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REVIEW 4 major objections 4 minor 44 references

Brezis-Nirenberg-type results for the anisotropic $p$-Laplacian

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

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Pith's one-line read This paper extends the classical Brezis-Nirenberg existence theorem to the anisotropic p-Laplacian, proving positive weak solutions under the same eigenvalue thresholds as the isotropic case.

desk verdict First anisotropic Brezis-Nirenberg paper has the right strategy and likely-correct results, but the key bubble estimates in Lemma 2.5 have reversed powers and the proofs as written do not establish the theorems. read the letter →

arxiv 2411.16257 v1 pith:GVQMJYHB submitted 2024-11-25 math.AP

Stefano Biagi , Francesco Esposito , Alberto Roncoroni , Eugenio Vecchi This is my paper · ORCID
classification math.AP MSC 35J6235B3335A1535J92
keywords anisotropicp-LaplacianBrezis-NirenbergproblemcriticalSobolevexponentpositivesolutionsmountainpasstheoremAubin-TalentifunctionsFinslergeometryPohozaevidentity
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

This paper extends the classical Brezis-Nirenberg theorem from the $p$-Laplacian to the anisotropic $p$-Laplacian $-\Delta^H_p$, where the Euclidean norm is replaced by a convex, positively homogeneous norm $H$. It proves that the critical Dirichlet problem $-\Delta^H_p u = u^{p^*-1}+\lambda u^{q-1}$, with $u>0$ in a smooth bounded domain $\Omega$ and $u=0$ on the boundary, admits a weak positive solution for $q=p$ under exactly the same eigenvalue thresholds as in the isotropic case, and for $p

What carries the argument

The load-bearing object is the family of anisotropic Aubin-Talenti functions $$U_{\mu,x_0}^H(x) = \left(\frac{\$mu^{{1/(p-1)}}$c_{n,p}}{\$mu^{{p/(p-1)}}$ + H_0(x-x_0)^{p/(p-1)}}\right)^{(n-p)/p},$$ where $H_0$ is the dual norm of $H$ and the constants are chosen so that these functions solve the critical anisotropic equation $-\Delta^H_p u = u^{p^*-1}$ on $\mathbb{R}^n$ and attain the sharp anisotropic Sobolev constant $S_H$. The proof estimates suitable truncated versions $\eta_\varepsilon$ of these bubbles in $L^p$, $L^{p^*}$, and $L^q$ norms, obtaining the asymptotic expansions of Lemma 2.5 (including a $|\log\varepsilon|$ term at the borderline dimension $n=p^2$). These estimates feed into Lemmas 3.1–3.3, which construct mountain-pass paths whose energy stays below $S_H^{n/p}/n$; combined with the Palais-Smale compactness lemma at levels below that threshold, they yield the existence theorems.

What would settle it

Take a fixed non-Euclidean norm $H$ satisfying (i)–(v) on a ball in $\mathbb{R}^n$, with $q=p$ and $n=p^2$, and compute the $L^p$-gradient and $L^{p^*}$ norms of the truncated bubble $\eta_\varepsilon$ numerically for very small $\varepsilon$: if the leading terms of (2.15)–(2.16) do not match $S_H^{n/p}$ with the stated constants, Lemma 2.5 fails. Alternatively, solve (1.1) on a ball in $\mathbb{R}^3$ for $p=2$ and $\lambda$ slightly above $\lambda_1^H(\Omega)-S_H|\Omega|^{-2/3}$: if no positive solution appears, the low-dimensional threshold in Theorem 1.3 is false.

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Extended reading notes

Core claim

The paper's central claim is that problem (1.1) admits a nontrivial positive weak solution in three regimes, stated as Theorems 1.2, 1.3, and 1.4. For $q=p$ and $n\ge p^2$, existence holds for every $\lambda\in(0,\lambda_1^H(\Omega))$; for $q=p$ and $p<n<p^2$, existence holds for $\lambda$ in $(\lambda_1^H(\Omega)-\Lambda,\lambda_1^H(\Omega))$ with $\Lambda=S_H|\Omega|^{-p/n}$; for $p<q<p^*$, existence holds for every $\lambda>0$ when $n>\kappa_{p,q}$, and for all $\lambda\ge\lambda_0$ when $n\le\kappa_{p,q}$, where $\kappa_{p,q}=p[q(p-1)+p]/(q(p-1)+p-p(p-1))$. These are found by replacing Talenti's bubbles with the explicit anisotropic bubbles of (2.10) and showing that truncated versions of them push the mountain-pass energy below the compactness threshold $S_H^{n/p}/n$. The paper also proves that no regular solutions exist in star-shaped domains when $\lambda\le 0$, via the anisotropic Pohozaev identity.

