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Global Compactness Result for a Br\'ezis-Nirenberg-Type Problem Involving Mixed Local Nonlocal Operator

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Pith's one-line read In a mixed local-nonlocal problem, bubbles stay purely local

desk verdict The global compactness theorem is real and mostly sound; the Coron application has a fillable gap in transferring the Palais–Smale condition to the sphere. read the letter →

arxiv 2504.15968 v2 pith:5ZKTONVC submitted 2025-04-22 math.AP math.FA

Souptik Chakraborty , Diksha Gupta , Shammi Malhotra , Konijeti Sreenadh This is my paper · ORCID
classification math.APmath.FA MSC 35B3335B3835J2035J61
keywords globalcompactnessprofiledecompositionPSsequencemixedlocalnonlocaloperatorcriticalSobolevexponentfractionalLaplacianannulardomainhigh-energysolutions
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 global compactness (profile decomposition) theorem for PS sequences of a critical exponent problem driven by the mixed local nonlocal operator $-\Delta+(-\Delta)^s$ with a linear spectral term on a smooth bounded domain. It shows that every bounded PS sequence decomposes, up to a subsequence, into a weak solution of the original problem plus finitely many rescaled bubbles, where each bubble solves the purely local equation $-\Delta U=|U|^{2^*-2}U$ in $\mathbb{R}^N$, and the energy of the sequence is the sum of the energy of the weak limit and the bubble energies. The proof rests on a compact embedding of the mixed space into the fractional Sobolev space $H^s$, which kills the fractional part of the remainder. As an application, the authors solve an annular-domain existence problem: when the domain has a hole, the problem with $\lambda=0$ admits a positive high-energy solution in sufficiently large dimension.

What carries the argument

The load-bearing mechanism is the compact embedding of Lemma 2.2: the space $X_0^{1,2}(\Omega)$, the completion of $C_c^\infty(\Omega)$ in the norm built from the gradient $L^2$ norm plus the fractional seminorm, embeds compactly into the fractional Sobolev space $H^s(\Omega)$. This forces $[u_k-u_0]_s\to0$ for the weakly convergent remainder, so the nonlocal part of the energy and of the equation disappears on the bubbling scales. That reduction converts the mixed problem into the classical local global compactness theorem, whose bubbles are the extremal functions for the sharp constant in the standard Sobolev inequality, i.e., solutions of the limiting equation (1.4). The same embedding is used in the application to show that the fractional seminorm of approximate bubbles vanishes as the cut-off radius tends to infinity.

What would settle it

A concrete check is to evaluate the inequality in Lemma 4.3, namely $[u_0]_s^2 < (2^{2/N}-1)\int_{\mathbb{R}^N}|\nabla u_0|^2\,dx$ for the standard bubble $u_0(x)=(1+|x|^2)^{-(N-2)/2}$, numerically for $N=3,4,5,\dots$ and a fixed $s\in(0,1)$. If it already holds in low dimensions, the restriction to large $N$ in the annular-domain theorem is not optimal; if it fails for some $N$, the claimed threshold $N_0(s)$ is real and can be located.

Watch

Extended reading notes

Core claim

The central discovery is Theorem 1.2: for every Palais-Smale (PS) sequence $\{u_k\}$ for the functional $I_{\lambda,s}$ at level $\beta$, there exist a weak solution $u_0$ of the original problem, a finite index $l$, sequences of scales and centers, and solutions $U_i$ of $-\Delta U_i=|U_i|^{2^*-2}U_i$ in $\mathbb{R}^N$ such that, up to a subsequence, $u_k = u_0 + \sum_{i=1}^l U_i^k + o(1)$ in $D^{1,2}(\mathbb{R}^N)$, with the scales and centers of distinct bubbles separating, and the energies adding: $I_{\lambda,s}(u_k) \to I_{\lambda,s}(u_0)+\sum_{i=1}^l I_\infty(U_i)$. The proof shows that the remainder $v_k=u_k-u_0$ has vanishing fractional seminorm $[v_k]_s\to0$, so $v_k$ becomes a PS sequence for the pure Laplacian functional at infinity, and the classical local global compactness theorem then applies to extract the bubbles. The fractional Laplacian therefore plays no role in the concentration profiles.

Load-bearing premise

The argument collapses if the mixed space $X_0^{1,2}(\Omega)$ did not embed compactly into the fractional Sobolev space $H^s(\Omega)$, because then the fractional part of the remainder would not vanish and the bubbles could be solutions of the mixed equation in $\mathbb{R}^N$ instead of purely local ones.

Editorial extensions

If this is right

  • Every PS sequence converges up to finitely many purely local bubbles, so compactness is lost only at integer multiples of the classical critical level.
  • The functional satisfies the Palais-Smale condition at all energies strictly below the first critical level, extending the Brezis-Nirenberg threshold to the mixed operator.
  • The decomposition yields a positive high-energy solution on an annular domain with a hole, provided the dimension is large and the radius ratio is large, with energy between the first and second critical levels.
  • Any sign-changing solution with lambda=0 has energy at least twice the first critical level, so the solution obtained is necessarily positive.

