On a Sobolev critical problem for the superposition of a local and nonlocal operator with the "wrong sign''

Enrico Valdinoci; Eugenio Vecchi; Serena Dipierro; Stefano Biagi

REVIEW 1 major objections 5 minor 11 references

On a Sobolev critical problem for the superposition of a local and nonlocal operator with the "wrong sign''

T0 review · 1 major / 5 minor · reviewed 2026-08-03 · deepseek-v4-flash

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Pith's one-line read A fractional Laplacian with the wrong sign still produces a nontrivial solution to the critical problem, with existence proved for every small coupling in dimensions at least five and for strong enough couplings in dimensions three and four

desk verdict Solid existence theorem for the wrong-sign mixed critical problem; main proof holds, with a sign typo in Definition 2.2 and a dropped normalization constant that need cleaning up. read the letter →

arxiv 2601.07521 v2 pith:QHE5VF5Z submitted 2026-01-12 math.AP

Stefano Biagi , Serena Dipierro , Enrico Valdinoci , Eugenio Vecchi This is my paper

classification math.AP MSC 35B3335R1135A1535A1649R05

keywords criticalSobolevexponentmixedlocal-nonlocaloperatorfractionalLaplacianwrongsignexistenceofweaksolutionsconstrainedminimizationcompactembeddingconstant

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 a critical equation on a bounded domain in which the standard Laplacian is combined with a fractional Laplacian carrying the opposite sign, so the operator is −Δ − γ(−Δ)^s. The authors aim to show that despite the 'wrong sign', the fractional term acts as a useful nonlocal perturbation: for every sufficiently small γ, a nontrivial weak solution exists in any bounded Lipschitz domain. They prove this in full for dimensions n≥5, and for dimensions 3 and 4 they prove existence for all γ above a certain (nonexplicit) threshold. The interest is that the critical problem, which for the pure Laplacian has no solution in a bounded domain without extra perturbations, becomes solvable merely by adding the wrong-sign fractional term.

What carries the argument

The carrying object is the energy functional Q_γ(u) = ∫|∇u|² − γ[u]_s², minimized on the unit sphere of L^{2*}. Two mechanisms do the work. First, a compactness step: if S(γ)<S_n, a minimizing sequence converges in the Hilbert space X^{1,2}(Ω), and the fractional seminorm converges strongly because X^{1,2}(Ω) embeds compactly into the fractional Sobolev space; a standard splitting lemma then shows the minimum is achieved. Second, the threshold comparison: either an explicit family of sharp-Sobolev-type test functions produces the asymptotic expansion [U_ε]_s² = ε^{2−2s}[V_1]_s² + O(ε²) that pushes Q below S_n, or a one-variable continuity argument on S(γ) forces the same conclusion near γ=C_

What would settle it

Numerically evaluate S(γ) on the unit ball in dimension 3 with s=1/2: a single γ<C_emb with S(γ)=S_n disproves Theorem 1.1(2), while an explicit bounded sequence in X^{1,2}(Ω) with non-convergent fractional seminorm would break Proposition 3.1.

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Extended reading notes

Core claim

The central claim is that the constrained minimum S(γ) = inf{∫|∇u|² − γ[u]_s² : ||u||_{L^{2*}}=1} falls strictly below the best constant S_n of the classical Sobolev inequality for a suitable range of γ, and that once it does, the minimum is attained and, after rescaling, solves the equation. In high dimensions n≥5, the drop is proved directly by testing with truncated sharp-Sobolev optimizers: the nonlocal term contributes a leading negative correction of order ε^{2−2s}, so the quotient is S_n − c γ ε^{2−2s} + o(ε^{2−2s}) < S_n for ε small. In dimensions 3 and 4 the same test is not decisive, and the proof instead shows that the function γ ↦ S(γ) is continuous, non-increasing, with limits S

Load-bearing premise

The proof stands on the compact embedding of the space X^{1,2}(Ω) into the fractional Sobolev space H^s(R^n); if that embedding were not compact, the fractional seminorm of a minimizing sequence could fail to converge strongly, and the constrained minimum in Proposition 3.1 would not be attained.

Editorial extensions

If this is right

For n≥5, problem (1.1) has a nontrivial weak solution for every γ in (0,C_emb); no extra perturbation term is needed to overcome the critical exponent.
For n=3,4, existence is guaranteed on the interval (γ*, C_emb), with γ*∈[0,C_emb) given by the first γ where S(γ) drops below S_n.
When the minimum is attained, the minimizer rescaled by S(γ)^{1/(2*−2)} is a solution; in particular, existence is obtained via a constrained variational principle rather than a fixed-point method.
The best constant S(γ) for the wrong-sign functional is attained, whereas the analogous constant for the local operator alone is not; the negative fractional term is precisely what restores compactness at the critical level.

