Energy asymptotics in the three-dimensional Brezis--Nirenberg problem

Hynek Kovarik; Rupert L. Frank; Tobias K\"onig

Energy asymptotics in the three-dimensional Brezis--Nirenberg problem

T0 review · 0 major / 3 minor · reviewed 2026-08-14 · deepseek-v4-flash

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Pith's one-line read For critical potentials, small perturbations lower the sharp Sobolev energy quadratically, with the coefficient set by a nonlocal Green-function average.

desk verdict Solid, careful analysis that nails the 3D Brezis–Nirenberg energy gap at order ε^2; the reliance on Druet's theorem is real but standard. read the letter →

arxiv 1908.01331 v1 pith:T6CZ375U submitted 2019-08-04 math.AP

Rupert L. Frank , Tobias König , Hynek Kovarik This is my paper · ORCID

classification math.AP MSC 35J2035J6035B4046E35

keywords criticalSobolevexponentRobinfunctionzerosetGreen'sregularpartenergyasymptoticsalmostminimizerconcentrationnonlocalperturbationfunctionalthree-dimensionalinequality

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 studies the sharp Sobolev minimization problem in a bounded three-dimensional domain when the potential is critical—meaning its infimum already equals the sharp Sobolev constant and any pointwise smaller potential lowers it—and is then perturbed by a small multiple of an arbitrary bounded function. It proves that the first-order energy change vanishes, so the leading effect is a quadratic drop in the perturbation parameter, whose coefficient is a nonlocal average of the perturbation against the square of the Dirichlet Green function, optimized over the zero set of the Robin function. It also proves that almost minimizers concentrate at the optimizing point, with a blow-up rate proportional to the inverse perturbation size. The result matters because it identifies the precise order at which a critical threshold turns into strict subcritical behavior and gives a nearly sharp criterion for when a perturbation lowers the energy.

What carries the argument

The driving object is the zero set of the Robin function $\varphi_a(x)=H_a(x,x)$, the diagonal of the regular part of the Green function of $-\Delta+a$ with Dirichlet boundary conditions. Criticality forces $\varphi_a\ge0$ with a nonempty zero set, and the paper shows concentration must occur there. The new mechanism is the nonlocal functional $Q_V(x)=\int_\Omega V(y)G_a(x,y)^2\,dy$: negative values of $Q_V$ on $N_a$ select the concentration point and set the energy coefficient, so the perturbation acts through a Green-weighted average rather than pointwise. Technically, the argument is carried by projected bubbles $\psi_{x,\lambda}=PU_{x,\lambda}-\lambda^{-1/2}(H_a(x,\cdot)-H_0(x,\cdot))$ and by the coercivity of the quadratic form $\int_\Omega(|\nabla v|^2+av^2-15U_{x,\lambda}^4v^2)\,dy$ on the orthogonal complement of the finite-dimensional tangent space of the bubble ansatz; three successive applications of this coercivity extract the remainder structure to order $\epsilon^2$.

What would settle it

Work in the unit ball with the constant potential $a=-\pi^2/4$, which is critical and has $N_a=\{0\}$. For any bounded $V$ with $q_V=\int_B V(y)\cos^2(\pi|y|/2)/|y|^2\,dy<0$, the theorem predicts $S(a+\epsilon V)-S=-(3/S)^{1/2}(2\pi^4)^{-1}q_V^2\epsilon^2+o(\epsilon^2)$. A high-precision numerical minimization of the quotient over $H_0^1(B)$ for several small $\epsilon$ should reproduce this exact quadratic slope; a systematically different slope would falsify the theorem.

Watch

Extended reading notes

Core claim

The paper's central claim is a second-order energy expansion for the critical Sobolev quotient under a small potential perturbation. Let $G_a$ be the Dirichlet Green function of $-\Delta+a$, let $H_a(x,y)=|x-y|^{-1}-G_a(x,y)$ be its regular part, and set $\varphi_a(x)=H_a(x,x)$. For a critical $a$, one has $\varphi_a\ge0$ and the zero set $N_a=\{\varphi_a=0\}$ is nonempty. The paper proves that if $Q_V(x)=\int_\Omega V(y)G_a(x,y)^2\,dy$ is negative somewhere on $N_a$, then $S(a+\epsilon V)<S$ for all $\epsilon>0$ and $$S(a+\epsilon V)-S=-\left(\frac{3}{S}\right)^{1/2}\frac{1}{8\$pi^{2}$}\sup_{x\in N_a(V)}\frac{Q_V(x)^2}{|a(x)|}\$epsilon^{2}$+o(\$epsilon^{2}$),$$ with $N_a(V)=\{x\in N_a: Q_V(x)<0\}$. Moreover, almost minimizers concentrate at a point attaining the supremum and satisfy $\epsilon\lambda\to4\pi^2|a(x_0)|/|Q_V(x_0)|$. In the complementary case the drop is only $o(\epsilon^2)$, and if $Q_V$ is strictly positive on $N_a$ the energy stays exactly at $S$ for all sufficiently small $\epsilon$.