Reading between the lines

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

  • The decomposition suggests a general principle: lower-order nonlocal terms are subcritical in the concentration regime, so any mixed operator whose nonlocal part scales like the fractional Laplacian will have purely local bubbles; this could extend to more general mixed operators with different fractional orders.
  • The large-dimension assumption in the annular-domain theorem is tied to the numerical inequality in Lemma 4.3; checking that inequality for low dimensions would pinpoint the exact threshold N0(s) and possibly allow a perturbation argument with a small coefficient on the fractional term to lower the required dimension.
  • Because the compact embedding into H^s is the structural fact, one could test whether the same decomposition holds on unbounded domains or with rough boundaries, where the embedding may fail and mixed bubbles could appear.
  • The energy additivity could be exploited for multiplicity results: if one can construct PS sequences with several well-separated bubbles, their limits should yield multiple positive solutions in suitable topological settings.
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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

2 major / 5 minor

Summary. The paper establishes a global compactness (profile decomposition) theorem for Palais–Smale sequences of the functional I_{λ,s}(u) = (1/2)ρ(u)^2 − (λ/2)∫|u|^2 − (1/2*)∫|u|^{2*} associated with the mixed local–nonlocal problem −Δu + (−Δ)^s u − λu = |u|^{2*−2}u in a bounded smooth domain Ω, with zero Dirichlet condition on R^N \ Ω. The main result, Theorem 1.2, asserts that any PS sequence decomposes, up to a subsequence, as u_k = u0 + Σ_{i=1}^l U_i^k + o(1) in D^{1,2}(R^N), where u0 solves the original problem, each U_i^k is a rescaled solution of the purely local critical equation −ΔU = |U|^{2*−2}U in R^N, the scales and centers of distinct bubbles separate, and energies add. The proof relies on a compact embedding X_0^{1,2}(Ω) ↪ H^s(Ω) (Lemma 2.2), which forces the fractional part of the remainder to vanish. As an application, Theorem 1.3 claims a Coron-type existence result: for domains containing a sufficiently thick annular region and for sufficiently large dimension N, a positive solution exists with energy between the first and second critical thresholds.

Significance. The global compactness theorem is a meaningful extension of Struwe's classical decomposition to the mixed local–nonlocal setting. The proof is structurally transparent and the key mechanism—the compact embedding into H^s—is correctly identified; the external input S_{N,s}=S_N from [8] is clearly acknowledged. The decomposition is parameter-free and the energy additivity is cleanly derived. The Coron-type application follows the standard deformation/topological framework and would be a valuable consequence if the proof is completed. The main weaknesses are concentrated in the statement's bubble scaling and in an unproved Palais–Smale transfer in the application; both are fixable and do not affect the validity of the core compactness mechanism.

major comments (2)
  1. [Section 4, proof of Theorem 1.3] The assertion 'By Theorem 1.2, the functional I_{0,s} satisfies the Palais–Smale condition (PS)_β on X_0^{1,2}(Ω) for 1/N S_N^{N/2} < β < 2/N S_N^{N/2}' is not a direct consequence of Theorem 1.2 and is false without the contradiction hypothesis. Under the assumption that no positive solution exists, the claim requires a short argument using the decomposition, Lemma 5.1, and the energy quantization of bubbles: any PS sequence at such a level would have u0 = 0 and exactly one bubble with energy (1/N)S_N^{N/2}, contradicting the strict inequality. The subsequent consequence 'Consequently, S(·;Ω) satisfies (PS)_β on M' also needs the standard link between constrained and unconstrained PS sequences (introducing Lagrange multipliers and rescaling v_k = S(u_k)^{1/(2*−2)} u_k to obtain a PS sequence for I_{0,s}). Both steps are omitted, and the deformation lemma is applied to S on M, so this gap is load-bearing for Theorem 1.3.
  2. [Theorem 1.2, statement of U_i^k] The definition U_i^k(x) := (R_i^k)^{-(N−2)/2} U_i((x−x_i^k)/R_i^k) does not solve the critical equation −ΔU = |U|^{2*−2}U in R^N; the correct concentrating bubble is R^{(N−2)/2} U(R(x−x0)) as used in Step 3 of the proof. As written, the statement attributes false scaling to the profiles. The statement should be corrected, for example by writing U_i^k(x) = (R_i^k)^{(N−2)/2} U_i(R_i^k(x−x_i^k)) or by introducing ε_i^k = 1/R_i^k and writing U_i^k(x) = (ε_i^k)^{-(N−2)/2} U_i((x−x_i^k)/ε_i^k). Harmonizing the statement with the proof is necessary for the theorem to be correct as stated.
minor comments (5)
  1. [Section 2.2, proof of Lemma 2.2] The choice of the intermediate exponent s' is not stated explicitly; the proof should specify 0 < s < s' < 1 before applying Lemma 2.1.
  2. [Lemma 4.1, estimate for I1] The inequality |y| ≥ (2R−t)/(1−t) improves to |y| ≥ 2R−1 for t close to 1, not to |y| ≥ 2R; the subsequent integral over R^N \ B_{2R} is harmless but the estimate should be adjusted or clarified.
  3. [Theorem 1.2 and Step 3 of its proof] The proof of Step 3 uses the scaling R_k^{(N−2)/2} v_0(R_k(·−x_k)) without relating it to the U_i^k notation in the theorem statement; after correcting the statement this link should be made explicit for readability.
  4. [References] Reference [8] is cited only as an arXiv preprint; the authors should update it to the published version if one exists.
  5. [Various] There are several typographical and formatting issues, including 'in volving' in the abstract, inconsistent use of 'P.S.' and 'PS', and the reference to Coron's paper is missing the period after 'Sér I Math'; a careful proofreading pass is recommended.

