{"id":"53fb1ed3-6e56-47e1-97a9-57ff5d1d9065","arxiv_id":"2411.13894","paper_version":1,"verdict":"CONDITIONAL","confidence":"MODERATE","novelty_score":6.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":2,"one_line_summary":"For a double critical curl-curl equation in R^3, ground states exist in several exponent regimes, and a negative coupling below a threshold forces the zero solution.","lead":"This paper proves existence and nonexistence results for a nonlinear Maxwell (curl-curl) equation in three dimensions with two competing critical singular terms. It shows that solvability flips depending on the sign and size of the coupling coefficient, and that solutions converge to the single-critical solution when the coefficient tends to zero.","discovery_kind":"extension","skeptic_critique":{"model":"deepseek-v4-flash","headline":"Unproved interpolation inequality Lemma 3.2 is load-bearing: Theorem 1.5's nonexistence threshold λ∗ and Lemma 3.10 both depend on it, but the lemma lacks a proof and the constant bar{S} is unspecified, so the claimed explicit threshold is not established.","rationale":"The reader's weakest assumption—Lemma 3.2 unproved—is indeed the principal load-bearing point. The paper's architecture is standard and the existence results for λ>0 and λ<0 with s1<s2 seem plausible, but Theorem 1.5's nonexistence threshold and parts of Theorem 1.4 rely on an unproved interpolation inequality with an unspecified constant. We also note the reversed inequality in Lemma 4.2 (the statement says λ<λ∗∗ while the proof only supports λ∈(λ∗∗,0)); this is a secondary issue, likely fixable, but it is a concrete sign error that should be corrected. Because these are gaps in proofs rather than demonstrated falsehoods of the main claims, the appropriate verdict remains CONDITIONAL: the paper should be accepted only after Lemma 3.2 is proved (or its constant made explicit) and the sign errors in Section 4 are fixed.","tokens_in":33757,"tokens_out":21698,"duration_ms":199866,"concrete_test":"Provide a complete proof of Lemma 3.2 for the exponent a0 used in (4.1), or produce a counterexample. A decisive check: compute the ratio R_n = B1(u_n)^{1/(6-2s1)} / ( A(u_n)^{a0/2} B2(u_n)^{(1-a0)/(6-2s2)} ) for a family u_n in X_SO concentrating at the origin with u_n vanishing linearly in |x'| on the axis. If sup_n R_n = ∞, Lemma 3.2 is false and Theorem 1.5 has no foundation; if sup_n R_n < ∞ for all such families, the missing proof should be supplied before the theorem's threshold λ∗ can be taken as established.","verdict_should_be":"UNCHANGED","load_bearing_attack":"Lemma 3.2 is stated as a 'changed version of the Caffarelli-Kohn-Nirenberg inequality' with no proof and an unspecified constant bar{S}. The proof of Theorem 1.5 (Section 4.1) applies it with the specific exponent a0 = (s1-s2)/((2-s2)(3-s1)) and then uses Young's inequality with a parameter γ to force the coefficients in (4.1) to have signs that yield <I'_λ(u),u> > 0 for all u. If Lemma 3.2 fails for this a0 on X_SO, or if bar{S} does not exist, then the signs in (4.1) cannot be guaranteed and the entire nonexistence conclusion (λ < λ∗ implies only zero solution) collapses. Lemma 3.2 is also used in Lemma 3.10 to infer B∞ > 0 in the existence proof for λ<0, 0<s1<s2<2, so the gap affects more than Theorem 1.5. The concern is not merely cosmetic: the a=1 case of Lemma 3.2 is just the Hardy-Sobolev inequality (2.2), but the a<1 interpolation with L^{6-2s2}(|x|^{-s2}) does not follow from (2.2) alone, since B2 can be much smaller than A; a nontrivial interpolation between X_SO and the weighted Lebesgue space is required. Without a proof (or a counterexample), the central claim that the solvability picture is 'complete up to an explicit threshold' is not established.","agreement_with_reader":"agree"},"referee_report":{"model":"deepseek-v4-flash","summary":"The paper studies the double critical Maxwell equation (1.1) with two Hardy-Hardy-Sobolev-Sobolev critical nonlinearities. After reducing the curl-curl problem to a scalar equation on the cylindrically symmetric space X_SO, the authors prove several existence results: ground states for λ>0 with 0≤s1<s2<2, for λ<0 with 0<s1<s2<2, and for 0≤s2<s1<2 with λ in (λ**,0). They also prove a nonexistence result for λ<λ*, where λ* is an explicit-looking negative constant depending on a constant from a stated interpolation inequality, and they