Load-bearing premise

The proof depends on a previously established result that, for a potential sitting exactly at the critical threshold, the minimization problem has no minimizer at all; without that result, the argument that almost minimizers must have zero weak limit, and hence the entire concentration decomposition, would collapse.

Editorial extensions

If this is right

If $N_a(V)\neq\emptyset$, the strict inequality $S(a+\epsilon V)<S$ holds for every $\epsilon>0$, and the energy drop is quadratic with the explicit leading coefficient given by the supremum of $Q_V^2/|a|$.
Any family of almost minimizers concentrates, along a subsequence, at a point in $N_a(V)$ maximizing $Q_V(x)^2/|a(x)|$, with the blow-up scale $\epsilon\lambda\to4\pi^2|a(x_0)|/|Q_V(x_0)|$.
If $N_a(V)=\emptyset$, the energy drop is only $o(\epsilon^2)$; if additionally $Q_V>0$ on $N_a$, the energy remains exactly at the Sobolev constant for all sufficiently small $\epsilon$.
The condition $N_a(V)\neq\emptyset$ is therefore almost necessary and sufficient for a small perturbation to lower the critical energy, with the only unresolved borderline being $\min_{N_a}Q_V=0$.
In the unit-ball example the whole formula becomes explicit: the coefficient is a universal constant times $q_V^2$, where $q_V=\int_B V(y)\cos^2(\pi|y|/2)/|y|^2\,dy$.

Reading between the lines

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

Because the coefficient is a Green-weighted average rather than a pointwise value, a sign-changing perturbation $V$ can lower the energy even where $V$ is locally positive, so the zero set of the Robin function acts as a nonlocal 'sensor' for the perturbation.
The iterative coercivity scheme is not obviously limited to two orders; pushing it further should produce the $\epsilon^3$ term and a systematic asymptotic expansion of $S(a+\epsilon V)$, likely with the same maximization principle applied to higher-order Green-function functionals.
The leftover borderline case $\min_{N_a}Q_V=0$ is a natural next target: there the $\epsilon^2$ coefficient vanishes, and one expects the next nonzero order to be governed by higher-order terms in the regular-part expansion around the zero set.
The same machinery could transfer to compact manifolds or to perturbations that depend on $\epsilon$ themselves, wherever a Dirichlet Green function and its Robin function are available.

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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

Short version: this is the right paper for the 3D case, and the proof holds up. The main theorem is the two-term expansion (1.6) with the ε^2 coefficient given by Q_V^2/|a|, plus the sharp selection rule for concentration points in Theorem 1.7. That is genuinely new: in N≥4 the first-order term survives, while here criticality kills it, so you need the quadratic correction. The paper works.

Where it earns credit: the upper bound is a clean trial-function argument; the lower bound is a three-stage bootstrap that refines the decomposition of almost minimizers, using coercivity twice more than previous arguments. The cancellation in Lemma 6.3, where the tangential components of the correction drop out, is the subtle step and it is handled carefully. The asymptotics for Theorem 1.4 are also a nice touch, showing the N_a(V)≠∅ condition is almost necessary.

Soft spots, in proportion: Lemma 2.6 is stated without proof; it is a routine analogue of Lemma 2.5, so this is minor, but a referee should ask for the proof. More importantly, the lower bound depends on Proposition 3.1, which in turn depends on Druet's non-attainment theorem and on Bahri–Coron. The stress-test note is right: if Druet's theorem were false, the decomposition would fail. But that theorem is a cornerstone of the field, and the paper identifies the dependence explicitly. This is a reliance on literature, not a hidden gap. The other heavy inputs—Esposito's coercivity and Rey's estimates—are standard and partly reproved.

Caveats: the estimates are numerous and not machine-checked, so confidence should be moderate, but I see no circular or fitted parameter anywhere. The constant in Lemma 2.7 has a typo ('As x→∞' should read 'As λ→∞'), trivial to fix.

For whom: anyone working on critical Sobolev problems, Brezis–Nirenberg asymptotics, or concentration of almost minimizers. It deserves a serious referee and, after minor revisions, publication.

