizing sequence concentrates and the infimum is not achieved by any function in $\\dot H^s(\\mathbb{R}^n)$, then the assumed compactness in Proposition 4.1(3) fails and the strict mountain-pass inequality $c<c^*$ collapses.","tokens_in":26817,"feed_emoji":"📐","tokens_out":9140,"duration_ms":88557,"temperature":0.7,"pith_summary":"This paper aims to prove that a fractional Laplacian equation carrying two critical nonlinearities—a Hardy–Sobolev term and a Riesz-potential (convolution) term—has at least one nonzero weak solution whenever the Hardy coefficient $\\gamma$ stays below the critical constant $\\gamma_H$ (and, when both singular exponents $\\alpha,\\beta$ are positive, even for negative $\\gamma$). Standard compactness fails because both nonlinearities sit at the critical exponent, so the paper introduces a weighted Morrey embedding chain and an improved Sobolev inequality to control the Palais–Smale sequences directly in $\\dot H^s(\\mathbb{R}^n)$, without truncation or extension to the upper half-space. If correct, this broadens the admissible parameters relative to earlier results and gives a simpler proof of known existence theorems. The same mechanism is claimed to transfer to the p-Laplacian analogue.","feed_headline":"Double-critical fractional PDE admits a nontrivial weak solution","feed_subtitle":"Weighted Morrey embeddings replace concentration-compactness and even allow negative Hardy coefficients.","key_machinery":"The engine of the proof is the embedding chain $$\\dot H^s(\\mathbb{R}^n) \\hookrightarrow $L^{{2^{*}}$_s(\\$\\alpha$)}(\\mathbb{R}^n,|y|^{-\\$\\alpha$}) \\hookrightarrow $L^{{p,\\frac{n-2s}}${2}p+pr}(\\mathbb{R}^n,|y|^{-pr}),$$ where the middle space is a weighted Morrey space: it controls, on every ball, the $p$-integral of $|u|^p|y|^{-pr}$ with a scale factor that is exactly dilation invariant. Proposition 1.3's improved Sobolev inequality, $$\\left(\\int_{\\mathbb{R}^n}\\frac{|u(y)|^{$2^{{*}}$_s(\\$\\alpha$)}}{|y|^\\$\\alpha$}dy\\right)^{1/$2^{{*}}$_s(\\$\\alpha$)} \\le C\\|u\\|^\\theta_{\\dot H^s(\\mathbb{R}^n)}\\|u\\|^{1-\\$\\theta$}_{$L^{{p,\\frac{n-2s}}${2}p+pr}(\\mathbb{R}^n,|y|^{-pr})},$$ then bounds the critical Hardy–Sobolev norm by a product of the energy norm and a Morrey norm. This is what rules out the vanishing of (PS) sequences and yields the minimizers of $S_\\mu$ and $\\Lambda$ used to define the mountain-pass threshold $c^*$.","core_discovery":"The central claim, Theorem 1.1, is that the equation $$(-\\$\\Delta$)^s u - \\gamma \\frac{u}{|x|^{2s}} = \\frac{|u|^{$2^{{*}}$_s(\\$\\beta$)-2}u}{|x|^\\$\\beta$} + [I_\\mu * F_\\$\\alpha$(\\cdot,u)](x) f_\\$\\alpha$(x,u), \\qquad u \\in \\dot H^s(\\mathbb{R}^n),$$ has at least one nontrivial weak solution for $s\\in(0,1)$, $0\\le\\alpha,\\beta<2s<n$, $\\mu\\in(0,n)$, provided either $\\alpha,\\beta>0$ and $\\gamma<\\gamma_H$, or $\\alpha\\beta=0$ and $0\\le\\gamma<\\gamma_H$. The supporting discovery is the embedding chain $$\\dot H^s(\\mathbb{R}^n) \\hookrightarrow $L^{{2^{*}}$_s(\\$\\alpha$)}(\\mathbb{R}^n,|y|^{-\\$\\alpha$}) \\hookrightarrow $L^{{p,\\frac{n-2s}}${2}p+pr}(\\mathbb{R}^n,|y|^{-pr})$$ and the improved Sobolev inequality built on it, which are strong enough to prevent the Palais–Smale sequence from vanishing and to produce minimizers for the variational constants that set the mountain-pass threshold.","pith_inferences":["Because the method relies only on dilation-invariant norm control, a natural testable extension is to systems or higher-order operators with several critical Riesz-potential terms; the p-Laplacian case is already sketched in the paper, but biharmonic or mixed-order analogues are not.","If the attainment of $\\Lambda(n,s,\\gamma,\\alpha)$ fails for some negative $\\gamma$, the existence claim could still be true through a different choice of mountain-pass path; the paper's proof, however, would need a new construction of $v_0$ that does not use the minimizer asserted in Proposition 4.1(3).","The sharp range of the exponent $\\theta$ in the improved Sobolev inequality could be probed numerically; classifying equality cases would give explicit threshold functions and possibly identify the extremals of the critical problem."],"forward_implications":["Every parameter set in case (I) of Theorem 1.1, including negative Hardy coefficients $\\gamma<0$, yields a nonzero weak solution in $\\dot H^s(\\mathbb{R}^n)$.","Every parameter set in case (II), where at least one of $\\alpha,\\beta$ is zero, yields the same conclusion for $0\\le\\gamma<\\gamma_H$; the fully unweighted case $\\alpha=\\beta=0$ is recovered through the Nehari-manifold route.","The improved Sobolev inequality (1.10) makes the earlier truncation and extension-space