establish asymptotic behavior of solutions as λ→0 in several regimes. The overall goal is a nearly complete solvability picture for (1.1).","tokens_in":34115,"tokens_out":15583,"duration_ms":140545,"significance":"If the main results were fully established, the paper would give a fairly complete answer to the sign-dependent solvability of a double critical Maxwell-type system and would partially address an open problem of Li-Lin type. The reduction to the scalar X_SO equation is standard and is handled cleanly, and several parts of the existence proofs follow classical Nehari-manifold and concentration-compactness arguments. The paper is also honest in Remark 1.7 about the cases that remain open. However, the central nonexistence theorem depends on Lemma 3.2, a nontrivial interpolation inequality that is stated without proof and with an unspecified constant; until that lemma is supplied with a complete proof and a quantitative or properly referenced constant, the threshold λ* and the claimed 'explicit' nonexistence result are not established. The paper does not appear to use fitted or circular parameters, and the constants are derived from known inequalities.","major_comments":[{"comment":"Lemma 3.2 is load-bearing for the main nonexistence theorem, but it is stated without any proof and with an unspecified constant \\bar{S}. The text says only that it is a 'changed version' of the Caffarelli-Kohn-Nirenberg inequality after 'a suitable transform' in [14]; neither the transform nor the derivation is given, so the admissible exponent range and the existence of a finite \\bar{S} cannot be verified. This lemma is used directly in the proof of Theorem 1.5 (Section 4.1, Eq. (4.1)) with the specific exponent a0=(s1-s2)/((2-s2)(3-s1)), and the signs in (4.1) determine the entire nonexistence conclusion. Moreover, the formula for λ* in Theorem 1.5 contains this same unspecified \\bar{S}, so the claim that λ* has an 'explicit expression' is not justified. The lemma is thus not a minor gap: without a proof (or a precise reference with conditions fully verified), the threshold result in Theorem 1.5 is unsupported.","section":"Section 3, Lemma 3.2"},{"comment":"There is a direction inconsistency in the truncation argument. Lemma 4.2 states that for S large there exists λ**=λ**(S)<0 such that for any λ<λ**, limsup_n ||u_n||<S/2. The proof, however, concludes at the end that inequality (4.15) fails for S>0 sufficiently large and 0>λ>-S^{2s2-6}, which is a condition on λ lying in a left-neighborhood of 0, i.e. λ∈(λ**,0), not λ<λ**. Lemma 4.3 then uses the conclusion on the interval (λ**,0), which is the opposite direction from the stated conclusion of Lemma 4.2. This makes the logical structure of the proof of Theorem 1.6 internally inconsistent as written.","section":"Section 4.2, Lemma 4.2 and Lemma 4.3"},{"comment":"The derivation of the key estimate (4.1) is not actually shown. The text says that from Lemma 3.2 with a=a0 one can 'directly obtain' the inequality, and then introduces a parameter γ with a specific definition, but the intermediate application of Young's inequality and the resulting exponents are not displayed. Since the sign conditions '1 - ... > 0' and 'λ + ... < 0' are exactly what makes the proof work, and since these signs depend on delicate choices of a0 and γ, the reader cannot check the algebra. This is particularly serious because the conclusion of Theorem 1.5 is the nonexistence for all λ<λ*, and a single sign error in (4.1) would invalidate it.","section":"Section 4.1, proof of Theorem 1.5"}],"minor_comments":[{"comment":"The first displayed identity after (2.4) repeats a limit with the value 1, but the preceding line has the same integral converging to S_{λ,s}(R^3); this appears to be a typo in the normalization condition.","section":"Section 2, Eq. (2.4)"},{"comment":"The proof writes 'there exists a sequence {λ_n>0}' even though Theorem 1.11 concerns λ<0; it should be λ_n<0. Also, the phrase 'as λ*<λ<0' seems to use λ* from the nonexistence theorem, whereas the intended interval is presumably λ**<λ<0 from Theorem 1.6.","section":"Section 5.4, proof of Theorem 1.11"},{"comment":"The notation S_s is introduced only for S_{0,s}(R^3), but Lemma 3.3 subsequently uses S_{s_1} and S_{s_2} without explicitly stating the identification; this should be made precise.","section":"Section 3, Lemma 3.3"},{"comment":"The list of open cases