Reading between the lines

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

[inference] The low-dimensional threshold γ* is probably the largest γ for which S(γ)=S_n; since the explicit test functions fail only because ε^{2−2s} is not small compared with ε^{n−2} when n=3,4, one would expect γ* to depend quantitatively on s and on Ω, and a numerical computation on a ball could locate it.
[inference] The same mechanism might yield existence for other wrong-sign combinations—e.g., two fractional Laplacians of different orders, or a wrong-sign lower-order term in a Dirichlet problem—whenever the sign-reversed term can lower the energy below the critical threshold without destroying coercivity.
[inference] The compact embedding of X^{1,2}(Ω) into the fractional space is load-bearing; if a non-compact domain or a rougher boundary weakened it, the attainment argument would need a different route even if the energy threshold were still below S_n.

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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: the main theorem is true and the proof is in good shape, but there are a few presentation slips, one in a definition, that need fixing.

New here is the sign: the mixed operator is -Δ - γ(-Δ)^s. All the cited mixed-operator critical problems use the plus sign. The main result separates high and low dimensions: n≥5 gives existence for every γ∈(0,C_emb); n=3,4 only for γ close to C_emb. That distinction is real and comes from the bubble expansion: for n≥5 the fractional correction ε^{2-2s} dominates the gradient error ε^{n-2}, so any γ>0 lowers the energy below S_n. In low dimensions that fails, and the implicit continuity/monotonicity argument is a sensible substitute.

What the paper does well: the variational setup is clean, and Proposition 3.1 is correct. I rechecked the algebra that drops the ||v_j||² term: the strict inequality S(γ)

Soft spots, in increasing order: - Definition 2.2 states the weak form with a plus sign: ⟨u,v⟩_ρ = ... + ∫∫, which is the equation for -Δ+(-Δ)^s, not for (1.1). The proof uses the correct form in (3.9). A sign typo in a definition is small but must be fixed. - In Lemma 3.6, the limit at γ→C_emb is written as S(γ)≤Q_γ(Φ_0)=C_emb-γ, but Φ_0 is not normalized in L^{2*}. The correct bound has denominator ||Φ_0||²_{L^{2*}}; since that denominator is fixed and positive, the conclusion stands. Harmless but false as written. - In the proof of Theorem 1.1(2), "(γ*, Sn)" should be "(γ*, Cemb)". - The compact embedding is cited rather than proved. It is standard and true, so I do not treat it as a gap.

Verdict: the paper deserves a serious referee. The result is solid, the flaws are typographical, and it extends the mixed local/nonlocal theory in a meaningful direction. Fix the definition and the normalization slip, and it should be accepted.

Referee Report

1 major / 5 minor

Summary. The paper proves an existence result for the critical mixed-order problem -Δu = γ(-Δ)^s u + |u|^{2*-2}u in a bounded Lipschitz domain Ω⊂R^n, with the fractional Laplacian appearing with the 'wrong sign'. The authors introduce the space X^{1,2}(Ω), the energy functional Q_γ(u)=∫|∇u|^2-γ[u]_s^2, and the constrained infimum S(γ) over the L^{2*} unit sphere. The main theorem states that S(γ) is attained below the Sobolev threshold S_n (and hence a nontrivial weak solution exists) for every γ∈(0,C_emb) when n≥5, and for γ near C_emb when n=3,4. The proof uses a Brezis-Nirenberg type compactness argument (Proposition 3.1), a sharp asymptotic expansion of the fractional seminorm of localized Talenti bubbles (Proposition 3.4), and a continuity/monotonicity argument for the low-dimensional case.

