{"id":"5f2f1049-38fb-4d10-ae48-4ded7f4307bc","arxiv_id":"2411.16257","paper_version":1,"verdict":"CONDITIONAL","confidence":"MODERATE","novelty_score":6.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":0,"one_line_summary":"The anisotropic p-Laplacian version of the Brezis-Nirenberg problem admits positive weak solutions under the same eigenvalue and dimension conditions as the isotropic case.","lead":"This paper shows that a classic nonlinear equation problem, the Brezis-Nirenberg problem, still has positive solutions when the Laplacian is replaced by a direction-dependent anisotropic version. The proof relies on sharp new estimates for the anisotropic versions of the functions that achieve the best Sobolev constant.","discovery_kind":"extension","skeptic_critique":{"model":"deepseek-v4-flash","headline":"Lemma 2.5's small-ε asymptotics have reversed powers; since Lemmas 3.1–3.3 invoke them, the energy bounds behind Theorems 1.2–1.4 are not established as written.","rationale":"I read the paper in good faith and the strategy is a sensible adaptation of the classical Brezis–Nirenberg compactness argument to the anisotropic p-Laplacian. The imported classification and sharp Sobolev inequality from [17] are standard and I found no concrete defect in that input; the shared author does not by itself make the result unreliable. However, the internal estimates that actually power the mountain-pass level do contain a systematic sign error: Lemma 2.5's displayed powers of ε are reversed relative to the scaling of ηε. This is not a subtle hypothesis issue but a directly checkable asymptotic; the subsequent equations (3.1)–(3.2) are consistent with the corrected signs, not with the stated lemma. This means the proof as submitted does not rigorously establish Theorems 1.2–1.4, although the errors appear mechanical and repairable. The reader's verdict of CONDITIONAL is therefore appropriate, and my independent concern supports that verdict rather than changing it. I would not recommend rejection or acceptance without revision; the natural next step is to correct the signs in Lemma 2.5 and re-verify Lemmas 3.1–3.3. The reader's own flagged issues (Lemma 3.2's mismatch and the q-range restriction in Lemma 3.3) are real and related, but the sign error in Lemma 2.5 is more load-bearing because it sits at the base of all three existence proofs.","tokens_in":24730,"tokens_out":28561,"duration_ms":252187,"concrete_test":"In the Euclidean case H=|·|, p=2, n=3, Ω=B_1, set ηε=(ε+|x|^2)^{−1/2}. Compute I(ε)=∫_{B_1}|∇ηε|^2 dx and J(ε)=∫_{B_1}ηε^6 dx as ε→0. Lemma 2.5 predicts I(ε)=C ε^{1/2}+O(1) and J(ε)=C ε^{3/2}+O(1); the change r=√ε s gives I(ε)∼c ε^{−1/2} and J(ε)∼c ε^{−3/2}. If the numerically observed or analytically derived powers are the negative ones, then (2.15)–(2.16) must be corrected, and Lemmas 3.1–3.3 should be rechecked against the corrected asymptotics.","verdict_should_be":"UNCHANGED","load_bearing_attack":"The paper's central energy mechanism rests on Lemma 2.5, but the displayed asymptotics (2.15) and (2.16) are not correct for the function ηε defined in (2.14). For ηε = φ(x)(ε + H0(−x)^{p/(p−1)})^{−(n−p)/p}, the change of variables x = ε^{(p−1)/p}y gives, for the main term, ∫ H(∇ηε)^p dx ∼ ε^{(p−n)/p} and ∫ ηε^{p*} dx ∼ ε^{−n/p}. Lemma 2.5 instead claims ε^{(n−p)/p} and ε^{n/p}. In the Euclidean case H=|·|, p=2, n=3, this is immediately visible: ηε=(ε+|x|^2)^{−1/2} satisfies ∫_{B_1}|∇ηε|^2 dx ∼ c ε^{−1/2} and ∫_{B_1}ηε^6 dx ∼ c ε^{−3/2}, whereas (2.15)–(2.16) predict the opposite powers. Consequently, the deductions (3.1)–(3.2) in Lemma 3.1 do not follow from (2.15)–(2.16) as written; the claimed O(ε^{(n−p)/p}) corrections are obtained only after the sign corrections are made. Since Lemma 3.2 and Lemma 3.3 also use these estimates to place the mountain-pass level below n^{−1}S_H^{n/p}, every existence theorem currently inherits this inconsistency. The flaw appears to be typographical and fixable, but the proof is not internally consistent in the submitted version.","agreement_with_reader":"partial"},"referee_report":{"model":"deepseek-v4-flash","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.","tokens_in":51,"tokens_out":20403,"duration_ms":359804,"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":[{"comment":"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.","section":"Section 2, Lemma 2.5, Eqs. (2.15)-(2.17), (2.23), (2.27)"},{"comment":"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.","section":"Section 3, Lemma 3.2 and