arguments unnecessary in the fractional setting, giving a direct proof of the corresponding existence results it extends.","Corollary 1.4 transfers the same machinery to the p-Laplacian equation with two critical exponents, producing Theorem 5.5."],"supporting_citations":[{"why":"Supplies the fractional Hardy–Sobolev inequalities and the attainment of $\\Lambda$ for $0\\le\\gamma<\\gamma_H$, the prior result this paper extends and simplifies.","marker":"[2]"},{"why":"Establishes the p-Laplacian double critical problem and the truncation method whose $\\alpha=0$ case the paper adapts; also the target of Corollary 1.4.","marker":"[20]"},{"why":"Treats the doubly critical fractional equation with $\\alpha=0$ by the Nehari-manifold method and supplies the limit equation used for the $\\alpha=\\beta=0$ case.","marker":"[41]"},{"why":"Provides the improved Sobolev embeddings and Morrey-space framework that Proposition 1.3 generalizes.","marker":"[4]"},{"why":"Gives the fractional Hardy inequality and the best constant $\\gamma_H$ that defines the admissible $\\gamma$ range.","marker":"[27]"},{"why":"Provides the Hardy–Littlewood–Sobolev inequality used to define and bound the convolution term $B_\\alpha$.","marker":"[35]"},{"why":"Weighted Riesz-potential inequality used in the proof of the improved Sobolev inequality (1.10).","marker":"[29]"}],"fun_headline_variants":["Nontrivial weak solution for fractional Laplacian with Hardy term","Improved Sobolev inequality yields solution to double-critical PDE","Weighted Morrey embeddings solve fractional Hardy problem","Negative Hardy coefficients allowed: existence of weak solution","Direct proof: nontrivial solution to critical fractional PDE"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"The load-bearing premise is that the variational constants $\\Lambda(n,s,\\gamma,\\alpha)$ and $\\Lambda(n,s,\\gamma,0)$ are actually attained in $\\dot H^s(\\mathbb{R}^n)$ for the full claimed range of $\\gamma$, including negative values; the paper asserts this by analogy with a proven case rather than supplying the full proof.","fun_headline_variants_meta":{"raw":{"variants":["Nontrivial weak solution for fractional Laplacian with Hardy term","Improved Sobolev inequality yields solution to double-critical PDE","Weighted Morrey embeddings solve fractional Hardy problem","Negative Hardy coefficients allowed: existence of weak solution","Direct proof: nontrivial solution to critical fractional PDE"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000409,"raw_usage":{"total_tokens":2369,"prompt_tokens":1441,"completion_tokens":928,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":1057,"completion_tokens_details":{"reasoning_tokens":848}},"tokens_in":1057,"tokens_out":928,"duration_ms":9238,"temperature":1.0,"reasoning_tokens":848,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-14T14:43:10.699207+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"Compute the Hardy–Sobolev ratio $\\Lambda(n,s,\\gamma,\\alpha)$ for a concrete negative $\\gamma$ (for example $n=3$, $s=1/2$, $\\alpha=1$, $\\gamma=-1$). If a minimizing sequence concentrates and the infimum is not achieved by any function in $\\dot H^s(\\mathbb{R}^n)$, then the assumed compactness in Proposition 4.1(3) fails and the strict mountain-pass inequality $c<c^*$ collapses.","supporting_citations":[{"cited_title":"Ghoussoub, S","cited_arxiv_id":null,"evidence_quote":"Supplies the fractional Hardy–Sobolev inequalities and the attainment of $\\Lambda$ for $0\\le\\gamma<\\gamma_H$, the prior result this paper extends and simplifies."},{"cited_title":"Filippucci, P","cited_arxiv_id":null,"evidence_quote":"Establishes the p-Laplacian double critical problem and the truncation method whose $\\alpha=0$ case the paper adapts; also the target of Corollary 1.4."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Treats the doubly critical fractional equation with $\\alpha=0$ by the Nehari-manifold method and supplies the limit equation used for the $\\alpha=\\beta=0$ case."},{"cited_title":"Palatucci, A","cited_arxiv_id":null,"evidence_quote":"Provides the improved Sobolev embeddings and Morrey-space framework that Proposition 1.3 generalizes."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Gives the fractional Hardy inequality and the best constant $\\gamma_H$ that defines the admissible $\\gamma$ range."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Provides the Hardy–Littlewood–Sobolev inequality used to define and bound the convolution term $B_\\alpha$."},{"cited_title":"Sawyer, R","cited_arxiv_id":null,"evidence_quote":"Weighted Riesz-potential inequality used in the proof of the improved Sobolev inequality (1.10)."}],"review_version":1}