Significance. If the result holds, it is a valuable contribution to the literature on critical problems for mixed local/nonlocal operators. The idea of using a positively weighted fractional Laplacian with the 'wrong sign' to lower the energy below the critical Sobolev threshold is natural and is implemented cleanly in the high-dimensional case by an explicit asymptotic expansion. The low-dimensional treatment via continuity of S(γ) and the limits S(γ)→S_n as γ→0, S(γ)→0 as γ→C_emb is elegant and avoids delicate expansions. The paper is generally well written and the variational structure is sound, but there is a sign error in Definition 2.2 and a reliance on a compact embedding from a preprint that should be made explicit.

major comments (1)

[Definition 2.2] Definition 2.2 is inconsistent with problem (1.1). The definition requires ⟨u,v⟩_ρ = ∫∇u∇v + ∫∫(u(x)-u(y))(v(x)-v(y))/|x-y|^{n+2s} = ∫|u|^{2*-2}uv, which is the weak form of -Δu + (-Δ)^s u = |u|^{2*-2}u, not of (1.1). The correct equation, with the factor -γ in front of the fractional part, appears only later in (3.9). Consequently, Proposition 3.1’s final step does not show that the constructed ũ satisfies Definition 2.2 as written. Please replace ⟨u,v⟩_ρ by a γ-dependent bilinear form B_γ(u,v)=∫∇u∇v - γ∫∫(u(x)-u(y))(v(x)-v(y))/|x-y|^{n+2s} and use this form in the definition (and in the proof) so that it matches (1.1) and the Lagrange multiplier equation. This is a local correction, but it is essential for the statement of the main theorem.

minor comments (5)

[Remark 2.1(ii)] The wording 'since this embedding is actually compact' is ambiguous: the compactness is for X^{1,2}(Ω) into H^s(R^n), not for H^1(R^n) into H^s(R^n) on the whole space. Also, the result is cited to the preprint [9, Theorem 1.3]. As this compactness is used in an essential way in Proposition 3.1 and in the existence of Φ0 in Remark 2.1(iii), the authors should state the theorem precisely and, if [9] is not yet in final form, provide a proof or a publicly available version.
[Proof of Theorem 1.1(2)] There is a typo: the interval in item ii) and in the final sentence should be (γ*, C_emb), not (γ*, S_n). As written, the notation S_n is used both as the Sobolev constant and as a right endpoint of a γ-interval.
[Lemma 3.6] At the beginning of Step II, 'We now prove the limits in (3.15)' should refer to (3.16), since (3.15) is the monotonicity assertion and (3.16) contains the limits.
[Proposition 3.1] Minor typo in Step I: 'for very j∈N' should be 'for every j∈N'.
[Low-dimensional proof] The sentence 'either S(J)=(0,S_n) or S(J)=(0,S_n]' could be slightly expanded to clarify that the endpoints 0 and S_n may or may not be reached because J is open; the subsequent definition of γ* is correct, but the notation would benefit from this clarification.

Circularity Check

0 steps flagged · score 0.0 of 10

No circular reasoning: the existence proof reduces S(γ)<S_n to a genuine asymptotic computation; self-citations supply external technical lemmas, not the theorem being proved.

full rationale

The derivation chain is: define C_emb via (2.6), define S(γ) via (3.2), prove attainment if S(γ)<S_n (Prop 3.1), then prove S(γ)<S_n. The attainment step uses the compact embedding X^{1,2}(Ω)↪H^s(R^n), cited to [9, Thm 1.3] in Remark 2.1(ii). This is a self-citation (Dipierro and Valdinoci are coauthors), and it is load-bearing because it yields [u_j-u]_s→0 in Step I. However, it is a functional-analytic embedding theorem whose assumptions do not include the target PDE existence; there is no quote or equation in the paper reducing it to the result being proved, so it does not constitute circularity under the stated rules. The key sub-threshold estimate for n≥5 (Prop 3.5) uses Lemma 3.2(iv) ([U_ε]_s^2=O(ε^{2-2s}), cited to [2]) and Proposition 3.4 ([U_ε]_s^2=ε^{2-2s}[V_1]_s^2+O(ε^2)), whose proof is a direct computation with the Aubin–Talenti rescaling, not a fitted parameter. No quantity is fitted to the predicted conclusion. The low-dimensional case uses monotonicity, continuity and the limits S(γ)→S_n and S(γ)→0, with the latter following from the achiever Φ0 of C_emb; again no circular reduction. The sign error in Definition 2.2 (⟨u,v⟩_ρ=... instead of ...−γ...) is an internal inconsistency: the proof's (3.9) uses the correct minus sign, so the constructed weak solution solves (1.1) rather than Definition 2.2 as printed. This is a correctness/typo concern, not a case of the conclusion being assumed in the premise. Overall, the central claim has independent variational content and no fitted input is renamed as a prediction.