includes 'λ<0, 0=s_1<s_2<2', which is consistent with the hypotheses of Theorem 1.4 requiring 0<s_1<s_2<2, but the paper does not explicitly point out that the boundary case s_1=0 is excluded from Theorem 1.4 for a reason.","section":"Remark 1.7"}],"recommendation":"major_revision","confidential_remarks":"The manuscript has a promising structure and a plausible set of results, but the central nonexistence threshold rests entirely on the unproved Lemma 3.2, and the truncation argument in Section 4.2 contains a clear direction error. I would recommend asking the authors to provide a complete proof of Lemma 3.2 (or a precise reference with all conditions verified), to correct the statement of Lemma 4.2, and to rewrite the derivation of (4.1) before the paper can be further considered."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"This paper studies a curl-curl equation in R3 with two critical nonlinearities with different singular weights. The genuine advance is the sign-dependent solvability picture: existence for λ>0 when s1<s2, for λ<0 when s1<s2, and for 0≤s2<s1<2 with λ in (λ**,0), plus a nonexistence threshold λ* below which only zero exists. That goes beyond the single-critical case in [20] and connects to a Li-Lin-type open problem, though in the vector setting.\n\nWhat the paper does well: the variational framework is standard but applied carefully. Theorems 1.3 and 1.4, the truncation existence theorem, and the asymptotic results are mostly checkable, and Remark 1.7 honestly lists the cases left open. The self-citations are contextual and do not carry the derivation.\n\nThe soft spots, in proportion. The first is Lemma 3.2, a stated-but-unproved interpolation inequality of Caffarelli-Kohn-Nirenberg type with an unspecified constant \\bar S. This is load-bearing for Theorem 1.5: the nonexistence threshold λ* is derived from it by choosing an interpolation exponent a0 and a Young parameter γ. If the lemma fails for that exponent on the X_SO space, the threshold and the whole nonexistence claim collapse. The a=1 case is just Hardy-Sobolev, but the a<1 interpolation between the X_SO norm and the weighted L^{6-2s2} norm does not follow immediately, since the latter weighted norm can be much smaller than the former. The stress-test note is accurate here; the gap is real.\n\nSecond, Lemma 4.2 states the λ-direction backwards. Its statement says the bound holds for λ < λ**, but the proof actually establishes it for λ close to zero, i.e. λ** < λ < 0, which is the direction Theorem 1.6 needs. This looks like a sign typo and should be easy to fix, but it should not be left.\n\nThird, the 'Furthermore problems' section announces analogous results without proofs. That is acceptable as a remark, but it should be labeled as such.\n\nOne more note: because \\bar S in Lemma 3.2 is unspecified, λ* is not truly explicit until that constant is pinned down. The paper advertises an explicit expression, so this matters.\n\nWho is this for? People working on critical curl-curl problems, singular weights, and threshold phenomena. The paper deserves a serious referee: the problem is interesting, most results are likely correct, and the gaps are fixable in principle. I would send it to review with a strong request: prove Lemma 3.2 or give a precise, verifiable reference, and correct the direction in Lemma 4.2. If those are done, it becomes a solid contribution. As it stands, I would not cite the threshold result.","headline":"Double-critical curl-curl results that are plausible and interesting, but the nonexistence threshold rests on an unproved interpolation inequality, so the advertised complete picture is not yet established.","tokens_in":34644,"tokens_out":5953,"would_cite":false,"duration_ms":53814,"reading_group":"maybe","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["35A15","35B33","35Q61"],"pacs":[],"model":"deepseek-v4-flash","headline":"The paper establishes a nearly complete existence/nonexistence picture for the double critical Maxwell equation, with an explicit negative threshold below which only the zero solution survives.","keywords":["double critical Maxwell equation","curl-curl operator","Hardy critical exponent","Hardy-Sobolev critical exponent","ground state solutions","nonexistence threshold","concentration-compactness","Nehari manifold"],"falsifier":"Find a single nonzero solution to (1.1) for some $\\lambda<\\lambda^*$ with $0\\le s_2<s_1<2$; Theorem 1.5 would be false. More directly, test Lemma 3.2 on a family of cylindrically symmetric functions concentrating near the singular axis: if the claimed interpolation bound at the stated exponent $a_0$ is violated, every result built on it loses its support.","tokens_in":33559,"feed_emoji":"⚡","tokens_out":5328,"duration_ms":109049,"temperature":0.7,"pith_summary":"The paper aims to settle, for the double critical Maxwell equation $\\nabla\\times(\\nabla\\times u)=|u|^{4-2s_1}u/|x|^{s_1}+\\lambda|u|^{4-2s_2}u/|x|^{s_2}$ on $\\mathbb{R}^3$, when nontrivial solutions exist in the cylindrically symmetric divergence-free class $D^F$. It proves ground states for $\\lambda>0$ with $0\\le s_1<s_2<2$, for $\\lambda<0$ with $0<s_1<s_2<2$, and for $0\\le s_2<s_1<2$ with $\\lambda^{**}<\\lambda<0$; and it proves that for $\\lambda<\\lambda^*$, with an explicit negative $\\lambda^*$, only the zero solution exists. The proofs reduce the curl-curl operator to a scalar equation on an $SO(2)$-invariant subspace, then use the Nehari manifold, mountain pass arguments, a changed Caffarelli–Kohn–Nirenberg inequality, and a truncation device. The paper also shows that ground states converge to the $\\lambda=0$ ground state as $\\lambda\\to 0$. If correct, this gives a nearly complete sign-and-exponent-dependent solvability picture and partially answers an open problem in the literature.","feed_headline":"Existence and nonexistence mapped for critical Maxwell equations","feed_subtitle":"Ground states for most exponent pairs; below an explicit negative threshold only zero survives.","key_machinery":"The central object is the subspace $X_{SO}$ of $X=\\{u\\in D^{1,2}(\\mathbb{R}^3): \\int |u|^2/|x'|^2\\,dx<\\infty\\}$ consisting of functions invariant under $SO(2)\\{I\\}$; on this subspace the curl-curl equation becomes the scalar equation $-\\Delta u+u/|x'|^2=|u|^{4-2s_1}u/|x|^{s_1}+\\lambda|u|^{4-2s_2}u/|x|^{s_2}$. The argument is carried by the Nehari manifold homeomorphism, the mountain pass theorem, concentration-compactness arguments, and a 'changed version of the Caffarelli–Kohn–Nirenberg inequality' (Lemma 3.2) stated for $X_{SO}$ with an unspecified best constant $\\bar S$. That inequality is what produces the explicit threshold $\\lambda^*$ in the nonexistence proof, while the truncation/cut-off method is what forces bounded Palais–Smale sequences in the reversed-order negative-$\\lambda$ case.","core_discovery":"The central claim is that solvability of the double critical Maxwell equation in $D^F$ is governed by the ordering of the two critical exponents together with the sign of $\\lambda$. When $\\lambda>0$ and $0\\le s_1<s_2<2$, or when $\\lambda<0$ and $0<s_1<s_2<2$, a nontrivial ground state exists. When $0\\le s_2<s_1<2$ and $\\lambda<0$, the behavior is thresholded: for $\\lambda<\\lambda^*$, where $\\lambda^*$ is given by an explicit formula involving the best constant of a changed Caffarelli–Kohn–Nirenberg inequality, no nontrivial solution exists; for $\\lambda^{**}<\\lambda<0$ with $\\lambda^*<\\lambda^{**}$, a nontrivial solution exists. In all covered regimes, solutions converge to a ground state of the $\\lambda=0$ problem as the coefficient tends to zero. The paper therefore gives a nearly complete existence table, leaving only three listed cases open.","pith_inferences":["The explicit threshold $\\lambda^*$ depends on an unknown best constant $\\bar S$; if Lemma 3.2 were sharpened, the true threshold might have a simpler or optimal form, and the gap between $\\lambda^*$ and $\\lambda^{**}$ might close.","The sign-splitting mechanism likely appears in the higher-dimensional and scalar analogues proposed in the paper: existence for $\\lambda>0$ regardless of exponent order, but a negative threshold when the more singular exponent is the larger one.","The remaining open cases, especially $\\lambda\\ge \\bar\\lambda$ with $s_2=2$, are the ones where the singular potential is strongest; those likely require a different compactness mechanism because the present arguments rely on strict inequality below $\\bar\\lambda$."],"forward_implications":["If the theorems are correct, the solvability table for the double critical Maxwell equation in $D^F$ is complete except for the three cases listed in Remark 1.7.","The explicit formula for $\\lambda^*$ gives a checkable criterion: below this negative threshold