Theorem 1.3"},{"comment":"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.","section":"Section 3, Lemma 3.3 versus Theorem 1.4"},{"comment":"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.","section":"Section 2, Lemma 2.9, Step i"}],"minor_comments":[{"comment":"The phrase 'pass to the limit for t→0+' should read 't→∞'; the limit variable is the Palais-Smale index, not a spatial scale.","section":"Section 2, Lemma 2.9, Step ii"},{"comment":"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.","section":"Section 3, Lemma 3.2, proof"},{"comment":"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.","section":"Section 3, Lemma 3.3, Case (i)"},{"comment":"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.","section":"Section 2, Lemma 2.5, proof of (2.18)"}],"recommendation":"major_revision","confidential_remarks":"The manuscript shares an author with [17], the source of the classification of anisotropic critical bubbles and the sharp Sobolev constant. This reliance is acceptable because [17] is a peer-reviewed external result and does not presuppose the present theorems; I see no circularity, though the editor may wish to be aware of the overlap. The paper fits the scope of the journal and the results are of genuine interest, but the internal inconsistencies described in the major comments are substantial enough that the current version should not be accepted without a careful revision and re-check of the proofs."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"The paper is the first to attack the Brezis-Nirenberg problem for the anisotropic p-Laplacian, and the overall strategy—mountain pass plus sharp estimates of truncated anisotropic Talenti bubbles—is the right one. The nonexistence result via the anisotropic Pohozaev identity is clean. But the central bubble estimates in Lemma 2.5 are not correct as written, and this is not a cosmetic issue.\n\nThe stress-test note is right. For ηε defined in (2.14), the change of variables gives the main terms of order ε^{(p-n)/p} for the H(∇·)^p integral and ε^{-n/p} for the L^{p*} integral. Lemma 2.5 states the opposite powers. The proof actually computes the correct exponent and then flips the sign in the final equality. This matters because Lemmas 3.1–3.3 use (2.15)–(2.16) to place the mountain-pass level below n^{-1}S_H^{n/p}. As written, the deductions (3.1)–(3.2) do not follow from the stated lemma. The good news is that after correcting the signs, (3.1)–(3.2) come out exactly as the authors need them. So this looks like a genuine typographical slip, but it has to be fixed before the existence theorems are justified.\n\nThere are two smaller issues the reader caught. Lemma 3.2 is stated for the truncated bubble vε but the proof only bounds the path along the first eigenfunction u1; the mountain-pass proof of Theorem 1.3 then invokes Lemma 3.2 for vε. This is a mismatch between statement and proof, again fixable by using the u1 path in the theorem. And Lemma 3.3 restricts q to p<q<p*-1 while Theorem 1.4 claims p<q<p*; the gap needs addressing.\n\nThe citation pattern looks fine. The classification of anisotropic bubbles and the sharp Sobolev constant are imported from [17], which is external and peer-reviewed; sharing an author does not by itself create a problem here.\n\nBottom line: the paper has the right architecture and the main results are probably true, but the submitted version has a load-bearing sign error in the key lemma plus two statement/proof mismatches. A serious referee should see it; I would recommend major revision rather than desk rejection.","headline":"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.","tokens_in":25615,"tokens_out":12210,"would_cite":false,"duration_ms":161725,"reading_group":"yes","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["35J62","35B33","35A15","35J92"],"pacs":[],"model":"deepseek-v4-flash","headline":"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.","keywords":["anisotropic p-Laplacian","Brezis-Nirenberg problem","critical Sobolev exponent","positive solutions","mountain pass theorem","Aubin-Talenti functions","Finsler geometry","Pohozaev identity"],"falsifier":"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.","tokens_in":24471,"feed_emoji":"📐","tokens_out":9437,"duration_ms":72002,"temperature":0.7,"pith_summary":"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 $p<q<p^*$ under the known dimensional threshold $\\kappa_{p,q}$. The linear case splits as expected: every $\\lambda \\in (0,\\lambda_1^H(\\Omega))$ works when $n\\ge p^2$, while the interval shrinks to $(\\lambda_1^H(\\Omega)-S_H|\\Omega|^{-p/n},\\lambda_1^H(\\Omega))$ when $p<n<p^2$. A nonexistence theorem rules out regular solutions on star-shaped domains when $\\lambda\\le 0$. The upshot is that the compactness-restoring mechanism of Brezis-Nirenberg operates unchanged once the Euclidean Aubin-Talenti bubbles are replaced by their anisotropic counterparts.","feed_headline":"Anisotropic p-Laplacian gets Brezis-Nirenberg solutions","feed_subtitle":"Classical eigenvalue thresholds carry over to Finsler geometry for linear and superlinear perturbations.","key_machinery":"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.","core_discovery":"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.","pith_inferences":["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."],"forward_implications":["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."],"supporting_citations":[{"why":"Supplies the original variational strategy, the mountain-pass framework, and the eigenvalue thresholds that the paper transfers to the anisotropic setting.","marker":"[10]"},{"why":"Provides the classification of positive finite-energy solutions of the critical anisotropic p-Laplacian and the sharp anisotropic Sobolev inequality with constant S_H, which give the bubbles (2.10) and the compactness threshold.","marker":"[17]"},{"why":"Gives the p-Laplacian analogue of the low-dimensional q=p result and the constant/path construction adapted for p<n<p^2.","marker":"[4]"},{"why":"Establishes the n≥p^2, q=p existence result for the p-Laplacian and the bubble-truncation technique used in Theorem 1.2.","marker":"[33]"},{"why":"Proves the p-superlinear p-Laplacian existence results and the dichotomous high/low dimension conditions used for Theorem 1.4.","marker":"[34]"},{"why":"Provides the anisotropic Pohozaev identity used to prove the nonexistence statement for λ≤0 on star-shaped domains.","marker":"[37]"}],"fun_headline_variants":["Brezis-Nirenberg thresholds hold for anisotropic p-Laplacian","Positive solutions for anisotropic p-Laplacian at critical growth","Anisotropic p-Laplacian: critical existence without compactness","Brezis-Nirenberg results extend to Finsler p-Laplacian"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"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.","fun_headline_variants_meta":{"raw":{"variants":["Brezis-Nirenberg thresholds hold for anisotropic p-Laplacian","Positive solutions for anisotropic p-Laplacian at critical growth","Anisotropic p-Laplacian: critical existence without compactness","Brezis-Nirenberg results extend to Finsler p-Laplacian"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000338,"raw_usage":{"total_tokens":1872,"prompt_tokens":952,"completion_tokens":920,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":568,"completion_tokens_details":{"reasoning_tokens":843}},"tokens_in":568,"tokens_out":920,"duration_ms":7197,"temperature":1.0,"reasoning_tokens":843,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-12T13:19:55.929745+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"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.","supporting_citations":[{"cited_title":"Br ´ ezis, L","cited_arxiv_id":null,"evidence_quote":"Supplies the original variational strategy, the mountain-pass framework, and the eigenvalue thresholds that the paper transfers to the anisotropic setting."},{"cited_title":"Ciraolo, A","cited_arxiv_id":null,"evidence_quote":"Provides the classification of positive finite-energy solutions of the critical anisotropic p-Laplacian and the sharp anisotropic Sobolev inequality with constant S_H, which give the bubbles (2.10) and the compactness threshold."},{"cited_title":"Arioli, F","cited_arxiv_id":null,"evidence_quote":"Gives the p-Laplacian analogue of the low-dimensional q=p result and the constant/path construction adapted for p<n<p^2."},{"cited_title":"Garc´ıa Azorero, I","cited_arxiv_id":null,"evidence_quote":"Establishes the n≥p^2, q=p existence result for the p-Laplacian and the bubble-truncation technique used in Theorem 1.2."},{"cited_title":"Garc´ıa Azorero, I","cited_arxiv_id":null,"evidence_quote":"Proves the p-superlinear p-Laplacian existence results and the dichotomous high/low dimension conditions used for Theorem 1.4."},{"cited_title":"Montoro, B","cited_arxiv_id":null,"evidence_quote":"Provides the anisotropic Pohozaev identity used to prove the nonexistence statement for λ≤0 on star-shaped domains."}],"review_version":1}