{"id":"93d61cee-5a85-49f6-9d18-f019bef2e16c","arxiv_id":"2506.07602","paper_version":1,"verdict":"CONDITIONAL","confidence":"MODERATE","novelty_score":7.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":0,"one_line_summary":"Nearly stationary functions for the Brezis-Nirenberg problem on bounded domains lie within a sharp, dimension-dependent distance of a solution plus bubbles, and the optimal exponents are identified.","lead":"This mathematics paper proves sharp quantitative stability estimates for the Brezis-Nirenberg problem, a classic equation tied to the critical Sobolev inequality on bounded domains. If correct, it gives the first optimal rates at which almost-solutions converge to a solution plus bubbles, including new exponents caused by the domain boundary.","discovery_kind":"new_application","skeptic_critique":{"model":"deepseek-v4-flash","headline":"Main theorems hinge on the unproved non-degeneracy of u0 (Assumption B); Lemma 4.2 and Prop 3.2 fail if the linearized operator has a kernel.","rationale":"I traced the logical chain of the paper and found that the main theorems are internally coherent: the compactness assumption in Theorem 1.1 makes |ξ_i−ξ_j| bounded, so the cross terms in Lemma 2.8 are comparable to q_ij and the induction in Proposition 4.4 works; the sign condition φ^3_λ<0 is used precisely to prevent sign competition; and the optimality constructions scale correctly (e.g., d∼δ^{1/2} in the boundary cancellation regime n=4,5,u0>0 gives the claimed t^{(n-2)/(n-1)}). I could not identify an internal contradiction. The load-bearing weakness is exactly the non-degeneracy of u0, as the reader flagged: it enters at the core of the linear theory (Lemma 4.2 and Proposition 3.2), is not proved in the paper, and Corollary 1.5 explicitly relies on it with the proof omitted. The paper's own candor about other limitations supports the trustworthiness of the estimates that are asserted, but the non-degeneracy dependence means the main theorems are conditional. Since the reader already assigned CONDITIONAL on this basis, my stress-test does not move the verdict.","tokens_in":66450,"tokens_out":20469,"duration_ms":220987,"concrete_test":"Verify whether the cited [36, Lemma 4.9] proves generic non-degeneracy of positive solutions of (1.1) for the operator −∆−λ−p u0^{p−1} on arbitrary smooth bounded domains for all λ∈(0,λ1). If the citation does not cover this setting, restate the theorems with non-degeneracy as an explicit hypothesis and either complete the proof of Corollary 1.5 (currently omitted in Remark 1.6) or withdraw it; as a further check, recompute the contradiction argument of Lemma 4.2 in the presence of a nonzero kernel element to see whether the conclusion ∥ϱ∥_{H^1_0} ≲ ∥h∥ actually fails.","verdict_should_be":"UNCHANGED","load_bearing_attack":"The stability estimates (1.10) and (1.12) are proved only for background solutions u0 satisfying Assumption B, in particular the non-degeneracy requirement that −∆−λ−p u0^{p−1} has trivial kernel in H^1_0(Ω). This hypothesis is used in two indispensable places: Lemma 4.2, where the contradiction argument passes to the limit and invokes non-degeneracy to conclude that the limit ϱ∞ vanishes, and Proposition 3.2, where a spectral gap for −∆−λ−2u0 (the n=6 case) is needed for the Green's-function bound. If u0 is degenerate, the linearized operator has a nontrivial kernel, and the blow-up argument produces a nonzero limit ϱ∞; the conclusion ∥ϱ∥_{H^1_0} ≲ ∥h∥ fails, so the final estimates (1.10) and (1.12) are not established. The paper cites [36, Lemma 4.9] for generic non-degeneracy but gives no proof, and Corollary 1.5—which assumes every positive solution is non-degenerate—has its proof explicitly omitted (Remark 1.6). Thus the main theorems are conditional on an unverified hypothesis, and the advertised 'first quantitative stability result on bounded domains' is not yet fully unconditional.","agreement_with_reader":"agree"},"referee_report":{"model":"deepseek-v4-flash","summary":"The paper proves sharp quantitative stability estimates