no nontrivial cylindrically symmetric divergence-free solution can exist.","As $\\lambda\\to 0$, ground states for $\\lambda\\ne 0$ converge in $D^F$ to a ground state of the $\\lambda=0$ problem, giving a continuous limit of the solution family.","The truncation method supplies existence in the reversed-order negative-$\\lambda$ case even though the functional lacks the usual mountain pass geometry.","The same pattern of results is claimed to extend to the higher-dimensional analogues posed as problems (P1) and (P2)."],"supporting_citations":[{"why":"Supplies the reduction from the curl-curl equation to the scalar problem on $X_{SO}$ and the known $\\lambda=0$ ground state that the paper extends.","marker":"[20]"},{"why":"Poses the open problem about double critical exponents that Theorems 1.5 and 1.6 partially answer.","marker":"[31]"},{"why":"Source of the interpolation inequality that Lemma 3.2 adapts for $X_{SO}$ and that determines the explicit threshold $\\lambda^*$.","marker":"[14]"},{"why":"Provides the truncation/cut-off technique used to obtain bounded Palais–Smale sequences in the reversed-order negative-$\\lambda$ existence proof.","marker":"[29]"},{"why":"The Brezis–Lieb lemma used to split energies of Palais–Smale sequences and identify limit solutions.","marker":"[12]"},{"why":"Supplies the Nehari manifold and mountain pass machinery that equates the three critical levels used throughout.","marker":"[51]"}],"fun_headline_variants":["Critical Maxwell equations: existence map for all sign-exponent regimes","Solvability threshold found for double critical Maxwell equations","Explicit lambda boundary separates existence from none for Maxwell","Ground states exist except below explicit negative lambda threshold","Exponent ordering decides solvability of double critical Maxwell"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"The explicit threshold $\\lambda^*$ and the nonexistence theorem rest on an interpolation inequality (Lemma 3.2) whose constant $\\bar S$ is left unspecified and whose proof is not given; if that inequality fails, the whole nonexistence picture collapses.","fun_headline_variants_meta":{"raw":{"variants":["Critical Maxwell equations: existence map for all sign-exponent regimes","Solvability threshold found for double critical Maxwell equations","Explicit lambda boundary separates existence from none for Maxwell","Ground states exist except below explicit negative lambda threshold","Exponent ordering decides solvability of double critical Maxwell"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000315,"raw_usage":{"total_tokens":1804,"prompt_tokens":980,"completion_tokens":824,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":596,"completion_tokens_details":{"reasoning_tokens":746}},"tokens_in":596,"tokens_out":824,"duration_ms":6561,"temperature":1.0,"reasoning_tokens":746,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-12T15:46:11.455293+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"Find a single nonzero solution to (1.1) for some $\\lambda<\\lambda^*$ with $0\\le s_2<s_1<2$; Theorem 1.5 would be false. More directly, test Lemma 3.2 on a family of cylindrically symmetric functions concentrating near the singular axis: if the claimed interpolation bound at the stated exponent $a_0$ is violated, every result built on it loses its support.","supporting_citations":[{"cited_title":"Gaczkowski, J","cited_arxiv_id":null,"evidence_quote":"Supplies the reduction from the curl-curl equation to the scalar problem on $X_{SO}$ and the known $\\lambda=0$ ground state that the paper extends."},{"cited_title":"Caffarelli, R","cited_arxiv_id":null,"evidence_quote":"Source of the interpolation inequality that Lemma 3.2 adapts for $X_{SO}$ and that determines the explicit threshold $\\lambda^*$."},{"cited_title":"Jeanjean, S","cited_arxiv_id":null,"evidence_quote":"Provides the truncation/cut-off technique used to obtain bounded Palais–Smale sequences in the reversed-order negative-$\\lambda$ existence proof."},{"cited_title":"Br´ ezis, E","cited_arxiv_id":null,"evidence_quote":"The Brezis–Lieb lemma used to split energies of Palais–Smale sequences and identify limit solutions."},{"cited_title":"Willem, Minimax Theorems","cited_arxiv_id":null,"evidence_quote":"Supplies the Nehari manifold and mountain pass machinery that equates the three critical levels used throughout."}],"review_version":1}