Referee Report

0 major / 3 minor

Summary. This paper studies the three-dimensional Brezis–Nirenberg problem for the functional S(a+εV) with a critical function a in the sense of Hebey and Vaugon. The main result, Theorem 1.3, gives a two-term asymptotic expansion of S(a+εV)-S with leading order -(3/S)^{1/2}(8π²)^{-1} sup_{x∈N_a(V)} Q_V(x)²/|a(x)| ε², where Q_V is defined via the Green's function G_a and N_a(V) is the subset of the Robin-function zero set where Q_V<0. Theorem 1.4 treats the complementary case N_a(V)=∅, showing S(a+εV)=S+o(ε²) and equality for small ε when Q_V>0 everywhere on N_a. Theorem 1.7 characterizes almost minimizers: they concentrate at maximizers of Q_V²/|a| in N_a(V), with ελ → 4π²|a(x_0)|/|Q_V(x_0)|. The upper bound is obtained by explicit trial functions ψ_{x,λ}; the lower bound is obtained through a three-stage bootstrap (Sections 4–6) using coercivity of a quadratic form on the orthogonal complement of the approximate kernel, together with a concentration-compactness argument for almost minimizers in Appendix B.

Significance. Assuming the main results are correct, this is a significant contribution: it computes a higher-order (ε²) asymptotic in the critical case where the first-order term cancels, and it shows that the coefficient is non-local (through Q_V) rather than pointwise in V. The proof is careful: the upper bound is parameter-free and the lower bound tracks error terms explicitly to o(ε²). The paper also gives precise concentration-point selection for almost minimizers, going beyond earlier works that identify only N_a. Strengths include the explicit two-sided bounds in Theorem 2.1, the transparent three-step refinement of the decomposition of almost minimizers, and the honest identification of the external input from Druet's non-attainment theorem in Proposition 3.1. I find no circularity or parameter fitting in the argument, provided the cited external theorem is sound.

minor comments (3)

[Section 2.2, Lemma 2.6] The proof of Lemma 2.6 is omitted: the text says the proof is similar but simpler than that of Lemma 2.5. Since this lemma is used in the denominator expansion (2.3) and in the estimate leading to Lemma 6.6, the authors should supply the short proof or give a precise reference. In its present form this is a missing support for a step in the main argument; I do not regard it as fatal, but it should be fixed before publication.
[Section 6.4, around (6.18)] The symbols 'opλ2q' and 'opǫλ´1q' in the display (6.18) and in the surrounding text should presumably read o(λ^{-2}) and o(ελ^{-1}), respectively. Please correct these typographical errors so the order estimates are unambiguous.
[Appendix B, Step 1] The proof that u_ε converges weakly to zero imports Druet's non-attainment theorem [12, Step 1]. The authors note this dependence and that the relevant part of [12] requires only a∈L^{3/2}. For completeness, please state the imported theorem precisely and confirm that the hypotheses of the present setting (in particular, criticality in the sense of Definition 1.1 and the regularity of a) match those of [12, Step 1]. This is a clarification request about a load-bearing literature result, not an assertion of circularity.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: the asymptotic coefficient in (1.6) is computed from independently defined Green's-function data, and the cited non-attainment theorem is external support rather than a recycled input.

full rationale

The paper's main formula (1.6) expresses the ǫ² coefficient through Q_V, φ_a and a(x), all defined from the Green's function of −Δ+a before any asymptotic claim is made. The upper bound in Section 2 evaluates the quotient on explicit trial functions; the lower bound in Sections 4–6 uses coercivity, concentration-compactness and a three-step bootstrap, with no parameter fitted to the target coefficient. Proposition 3.1 and Appendix B establish the asymptotic form of almost minimizers using the external theorem of Druet that S(a) is not attained for critical a; this is explicitly identified and is a standard literature result, not a self-citation and not an input equivalent to the paper's conclusion. The paper also cites Esposito's coercivity lemma as an external tool, and the only self-citation (Ekholm–Frank–Kovařík, reference [14]) appears in an introductory comparison and is not load-bearing. No step was found in which a prediction reduces by construction to a fitted parameter or to a prior claim by the same authors, so the derivation is self-contained relative to its stated external inputs.

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

No free parameters appear in the proof. The main external inputs are Druet's non-attainment theorem, Esposito's coercivity lemma, and Rey's projection estimates, all from the cited literature and independent of the target result. The paper introduces no new postulated physical entities.