for almost solutions of the Brezis–Nirenberg problem (1.1) in a smooth bounded domain. Under a closeness assumption on u (Assumption B) and a non-degeneracy assumption on the background solution u0, Theorems 1.1 and 1.3 show that the H^1_0 distance from u to a profile u0 plus projected bubbles is controlled by a dimension-dependent function ζ(Γ(u)), where Γ(u) is the H^{-1} norm of the equation's residual. The optimality of each displayed ζ is also asserted, with explicit constructions in Sections 4.2 and 5. The exponents include new sublinear regimes (e.g., t^{3/4} for n=5, u0=0; t|log t|^{1/2} for n=6; boundary-regime exponents t^{(n-2)/(n-1)} and t^{(n+2)/(2(n-1))}). The proof is a long chain of interaction estimates, a weighted linear theory for n=6 (Section 3), and blow-up-based linear estimates (Lemma 4.2).","tokens_in":66583,"tokens_out":12160,"duration_ms":145796,"significance":"If the main theorems are correct, this is the first quantitative stability result for the Sobolev inequality on bounded domains in the presence of the Brezis–Nirenberg linear term, and the new exponents genuinely reflect the interplay of the background solution, the linear term, bubble interactions, and the boundary. The paper is technically substantial: it introduces a representation-formula-based linear theory for n=6, a direct blow-up argument that avoids coercivity inequalities, and explicit optimality constructions with verifiable lower bounds. The main caveat is that the results are conditional on the non-degeneracy of u0 (Assumption B), which is not proved in the paper, and Corollary 1.5’s proof is omitted entirely. These issues currently prevent the paper from being fully unconditional.","major_comments":[{"comment":"The non-degeneracy of u0 (Assumption B) is load-bearing: Proposition 3.2 uses it to obtain the Green's function bound for -Δ-λ-2u0, and Lemma 4.2 uses it to force the limit ϱ∞ to vanish in the contradiction argument. The paper cites [36, Lemma 4.9] for generic non-degeneracy but neither states the lemma nor verifies its hypotheses for solutions of (1.1). Moreover, Remark 1.6 states that the proof of the advertised Corollary 1.5 is omitted. Thus the main theorems are conditional on an unproved hypothesis, and the application in Corollary 1.5 is not established within the paper. Please provide a precise statement and proof of the non-degeneracy property (or explicitly reformulate the theorems as conditional on it), and include the proof of Corollary 1.5.","section":"§1.2 (Assumption B), §3 (Prop. 3.2), §4.1 (Lemma 4.2)"},{"comment":"The optimality construction in the linear case selects ν points ξ_i with d(ξ_i,∂Ω)≳1 and |ξ_i-ξ_j|≳1 for all i≠j. In a fixed bounded domain, for arbitrarily large ν such a configuration may not exist, while the theorem allows ν up to the energy bound. This means the proof of optimality does not cover all ν admitted by the theorem, or the statement needs to be qualified. For the exponents in (1.11) that are linear, a single-bubble construction (ν=1) already yields the sharp rate in most cases, so a clarifying remark or a modified construction would resolve this gap.","section":"§4.2, Case 1 (sharpness of ζ(t)=t)"},{"comment":"In the boundary-regime proof of Theorem 1.3, the argument repeatedly splits into subcases based on inequalities such as b_n λ δ_1^2 > c_n φ(ξ_1) δ_1^{n-2}. The connection between these sign conditions and the resulting estimate on ∥ρ∥_{H^1_0} is terse; for instance, the derivation of κ_1^{n-1} ≲ ∥f∥_{H^{-1}} + δ_1^{(n+2)/(n-2)+2} in the equality case would benefit from an explicit display of the projection identities used. Please expand these passages so the dependency of the final exponent on the sign of the projection is transparent.","section":"§5, Step 1, especially the case n≥7"}],"minor_comments":[{"comment":"There is a typographical error in reference [40]: \"Brez ´ ıs\" should be \"Brezis\". Also, the hyphenation of \"Brezis-Nirenberg\" is inconsistent (e.g., \"Brezis–Nirenberg\" vs. \"Brezis-Nirenberg\"); please unify.","section":"References"},{"comment":"The