Load-bearing premise

The results rest on the imported classification of all positive finite-energy solutions of the critical anisotropic $p$-Laplacian equation and on the sharp anisotropic Sobolev inequality with best constant $S_H$; if that classification or constant needs hypotheses beyond the stated convexity and regularity conditions, the energy thresholds used in the proof would not follow as written.

Editorial extensions

If this is right

  • For the $p$-linear perturbation with $n\ge p^2$, any $\lambda$ between $0$ and the first eigenvalue produces a positive weak solution.
  • In low dimensions $p<n<p^2$, existence requires $\lambda$ in an interval ending at the first eigenvalue, mirroring the classical Laplacian behaviour where small $\lambda$ fails.
  • For $p<q<p^*$, every $\lambda>0$ is admissible in dimensions above $\kappa_{p,q}$, while dimensions at or below the threshold need $\lambda$ sufficiently large.
  • The anisotropic Pohozaev identity excludes $\lambda\le 0$ on star-shaped domains, so the sign condition in the existence theorems is necessary for regular solutions.
  • The asymptotic expansions for truncated anisotropic bubbles are stated as reusable estimates for further critical problems in Finsler geometry.

Reading between the lines

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

  • Beyond the paper, the same energy-threshold method should transfer to other Finsler-type operators, such as anisotropic $(p,q)$-Laplacians, whenever a sharp Sobolev inequality with explicit extremals is available.
  • Beyond the paper, the $|\log\varepsilon|$ asymptotics at $n=p^2$ suggest that the borderline dimension has logarithmically small energy gaps, which might imply quantitative stability or nondegeneracy properties that could be checked by direct computation.
  • Beyond the paper, the dependence on the bubble family and $S_H$ suggests that a sharper lower bound on the mountain-pass energy could yield a precise value of $\lambda_0$ in the superlinear low-dimensional case.
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Editorial analysis

A structured set of objections, weighed in public.

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

Referee Report

4 major / 4 minor

Summary. The paper studies the anisotropic Brezis-Nirenberg problem (1.1) for the anisotropic p-Laplacian associated with a convex, positively homogeneous norm H. It proves existence of positive weak solutions for the critical-plus-lower-order problem in three regimes: q=p with n≥p^2 and λ∈(0,λ_1^H); q=p with p<n<p^2 and λ∈(λ_1^H-S_H|Ω|^{-p/n}, λ_1^H); and p<q<p^* with dimension-dependent thresholds κ_{p,q}, using a mountain-pass argument. It also proves a nonexistence result for λ≤0 on star-shaped domains via the anisotropic Pohozaev identity. The main technical novelty is a set of asymptotic estimates for truncated anisotropic Aubin-Talenti bubbles, combined with a Palais-Smale compactness lemma below the critical energy S_H^{n/p}/n.

Significance. If the results are correct, the paper gives a natural and valuable extension of the classical Brezis-Nirenberg theory to Finsler/anisotropic geometry. The overall strategy is sound: the variational formulation, the use of the anisotropic Sobolev inequality, and the compactness threshold are all standard in spirit and imported correctly from the literature. The paper relies on the classification and sharp constant from [17], which is external and peer-reviewed; I see no circularity. However, several load-bearing statements in the submitted version are internally inconsistent: the scaling exponents in Lemma 2.5 are reversed, Lemma 3.2 proves a different path estimate from the one it states and uses, Lemma 3.3 covers a narrower q-range than Theorem 1.4 claims, and the Palais-Smale boundedness in Lemma 2.9 has a gap when q=p. These issues appear fixable and do not, in my assessment, invalidate the intended results, but they must be corrected before the argument is complete.

