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Sharp quantitative stability estimates for the Brezis-Nirenberg problem

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Pith's one-line read 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…

desk verdict 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. read the letter →

arxiv 2506.07602 v1 pith:WM24KAE5 submitted 2025-06-09 math.AP

Haixia Chen , Seunghyeok Kim , Juncheng Wei This is my paper
classification math.AP MSC 35A2335B3535J08
keywords quantitativestabilitySobolevinequalityBrezis-NirenbergproblemStruwedecompositionprojectedbubblescriticalexponentboundaryeffectsbubblinganalysis
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 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.

What carries the argument

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).

What would settle it

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 $\delta\to0$.

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Extended reading notes

Core claim

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.

Load-bearing premise

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.

Editorial extensions

If this is right

  • 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.

Reading between the lines

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

  • 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.
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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

3 major / 4 minor

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).

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 (3)
  1. [§1.2 (Assumption B), §3 (Prop. 3.2), §4.1 (Lemma 4.2)] 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.
  2. [§4.2, Case 1 (sharpness of ζ(t)=t)] 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.
  3. [§5, Step 1, especially the case n≥7] 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.
minor comments (4)
  1. [References] 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.
  2. [§4.1, Proposition 4.4] 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.
  3. [§2.3, Lemma 2.5] 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).
  4. [§3, Definition 3.1] 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|.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: the stability exponents are derived from the interaction estimates and sharpness constructions, with no fitted parameter renamed as a prediction.

full rationale

The paper's central estimates (1.10) and (1.12) are derived by quantifying Struwe's decomposition: after fixing the projected bubbles P Ui as minimizers, the remainder rho satisfies equation (2.1), and the proof bounds ||rho|| in terms of ||f|| plus the interaction terms I1, I2, I3. The exponents in (1.11) and (1.13) emerge from comparing the sizes of these terms, e.g. (4.14), (4.15), (5.3), and the case analysis following (5.4), rather than being imposed by any fitting constant. Optimality is verified by constructing functions whose distance to the profile is comparable to zeta(Gamma(u)), so the lower bound is an independent computation. Assumption B's non-degeneracy of u0 is a stated hypothesis, not an input secretly equivalent to the conclusion; it is used in the linear blow-up arguments (Lemma 4.2, Proposition 3.2) and is cited to [36, Lemma 4.9] only for generic validity. The self-citations to [14,15] describe proof blueprints and techniques, but the present estimates are carried out in full within the paper, and no load-bearing step reduces to an unverified self-citation. The omitted proof of Corollary 1.5 is a gap in exposition, not a circular reduction, and Remark 1.6 explicitly flags the added assumption. Consequently, no specific circular step can be exhibited, and the appropriate score is 0.

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

The load-bearing inputs of the central claim are: (i) the closeness profile of Assumption B, (ii) non-degeneracy of u_0, (iii) the sign condition phi^3_lambda < 0 for the n=3 multi-bubble interior case, and (iv) the standard analytic toolkit (Struwe compactness, bubble non-degeneracy, elliptic regularity). No constants are fitted to data: all exponents are derived from interaction estimates among u_0, bubbles, the linear term lambda u, and the boundary, and sharpness is proved by explicit examples. The paper's own prior works [14, 15] supply the proof blueprint, particularly the n=6 linear theory, but the estimate's exponents are not assumed in the hypotheses. The main caveat is that non-degeneracy is treated as generic rather than proved here, and the global corollary assumes it for all solutions.

assumptions (5)
  • standard math Struwe's global compactness (Theorem A): any bounded-energy approximate solution sequence converges, up to bubbles, to a solution u_0 of (1.1).
    Stated as Theorem A in Section 1.1 and cited to [52, 9, 4]; it defines the profile that the stability estimates measure.
  • standard math Aubin-Talenti bubbles are non-degenerate: the kernel of -Delta v - p U^{p-1} v in D^{1,2}(R^n) is spanned by the n+1 variations Z^0, Z^k.
    Invoked in the linear theory (Lemma 4.2, Proposition 3.2) and in the orthogonality conditions of (2.1); a classical result cited to [3, 54].
  • domain assumption The positive solution u_0 of (1.1) is non-degenerate (Assumption B).
    Required for the Fredholm-alternative and blow-up-contradiction arguments; the paper cites [36, Lemma 4.9] for genericity but gives no proof.
  • ad hoc to paper For n=3, u_0=0, nu>=2, the condition phi^3_lambda(xi_i) < 0 holds (Theorem 1.1 hypothesis).
    Imposed so that the leading terms in Lemmas 2.7 and 2.8 do not cancel (Remark 1.2(4)); not a theorem, a hypothesis.
  • domain assumption Every positive solution of (1.1) is non-degenerate (Corollary 1.5).
    Needed because Corollary 1.5 removes the a priori choice of u_0; the proof is omitted (Remark 1.6).

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Pith. "Pith review of Sharp quantitative stability estimates for the Brezis-Nirenberg problem." pith.science (2026). https://pith.science/paper/WM24KAE5

@misc{pith2026250607602,
  author       = {Pith},
  title        = {Pith review of: Sharp quantitative stability estimates for the Brezis-Nirenberg problem},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/WM24KAE5}},
  note         = {Machine review of arXiv:2506.07602}
}
abstract

We study the quantitative stability for the classical Brezis-Nirenberg problem associated with the critical Sobolev embedding $H^1_0(\Omega) \hookrightarrow L^{\frac{2n}{n-2}}(\Omega)$ in a smooth bounded domain $\Omega \subset \mathbb{R}^n$ ($n \geq 3$). To the best of our knowledge, this work presents the first quantitative stability result for the Sobolev inequality on bounded domains. A key discovery is the emergence of unexpected stability exponents in our estimates, which arise from the intricate interaction among the nonnegative solution $u_0$ and the linear term $\lambda u$ of the Brezis--Nirenberg equation, bubble formation, and the boundary effect of the domain $\Omega$. One of the main challenges is to capture the boundary effect quantitatively, a feature that fundamentally distinguishes our setting from the Euclidean case treated in \cite{CFM, FG, DSW} and the smooth closed manifold case studied in \cite{CK}. In addressing a variety of difficulties, our proof refines and streamlines several arguments from the existing literature while also resolving new analytical challenges specific to our setting.

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