proof of estimate (4.17) is delegated to \"the same reasoning as in [21, Lemma 2.3]\" without presenting the induction step. Since the setting here involves the additional terms from Lemma 2.8 and the condition φ^3_λ(ξ_i)<0, please include a sketch of the induction or at least state explicitly how the hypotheses are used at each step.","section":"§4.1, Proposition 4.4"},{"comment":"In the display after (2.10), the exponent for the n≥7 interaction term is δ_i^{-(n+2)/2} R^{2-n} in the first line but later the expression uses R^{-4}; please verify that the two forms are consistent after the change of variables and the definition of R in (2.3).","section":"§2.3, Lemma 2.5"},{"comment":"The weight functions w_{3i}^{in} and w_{3i}^{out} have a factor δ_i^2/d(ξ_i,∂Ω)^4 and δ_i/d(ξ_i,∂Ω)^3 respectively; in the proof of Proposition 3.3 the estimate (w_{3i}^{out})^2 / v_{3i}^{out} ≲ (δ_i/d(ξ_i,∂Ω))^4 contains a factor 1/|x_i| which is later dropped. Please add a short justification for this bound uniformly in |x_i|.","section":"§3, Definition 3.1"}],"recommendation":"major_revision","confidential_remarks":"The paper is extremely long and technically dense; I cannot verify every estimate in the allotted time, but the overall strategy is credible and the new exponents are surprising. The main concern is the unproved non-degeneracy assumption on u0 (Assumption B) and the explicitly omitted proof of Corollary 1.5. The authors should also clarify the optimality claim for large ν in the linear case, as discussed in Major Comment 2. These issues are fixable but are load-bearing for the advertised unconditional results."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"The paper is a real advance. It gives the first quantitative stability theorem for the Sobolev inequality on bounded domains in the critical-point setting, and it identifies genuinely new optimal exponents that come from the boundary and from the linear term—t^{3/4} for n=5, u0=0, t|log t|^{1/2} for n=6, and the boundary exponents in Theorem 1.3. The optimality constructions are explicit and the paper is unusually honest about its own limitations: it flags non-optimal intermediate bounds, an open multi-bubble boundary case, and missing pointwise estimates. That candor makes the asserted claims more credible.\n\nThe soft spots are real but not disqualifying. The central condition is non-degeneracy of the background solution u0 (Assumption B). The reader is right that this is load-bearing: Proposition 3.2 and Lemma 4.2 use it in essential ways, and if u0 is degenerate the linearized operator has a kernel and the estimates as stated are not established. The paper cites generic non-degeneracy but supplies no proof. That makes the main theorems conditional rather than fully unconditional, and the advertised \"first quantitative stability result on bounded domains\" should carry that qualifier. Separately, the proof of Corollary 1.5 is explicitly omitted, which is a minor gap since the corollary is an application rather than a main theorem, but it is still a missing proof in the text.\n\nI did not verify the long interaction estimates in Lemmas 2.5–2.8 and Section 3 line by line; they are dense but consistent with the established program and with the cited literature. The blow-up-based linear estimate in Lemma 4.2 is structurally sound, and the contradiction argument is standard. The new exponents emerge from the competition between interaction terms, and the sharpness sections construct functions whose distance-to-profile is comparable to the claimed zeta. No circularity or fitted constants.