major comments (4)
  1. [Section 2, Lemma 2.5, Eqs. (2.15)-(2.17), (2.23), (2.27)] The asymptotic exponents in Lemma 2.5 are reversed. For ηε(x)=φ(x)(ε+H0(-x)^{p/(p-1)})^{-(n-p)/p}, the change of variables x=ε^{(p-1)/p}y gives ∫ H(∇ηε)^p dx ∼ ε^{(p-n)/p}, not ε^{(n-p)/p}; ∫ ηε^{p*} dx ∼ ε^{-n/p}, not ε^{n/p}; and for n>p^2, ∫ ηε^p dx ∼ ε^{(p^2-n)/p}, not ε^{(n-p^2)/p}. The proof itself shows this: in (2.23) the intermediate expression has ε^{(p-n)/p} before being rewritten as ε^{(n-p)/p}. Because Lemma 3.1 and Lemma 3.3 use the displayed formulas of Lemma 2.5 to derive (3.1)-(3.2) and the L^q lower bound, the chain of estimates is not internally consistent as written. The final vε estimates in Section 3 are nevertheless recovered after correcting the signs, so this is a fixable but mandatory correction.
  2. [Section 3, Lemma 3.2 and Theorem 1.3] Lemma 3.2 is stated for the truncated bubble vε, but its proof computes sup_{t≥0} J_{p,λ}(tu1) along the first eigenfunction u1 and does not mention vε. Consequently, the statement 'there exists ε small such that sup_{t≥0} J_{p,λ}(tvε) < S_H^{n/p}/n' is not proved. In the proof of Theorem 1.3, Lemma 3.2 is invoked precisely to bound the mountain-pass level of paths ending at ¯t vε, so the level bound for the chosen path is missing. The proof can be repaired by taking the path along u1 directly, but as written the argument is incomplete.
  3. [Section 3, Lemma 3.3 versus Theorem 1.4] Theorem 1.4 claims existence for every p<q<p^*, but Lemma 3.3 is stated and proved only for p<q<p^*-1. The interval q∈[p^*-1,p^*) is not covered by the stated lemma, so Theorem 1.4 is not proved in its full claimed range. The lower bound (2.18) in Lemma 2.5 is also stated only for p<q<p^*-1, although the proof of (2.18) appears to work for all q<p^*. The authors should either extend the proofs to the full range p<q<p^* or restrict the statement of Theorem 1.4 accordingly.
  4. [Section 2, Lemma 2.9, Step i] The proof of boundedness of the Palais-Smale sequence uses the inequality qJ_{q,λ}(u_t)-J'_{q,λ}(u_t)[u_t] ≥ ((q-p)/p)‖u_t‖_{H,p}^p. For q=p the coefficient vanishes and the inequality provides no bound. A separate argument is needed for the q=p case, for example using J'_{q,λ}(u_t)[u_t]=o(1)‖u_t‖ together with the Sobolev and Hölder inequalities to control ‖u_t‖_{H,p}. Since Lemma 2.9 is applied for q=p in Theorems 1.2 and 1.3, this gap must be filled.
minor comments (4)
  1. [Section 2, Lemma 2.9, Step ii] The phrase 'pass to the limit for t→0+' should read 't→∞'; the limit variable is the Palais-Smale index, not a spatial scale.
  2. [Section 3, Lemma 3.2, proof] The displayed formula for max_{x≥0}(ax-bx^{n/(n-p)}) has a garbled exponent: the second factor should be raised to (n-p)/p, and the final value should be a^{(n-p)/p} times the appropriate constant; please correct the typography.
  3. [Section 3, Lemma 3.3, Case (i)] The equivalence between n>κ_{p,q} and (n-p)/p>β_{p,q,n} is stated without derivation; a short algebraic verification at this point would help the reader confirm the threshold.
  4. [Section 2, Lemma 2.5, proof of (2.18)] The lower bound (2.18) is obtained by restricting the integral to a fixed H0-ball and then letting the scaled ball grow; this is valid, but the sentence 'provided ε>0 is sufficiently small' should explicitly justify that the scaled integration domain contains a fixed ball independent of ε.