assumptions (5)

standard math Sharp Sobolev inequality with constant S = 3(π/2)^{4/3} and classification of optimizers U_{x,λ}(y) = λ^{1/2}/(1+λ^2|y-x|^2)^{1/2} as the only positive extremals.
Used as the benchmark in (1.1) and as the ansatz in the upper and lower bounds; invoked in Section 1.1 and Appendix B Step 2 via Lions, Talenti, Aubin, Rodemich, Rosen.
domain assumption Druet's non-attainment theorem: for critical a, S(a) is not attained.
Used in Appendix B, Step 1, to rule out a nonzero weak limit u_0 of the almost minimizers via equality in Brézis-Lieb around (B.1); without this, the concentration profile in Proposition 3.1 would not follow.
domain assumption Coercivity of the quadratic form on the orthogonal complement T^⊥_{x,λ}: Lemma 4.3, due to Esposito [15, Lem. 2.2], with the underlying non-degeneracy inequality from Rey [23, (D.1)].
The three-stage refinement of almost minimizers in Sections 4, 5, and 6 rests on this bound; the paper gives only a sketch and refers to [15] for details.
domain assumption Rey's projection estimates [23, Prop. 1] for PU_{x,λ} and the regular part H_0, including bounds (2.7), (2.18), and (5.10).
Used throughout to control boundary effects and the finite-dimensional tangential part T_{x,λ}; for example, (2.18) bounds ||f_{x,λ}||_8 and Appendix A uses (B.2)-(B.7).
domain assumption Regularity hypotheses: Ω open bounded with C^2 boundary, a ∈ C(Ω̄) ∩ C^1(Ω), V ∈ L∞(Ω), and Assumption 1.2 including a(x) < 0 on N_a.
Stated as Assumption 1.2; the C^2 boundary enters through the lower bound on dist(y,∂Ω) φ_0(y) in Proposition 4.1, and C^1(a) is used in Lemma 2.5 Step 2.

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Pith. "Pith review of Energy asymptotics in the three-dimensional Brezis--Nirenberg problem." pith.science (2026). https://pith.science/paper/T6CZ375U

@misc{pith2026190801331,
  author       = {Pith},
  title        = {Pith review of: Energy asymptotics in the three-dimensional Brezis--Nirenberg problem},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/T6CZ375U}},
  note         = {Machine review of arXiv:1908.01331}
}

abstract

For a bounded open set $\Omega\subset\mathbb R^3$ we consider the minimization problem $$ S(a+\epsilon V) = \inf_{0\not\equiv u\in H^1_0(\Omega)} \frac{\int_\Omega (|\nabla u|^2+ (a+\epsilon V) |u|^2)\,dx}{(\int_\Omega u^6\,dx)^{1/3}} $$ involving the critical Sobolev exponent. The function $a$ is assumed to be critical in the sense of Hebey and Vaugon. Under certain assumptions on $a$ and $V$ we compute the asymptotics of $S(a+\epsilon V)-S$ as $\epsilon\to 0+$, where $S$ is the Sobolev constant. (Almost) minimizers concentrate at a point in the zero set of the Robin function corresponding to $a$ and we determine the location of the concentration point within that set. We also show that our assumptions are almost necessary to have $S(a+\epsilon V)<S$ for all sufficiently small $\epsilon>0$.

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Pith tools

Reviewed August 14, 2026 · model on record in the stance chip above.

[1] [1]

M. Amar, A. Garroni, Γ -convergence of concentration problems. Ann. Sc. Norm. Super. Pisa Cl. Sci. (5) 2 (2003), no. 1, 151–179

work page 2003

[2] [2]

F. V. Atkinson, L. A. Peletier, Elliptic equations with nearly critical growth . J. Diﬀerential Equations 70 (1987), no. 3, 349–365

work page 1987

[3] [3]

Aubin, Problèmes isoperimétriques et espaces de Sobolev

Th. Aubin, Problèmes isoperimétriques et espaces de Sobolev . J. Diﬀer. Geometry 11 (1976), 573–598. 42 RUPERT L. FRANK, TOBIAS KÖNIG, AND HYNEK KOV AŘÍK

work page 1976

[4] [4]

Bahri, Critical points at inﬁnity in some variational problems

A. Bahri, Critical points at inﬁnity in some variational problems . Pitman Research Notes in Mathematics Series,

work page

[5] [5]

Bahri, J.-M

A. Bahri, J.-M. Coron, On a nonlinear elliptic equation involving the critical Sob olev exponent: the eﬀect of the topology of the domain . Comm. Pure Appl. Math. 41 (1988), no. 3, 253–294

work page 1988

[6] [6]

Brézis, Elliptic equations with limiting Sobolev exponents—the im pact of topology

H. Brézis, Elliptic equations with limiting Sobolev exponents—the im pact of topology. Frontiers of the mathemat- ical sciences: 1985 (New York, 1985). Comm. Pure Appl. Math. 39 (1986), no. S, suppl., S17–S39

work page 1986

[7] [7]

Brézis, E

H. Brézis, E. H. Lieb, A relation between pointwise convergence of functions and c onvergence of functionals . Proc. Amer. Math. Soc. 88 (1983), no. 3, 486–490

work page 1983

[8] [8]

Brézis, L

H. Brézis, L. Nirenberg, Positive solutions of nonlinear elliptic equations involv ing critical Sobolev exponents . Comm. Pure Appl. Math. 36 (1983), 437–477

work page 1983