\n\nBottom line: this is serious work that deserves referee time. The referee should focus on the non-degeneracy hypothesis, the omitted proof of Corollary 1.5, and the key interaction estimates. I would bring it to a reading group and would cite it if I worked on stability of critical Sobolev inequalities.","headline":"Genuinely new sharp stability exponents for the Brezis–Nirenberg problem on bounded domains, but the main theorems are conditional on an unproved non-degeneracy assumption and the proof of Corollary 1.5 is omitted.","tokens_in":67312,"tokens_out":1287,"would_cite":true,"duration_ms":20098,"reading_group":"yes","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["35A23","35B35","35J08"],"pacs":[],"model":"deepseek-v4-flash","headline":"This paper proves sharp, dimension-dependent stability estimates for almost-solutions of the Brezis-Nirenberg problem on bounded domains: the $H^1_0$ distance to a solution plus projected bubbles is controlled by an optimal function of…","keywords":["quantitative stability","Sobolev inequality","Brezis-Nirenberg problem","Struwe decomposition","projected bubbles","critical Sobolev exponent","boundary effects","bubbling analysis"],"falsifier":"Take the unit ball in $\\mathbb{R}^5$, set $u_0=0$ and $\\lambda\\in(\\lambda_*,\\lambda_1)$, build the one-bubble plus perturbation example of Section 4.2 with scale $\\delta\\to0$, and compute the ratio of the $H^1_0$ distance to $\\Gamma(u)^{3/4}$; the claimed dimension-five exponent is correct exactly if this ratio stays bounded above and below by positive constants as $\\delta\\to0$.","tokens_in":66080,"feed_emoji":"🎯","tokens_out":14640,"duration_ms":151571,"temperature":0.7,"pith_summary":"This paper proves a sharp quantitative stability theorem for the Brezis-Nirenberg problem on smooth bounded domains. It says that if a nonnegative function $u$ is close, in $H^1_0$, to a genuine solution $u_0$ plus a finite sum of projected bubbles, then the error is controlled by an explicit function $\\zeta$ of the residual $\\Gamma(u)=\\|\\Delta u+\\lambda u+u^p\\|_{(H^1_0)^*}$, and no smaller function can replace $\\zeta$. The exponents are dimension-dependent and change when the bubble centers are allowed to approach the boundary, giving new rates such as $t^{3/4}$ in dimension five with $u_0=0$, $t|\\log t|^{1/2}$ in dimension six, and $t^{(n+2)/(2(n-1))}$ for boundary bubbles in higher dimensions. The authors present this as the first quantitative stability result for the Sobolev inequality on bounded domains, with the new boundary-dependent rates arising from the interaction of the linear term $\\lambda u$, the background solution $u_0$, and the bubbles with the boundary.","feed_headline":"Near-solutions of the Brezis-Nirenberg problem are sharply stable","feed_subtitle":"The residual controls the distance to a true solution plus bubbles, with new boundary-dependent rates.","key_machinery":"The machinery rests on three pieces. First, the residual functional $\\Gamma(u)=\\|\\Delta u+\\lambda u+u^p\\|_{(H^1_0)^*}$ measures how far an approximate solution is from solving the equation, and every estimate is phrased as distance $\\leq C\\zeta(\\Gamma)$. Second, the projected bubbles $P U_{\\delta,\\xi}$ are the standard critical bubbles adjusted to vanish on the boundary, either by solving $-\\Delta(PU)=U^p$ with zero boundary data or by solving its $\\lambda$-modified version $-\\Delta(PU)-\\lambda(PU)=U^p$; their dilation and translation modes $P Z^0=\\delta\\partial_\\delta P U$ and $P Z^k=\\delta\\partial_{\\xi_k}P U$ are the test functions that convert profile geometry into information about bubble scales. Third, the boundary enters through the function $\\varphi(\\xi)=H(\\xi,\\xi)$ (or its $\\lambda$-perturbation $\\varphi_\\lambda^n(\\xi)$), which behaves like $(2d(\\xi,\\partial\\Omega))^{-(n-2)}$ near the boundary; its gradient controls the translation-mode projections, while the dilation projection carries the leading boundary term $-\\delta^{n-2}/d(\\xi,\\partial\\Omega)^{n-2}$, and balancing these terms produces the boundary-dependent exponents in (1.13).","core_discovery":"The central claim is that the $H^1_0$ distance from an almost-solution to the set of profiles $\\{u_0+\\sum_{i=1}^{\\nu}P U_i\\}$ is bounded by $C\\zeta(\\Gamma(u))$, where $\\zeta$ is the piecewise function in (1.11) for interior centers and (1.13) for the single-bubble boundary case, and every displayed exponent is optimal. Theorem 1.1 handles any number of bubbles whose centers remain