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

No free parameters are fitted to data; C_emb and γ* are defined by variational principles, not chosen by hand. The axioms are standard Sobolev-space facts, cited estimates, and the domain regularity assumption.

assumptions (5)

standard math Sharp Sobolev inequality: S_n ||u||_{2*}^2 ≤ ||∇u||², with equality at Aubin–Talenti bubbles.
Used in Lemma 3.2(i),(iii) and in the upper/lower bounds for S(γ); classical Talenti result [11].
standard math Brezis–Lieb lemma for L^{2*} convergence.
Used in Proposition 3.1 Step I to split ||u_j||_{2*}^{2*} into ||u|| and ||v_j|| parts; cited [5].
standard math Compact embedding X^{1,2}(Ω) ↪ H^s(R^n).
Needed to pass from weak convergence to strong convergence of [u_j−u]_s; cited to [9] (arXiv preprint by the authors).
standard math Asymptotic estimates for the Gagliardo seminorm of localized Aubin–Talenti functions: [U_ε]_s²=O(ε^{2−2s}) and cross-terms O(ε^{n−2}).
Lemma 3.2(iv) cites [2, Lemma 4.10]; Proposition 3.4 cites [10, Prop 21] for cross-terms. These expansions are load-bearing for the high-dimensional competitor.
domain assumption Bounded Lipschitz domain Ω; by translation assume 0∈Ω for the competitor construction.
Theorem 1.1 assumes Ω open bounded with Lipschitz boundary; the competitor in Proposition 3.5 requires B_{4δ}⊂Ω, which is WLOG.

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Pith. "Pith review of On a Sobolev critical problem for the superposition of a local and nonlocal operator with the "wrong sign''." pith.science (2026). https://pith.science/paper/QHE5VF5Z

@misc{pith2026260107521,
  author       = {Pith},
  title        = {Pith review of: On a Sobolev critical problem for the superposition of a local and nonlocal operator with the "wrong sign''},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/QHE5VF5Z}},
  note         = {Machine review of arXiv:2601.07521}
}

abstract

We study a critical problem for an operator of mixed order obtained by the superposition of a Laplacian with a fractional Laplacian. The main novelty is that we consider a mixed operator of the form $-\Delta- \gamma(-\Delta)^s$, namely we suppose that the fractional Laplacian has the ``wrong sign'' and can be seen as a nonlocal perturbation of the purely local case, which is needed to produce a nontrivial solution of the critical problem.

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Works this paper leans on

11 extracted references · 2 linked inside Pith

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Pith tools

Reviewed August 3, 2026 · model on record in the stance chip above.

[1] [9]

Dipierro, E

S. Dipierro, E. Proietti Lippi, C. Sportelli, E. Valdinoci,Optimal embedding results for frac- tional Sobolev spaces, arXiv:2411.12245, (2024). 4

arXiv 2024

[2] [1]

Barrios, L.M

B. Barrios, L.M. Del Pezzo, A. Quaas,Mixed local and nonlocal laplacian without standard critical exponent for Lane-Emden equation, preprint.https://arxiv.org/abs/2507.122582

arXiv

[3] [2]

Biagi, S

S. Biagi, S. Dipierro, E. Valdinoci, E. Vecchi,A Brezis-Nirenberg type result for mixed local and nonlocal operators, NoDEA Nonlinear Differential Equations Appl.32, (2025), art. 62. 2, 3, 4, 8, 9, 10

2025

[4] [3]

Biagi, E

S. Biagi, E. Vecchi,Multiplicity of positive solutions for mixed local-nonlocal singular critical problems, Calc. Var. Partial Differ. Equ.63, (2024), Article 221. 3

2024

[5] [4]

Biagi, E

S. Biagi, E. Vecchi,On the existence of a second positive solution to mixed local-nonlocal concave-convex critical problems, Nonlinear Anal.256, (2025), 113795. 3

2025

[6] [5]

Brezis, E

H. Brezis, E. Lieb,A relations between pointwise convergence of functions and convergence of integrals, Proc. Amer. Math. Soc.88, (1983), 486–490. 6

1983

[7] [6]

Brezis, L

H. Brezis, L. Nirenberg,Positive solutions of nonlinear elliptic equations involving critical Sobolev exponents, Comm. Pure Appl. Math.36(1983), no. 4, 437–477. 3, 5, 8

1983

[8] [7]

da Silva, A

J.V. da Silva, A. Fiscella, V.A. Blanco Viloria,Mixed local-nonlocal quasilinear problems with critical nonlinearities, J. Differential Equations408, (2024), 494–536. 9

2024