Circularity Check

0 steps flagged · score 2.0 of 10

No significant circularity: the derivation imports an external classification result and sharp Sobolev constant, and the one overlapping-author citation is independent support.

full rationale

The paper's derivation chain is not circular. The central test functions in Lemma 2.5 are the anisotropic Aubin-Talenti bubbles imported from Ciraolo-Figalli-Roncoroni [17], where their explicit form and the sharp anisotropic Sobolev constant S_H are proved. That theorem is external, peer-reviewed, and does not presuppose the present existence results; it is parameter-free and falsifiable. Although [17] shares an author with the present paper, per the reviewing rules this is real evidence and does not raise the circularity score. No parameter is fitted to the target result: the eigenvalue thresholds λ_1^H(Ω) and dimension thresholds κ_{p,q} are independent inputs, and the mountain-pass level S_H^{n/p}/n is the standard Sobolev threshold. The compactness lemma (2.9) and geometry lemma (2.6) are standard. I note as a correctness flag (not a circularity) that the displayed asymptotics (2.15)-(2.16) in Lemma 2.5 appear to have reversed powers relative to the change of variables x = ε^{(p-1)/p}y performed in the proof; this affects the internal consistency of the submitted proof but does not make the argument circular. Overall, no circular step reduces a claimed prediction to an input by construction.

Assumptions & free parameters 0 free parameters · 5 assumptions · 0 invented entities

There is no curve fitting or numerical calibration. The only constants are geometric, such as the best Sobolev constant S_H, the first eigenvalue lambda_1^H, and the volume |Omega|, and thresholds derived by inequalities. The main imported ingredient is the classification of anisotropic bubbles from [17], which shares an author with this paper but is a published external result.

assumptions (5)
  • domain assumption The norm H satisfies (i)-(v), including uniform convexity of the Wulff shape B^H_1, which implies the uniform ellipticity condition (v').
    This is the setting of the paper; it guarantees the regularity of the dual norm H0 and the validity of the anisotropic Sobolev machinery.
  • domain assumption The classification of all positive finite-energy solutions of the critical anisotropic p-Laplacian equation (2.9), and the explicit form of the extremals U^H_mu,x0 in (2.10), are correct as stated in [17].
    Imported at the start of Section 2.ii; this is the nontrivial input that supplies the bubbles and the sharp constant S_H used in all energy estimates.
  • domain assumption The anisotropic Sobolev inequality (2.11) holds with best constant S_H equal to the one computed in [17].
    Used throughout to identify the critical energy level S_H^{n/p}/n.
  • domain assumption The anisotropic Pohozaev identity in [37, Theorem 1.2] is valid for the regular solutions considered in Theorem 1.5.
    Used in the proof of the nonexistence result.
  • standard math The strong maximum principle and Hopf boundary lemma for the anisotropic p-Laplacian apply to the weak solutions produced by the mountain pass theorem.
    Imported from [41] and [13] to conclude that the critical point is a positive solution.

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Pith. "Pith review of Brezis-Nirenberg-type results for the anisotropic $p$-Laplacian." pith.science (2026). https://pith.science/paper/GVQMJYHB

@misc{pith2026241116257,
  author       = {Pith},
  title        = {Pith review of: Brezis-Nirenberg-type results for the anisotropic $p$-Laplacian},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/GVQMJYHB}},
  note         = {Machine review of arXiv:2411.16257}
}
abstract

In this paper we consider a quasilinear elliptic and critical problem with Dirichlet boundary conditions in presence of the anisotropic $p$-Laplacian. The critical exponent is the usual $p^{\star}$ such that the embedding $W^{1,p}_{0}(\Omega) \subset L^{p^{\star}}(\Omega)$ is not compact. We prove the existence of a weak positive solution in presence of both a $p$-linear and a $p$-superlinear perturbation. In doing this, we have to perform several precise estimates of the anisotropic Aubin-Talenti functions which can be of interest for further problems. The results we prove are a natural generalization to the anisotropic setting of the classical ones by Brezis-Nirenberg \cite{BN}.

Discussion (0). Continue with ORCID to comment.

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