in a compact subset of $\\Omega$; Theorem 1.3 completely treats one bubble whose center may approach $\\partial\\Omega$; Corollary 1.5 turns the estimate into a global statement under the assumption that all positive solutions are non-degenerate and the energy is at most $(3/2)S_0^{n/2}$. The proof writes $u=u_0+\\sum P U_i+\\rho$, tests the equation for $\\rho$ against the projected dilation and translation modes of the bubbles, and uses a linear theory, in dimension six a weighted-norm and representation-formula argument, to control the main part of $\\rho$. Optimality is shown by constructing explicit nonnegative functions for which $\\Gamma(u)$ is small and the distance is comparable to $\\zeta(\\Gamma(u))$; the authors point out that the choice of projected bubble, whether it absorbs the linear term or not, can change the sharp exponent.","pith_inferences":["The multi-bubble case with centers approaching the boundary is left open; the paper identifies the obstruction as the competition between the boundary term $-\\delta^{n-2}/d^{n-2}$ and the bubble-bubble interaction $Q$, so a new mixed exponent could interpolate between (1.11) and (1.13).","The same strategy of testing by projected modes should transfer to other critical problems with a boundary, where one would predict similar boundary-driven modifications of the known Euclidean exponents.","For numerical approximation, the sharp $\\zeta$ gives a computable a posteriori criterion: if an approximate solution has $H^1_0$ error decaying slower than $\\zeta(\\Gamma)$, it cannot be near the genuine solution-bubble set, so the residual is the right quantity to monitor.","The projection dependence found in dimension five suggests that the distance to profiles is convention-dependent in low dimensions; applications should fix the projection rule first, or the stability rate itself is not well defined."],"forward_implications":["In the ranges $n=3,4$ with any number of bubbles, $n=5$ with $u_0>0$, and $n\\geq7$ with a single interior bubble, the distance-to-profile estimate is linear in $\\Gamma(u)$, meaning no logarithmic or fractional loss occurs.","In dimension five with $u_0=0$, dimension six, and $n\\geq7$ with several bubbles, the optimal rates are sublinear: $t^{3/4}$, $t|\\log t|^{1/2}$, and $t^{(n+2)/(2(n-2))}$, respectively.","For a single bubble near the boundary, the optimal rate shifts to $t^{(n+2)/(2(n-1))}$ for $n\\geq7$ and $t^{(n-2)/(n-1)}$ in dimensions four with $u_0>0$ and five, so the boundary itself changes the stability rate.","If every positive solution of (1.1) is non-degenerate, Corollary 1.5 gives a global stability statement for functions of bounded energy: a small residual $\\Gamma(u)$ forces quantitative closeness to some solution plus at most one projected bubble.","The sharp exponent can depend on which projected bubble is used; absorbing the linear term into the projection in dimension five with $u_0=0$ upgrades $t^{3/4}$ to linear stability, and both rates are sharp."],"supporting_citations":[{"why":"Supplies the sharp single-bubble stability estimate in R^n whose testing strategy and linearization the paper adapts to bounded domains.","marker":"[19]"},{"why":"Establishes sharp multi-bubble stability in R^n for 3≤n≤5, providing the baseline exponents and the treatment of bubble interactions.","marker":"[30]"},{"why":"Provides the sharp Struwe-decomposition estimates for n≥6, including the n=6 log-loss and the algebraic exponent for n≥7, and the bubble-tree and weighted-norm techniques reused here.","marker":"[21]"},{"why":"Supplies the sharp stability framework for the Yamabe problem on closed manifolds, including weighted-norm linear theory and contradiction arguments adapted to the Brezis-Nirenberg setting.","marker":"[14]"},{"why":"Contributes the representation-formula linear theory for the dimension-six case and the fixed-point construction of the main part of the error.","marker":"[15]"},{"why":"Gives Struwe's global compactness decomposition, the qualitative statement the paper makes quantitative.","marker":"[52]"},{"why":"Cited