On the double critical Maxwell equations

Cong Wang; Jiabao Su

REVIEW 3 major objections 4 minor 52 references

On the double critical Maxwell equations

T0 review · 3 major / 4 minor · reviewed 2026-08-12 · deepseek-v4-flash

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Pith's one-line read 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.

desk verdict 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. read the letter →

arxiv 2411.13894 v1 pith:RC74FMJV submitted 2024-11-21 math.AP

Cong Wang , Jiabao Su This is my paper · ORCID

classification math.AP MSC 35A1535B3335Q61

keywords doublecriticalMaxwellequationcurl-curloperatorHardyexponentHardy-Sobolevgroundstatesolutionsnonexistencethresholdconcentration-compactnessNeharimanifold

verification ladder T0 review T1 audit T2 compute T3 formal

The pith

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The reading

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

What carries the argument

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.

What would settle it

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.

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Extended reading notes

Core claim

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.

Load-bearing premise

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.

Editorial extensions

If this is right

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).

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

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 $\bar\lambda$.

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Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Desk Editor's Note

read the 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

What 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.

The 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.

Second, 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.

Third, the 'Furthermore problems' section announces analogous results without proofs. That is acceptable as a remark, but it should be labeled as such.

One 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.

Who 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.

Referee Report

3 major / 4 minor

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).

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 (3)

[Section 3, Lemma 3.2] 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 4.2, Lemma 4.2 and Lemma 4.3] 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 4.1, proof of Theorem 1.5] 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.

minor comments (4)

[Section 2, Eq. (2.4)] 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 5.4, proof of Theorem 1.11] 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 3, Lemma 3.3] 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.
[Remark 1.7] 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.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: the solvability thresholds are derived from external inequalities, not from the paper's own outputs, and self-citations are contextual.

full rationale

The derivation chain is not circular. The existence results in Theorems 1.2, 1.3, and 1.4 are proved by variational arguments (Nehari manifold, mountain pass theorem, concentration-compactness) using standard external tools: the Hardy-Sobolev inequality (2.2), the ground state result of Gaczkowski-Mederski-Schino [20], and Willem's minimax theory [51]. The nonexistence threshold λ* in Theorem 1.5 is obtained by applying the interpolation inequality of Lemma 3.2 with a specific exponent a0 and then optimizing a Young-inequality parameter γ; the threshold is therefore a consequence of the stated inequality, not a quantity fitted to solutions of (1.1) and then renamed as a prediction. Theorem 1.6 uses the Jeanjean-Le Coz truncation method, again with external inequalities. The self-citations [48]-[50] appear only as contextual references to related scalar double-critical problems and do not carry any load-bearing step. The one notable weakness is Lemma 3.2 itself: it is stated without proof, refers to a 'changed version' of the Caffarelli-Kohn-Nirenberg inequality, and leaves the best constant ar{S} unspecified. The formula for λ* and the sign analysis in (4.1) are conditional on this lemma, so a failure of Lemma 3.2 would invalidate Theorem 1.5 and parts of the Section 4 existence proof. That is a substantive correctness or rigor concern about an auxiliary external inequality, but it is not circular: the conclusion 'λ<λ* implies only the zero solution' does not reduce to the hypothesis 'Lemma 3.2 holds' by construction, and no fitted parameter is disguised as a prediction. Accordingly the appropriate circularity score is 0.

Assumptions & free parameters 2 free parameters · 4 assumptions · 0 invented entities

The paper introduces no new physical entities. The function spaces X, X_SO, D_F and the thresholds lambda*, lambda** are mathematical constructs built from existing theory. The main load-bearing input is Lemma 3.2, which is a quoted but unproved inequality, plus the symmetry reduction and compact embedding assumptions.

free parameters (2)

a0 = a0 = (s1-s2)/((2-s2)(3-s1))
Chosen by hand within the allowed range in Lemma 3.2 to make the nonexistence proof go through. It is an ad hoc interpolation exponent, not derived from data or a uniqueness theorem.
gamma (in Theorem 1.5 proof) = Algebraic expression in s1, s2, lambda, and bar{S}, chosen to make the two bracketed coefficients in (4.1) sign-definite
Auxiliary parameter introduced in the proof of Theorem 1.5 to force the coefficient of the ||u||^2 term positive and the coefficient of the weighted L^p term negative when lambda < lambda*.

assumptions (4)

domain assumption Lemma 3.2: the modified Caffarelli-Kohn-Nirenberg interpolation inequality holds on X_SO with the stated exponent range and some finite constant bar{S}.
Stated as a lemma but not proved; says it follows from a 'suitable transform' of functions in [14]. The nonexistence threshold lambda* in Theorem 1.5 depends directly on this inequality.
domain assumption The symmetry reduction: u in D_F solves (1.1) if and only if u in X_SO solves (1.17).
Invoked throughout via Remark 1.12 and the identities in (1.14); stated as a direct computation citing [20]. If false, all theorems in D_F would not follow from the X_SO results.
domain assumption Compact embeddings X_SO into weighted L^{6-2s_i}(B_{r,R}; |x|^{-s_i}) on annuli away from the singular axis hold.
Used in Lemmas 3.4, 3.8, and 4.4, and in the asymptotic proofs, to show vanishing of mass on annuli. Stated without proof.
domain assumption The known ground state result of [20] for the equation with s1=0, lambda=0 is available and supplies the baseline energy m0 in Section 5.
The asymptotic theorems converge to the ground state of the lambda=0 problem, whose existence is taken from [20] rather than re-proved here.

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Pith. "Pith review of On the double critical Maxwell equations." pith.science (2026). https://pith.science/paper/RC74FMJV

@misc{pith2026241113894,
  author       = {Pith},
  title        = {Pith review of: On the double critical Maxwell equations},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/RC74FMJV}},
  note         = {Machine review of arXiv:2411.13894}
}

abstract

In this paper, we focus on (no)existence and asymptotic behavior of solutions for the double critical Maxwell equation involving with the Hardy, Hardy-Sobolev, Sobolev critical exponents. The existence and noexistence of solutions completely depend on the power exponents and coefficients of equation. On one hand, based on the concentration-compactness ideas, applying the Nehari manifold and the mountain pass theorem, we prove the existence of the ground state solutions for the critical Maxwell equation for three different scenarios. On the other hand, for the case $\lambda<0$ and $0\leq s_2<s_1<2$, which is a type open problem raised by Li and Lin. Draw support from a changed version of Caffarelli-Kohn-Nirenberg inequality, we find that there exists a constant $\lambda^*$ which is a negative number having explicit expression, such that the problem has no nontrivial solution as the coefficient $\lambda<\lambda^*$. Moreover, there exists a constant $\lambda^*<\lambda^{**}<0$ such that, as $\lambda^{**}<\lambda<0$, the equation has a nontrivial solution using truncation methods. Furthermore, we establish the asymptotic behavior of solutions of equation as coefficient converges to zero for the all cases above.

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