as the source that positive solutions of (1.1) are generically non-degenerate, the condition Assumption B requires when u0>0.","marker":"[36]"},{"why":"Provides the projected-bubble expansion and identifies the function controlling boundary behavior in the estimates.","marker":"[51]"}],"fun_headline_variants":["Brezis-Nirenberg stability: sharp exponents with boundary","Optimal rates for near-solutions to Brezis-Nirenberg","Boundary effects yield new stability exponents","Quantitative stability for Brezis-Nirenberg on bounded domains","Sharp control of distance to bubbles in Brezis-Nirenberg"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"The load-bearing premise is that, whenever the comparison solution $u_0$ is positive, it is an isolated critical point: the linearized equation about $u_0$ has no nonzero solution in $H^1_0$, and if that fails the estimates can collapse.","fun_headline_variants_meta":{"raw":{"variants":["Brezis-Nirenberg stability: sharp exponents with boundary","Optimal rates for near-solutions to Brezis-Nirenberg","Boundary effects yield new stability exponents","Quantitative stability for Brezis-Nirenberg on bounded domains","Sharp control of distance to bubbles in Brezis-Nirenberg"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000528,"raw_usage":{"total_tokens":2598,"prompt_tokens":1045,"completion_tokens":1553,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":661,"completion_tokens_details":{"reasoning_tokens":1468}},"tokens_in":661,"tokens_out":1553,"duration_ms":14428,"temperature":1.0,"reasoning_tokens":1468,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-07T05:31:16.948169+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"Take the unit ball in $\\mathbb{R}^5$, set $u_0=0$ and $\\lambda\\in(\\lambda_*,\\lambda_1)$, build the one-bubble plus perturbation example of Section 4.2 with scale $\\delta\\to0$, and compute the ratio of the $H^1_0$ distance to $\\Gamma(u)^{3/4}$; the claimed dimension-five exponent is correct exactly if this ratio stays bounded above and below by positive constants as $\\delta\\to0$.","supporting_citations":[{"cited_title":"Ciraolo, A","cited_arxiv_id":null,"evidence_quote":"Supplies the sharp single-bubble stability estimate in R^n whose testing strategy and linearization the paper adapts to bounded domains."},{"cited_title":"Figalli and F","cited_arxiv_id":null,"evidence_quote":"Establishes sharp multi-bubble stability in R^n for 3≤n≤5, providing the baseline exponents and the treatment of bubble interactions."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Provides the sharp Struwe-decomposition estimates for n≥6, including the n=6 log-loss and the algebraic exponent for n≥7, and the bubble-tree and weighted-norm techniques reused here."},{"cited_title":"Sharp quantitative stability of the Yamabe problem","cited_arxiv_id":"2404.13961","evidence_quote":"Supplies the sharp stability framework for the Yamabe problem on closed manifolds, including weighted-norm linear theory and contradiction arguments adapted to the Brezis-Nirenberg setting."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Contributes the representation-formula linear theory for the dimension-six case and the fixed-point construction of the main part of the error."},{"cited_title":"Struwe,A global compactness result for elliptic boundary value problems involving limiting nonlinearities, Math","cited_arxiv_id":null,"evidence_quote":"Gives Struwe's global compactness decomposition, the qualitative statement the paper makes quantitative."},{"cited_title":"Jin and J","cited_arxiv_id":null,"evidence_quote":"Cited as the source that positive solutions of (1.1) are generically non-degenerate, the condition Assumption B requires when u0>0."},{"cited_title":"Rey,The role of the Green ’s function in a nonlinear elliptic equation involving the critical Sobolev exponent, J","cited_arxiv_id":null,"evidence_quote":"Provides the projected-bubble expansion and identifies the function controlling boundary behavior in the estimates."}],"review_version":1}