Circularity Check

0 steps flagged · score 0.0 of 10

No circularity: the mixed-operator compactness theorem reduces to external Struwe/Leoni results and contains no fitted parameters.

full rationale

The central derivation (Theorem 1.2) is not circular. Its load-bearing ingredients are external: Lemma 2.2 is proved from Leoni's interpolation estimate [33, Cor. 6.19/Thm 6.21], Step 3 invokes Struwe's compactness lemma [48, Lemma 3.3] on the local PS sequence v_k, and the bubble-interaction orthogonality is taken from Bahri-Coron [5]. The identity S_N,s = S_N is imported from [8], which is external to the present authors. The key structural step is the compact embedding X_0^{1,2}(Ω) ↪ H^s(Ω), which forces [v_k]_s -> 0; hence the fractional part of the mixed operator disappears in the limit and the bubbles solve the purely local equation (1.4). This is a genuine reduction, not an equivalence with the paper's own hypotheses. The energy estimate in Lemma 4.3 is an explicit computation using classical sharp constants, with no fitted parameter. The only self-citation is [7] (Bhakta-Chakraborty-Pucci), used as a literature pointer in the introduction and never as a premise; it is not load-bearing. The proof of Theorem 1.3 contains a real gap: the assertion that S(·;Ω) inherits (PS)_β on M from I_{0,s} requires a Lagrange-multiplier verification that is omitted. This is a correctness/rigor issue, not circularity, and it does not affect the circularity score. Overall the paper is self-contained against external benchmarks and no circular step can be exhibited.

Assumptions & free parameters 0 free parameters · 6 assumptions · 0 invented entities

No free parameters or invented entities. The central claim rests on standard tools: Struwe's decomposition, interpolation inequalities, and Ekeland and deformation arguments, plus one external theorem from [8] about the best constant.

assumptions (6)
  • standard math Struwe's global compactness lemma (Lemma 3.3 of [48]) for the classical Laplacian critical problem
    Used in Step 3 of the proof of Theorem 1.2 to extract the first bubble from the reduced PS sequence v_k in H_0^1(Ω).
  • standard math Compact embedding H^1_0(Ω) into H^s(Ω) via interpolation (Leoni [33], Corollary 6.19 and Theorem 6.21)
    Basis of Lemma 2.2, which makes the fractional part of the remainder vanish.
  • standard math Ekeland's variational principle and the deformation lemma (Struwe [48], Theorem II.3.11)
    Used in the proof of Theorem 1.3 to convert a minimizing sequence into a PS sequence and to flow the bubble family to low energy.
  • standard math Brezis-Lieb lemma and Vitali convergence theorem
    Used to split L^p norms in Step 2 of Theorem 1.2 and in the energy computations of Lemma 5.1.
  • domain assumption Theorem 1.1 of [8]: the best constant S_{N,s} for the mixed operator equals the classical constant S_N and is not attained
    Used to rule out ground states and to know that the relevant bubbles are the classical Aubin-Talenti bubbles; this is an external theorem the paper relies on.
  • standard math Strong maximum principle and regularity for mixed operators from Garain-Kinnunen [23, Theorem 8.4]
    Used in Theorem 1.3 to conclude a nonnegative critical point is either zero or strictly positive.

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Cite this review

Pith. "Pith review of Global Compactness Result for a Br\'ezis-Nirenberg-Type Problem Involving Mixed Local Nonlocal Operator." pith.science (2026). https://pith.science/paper/5ZKTONVC

@misc{pith2026250415968,
  author       = {Pith},
  title        = {Pith review of: Global Compactness Result for a Br\'ezis-Nirenberg-Type Problem Involving Mixed Local Nonlocal Operator},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/5ZKTONVC}},
  note         = {Machine review of arXiv:2504.15968}
}
abstract

This paper investigates the profile decomposition of Palais-Smale sequences associated with a Brezis-Nirenberg type problem involving a combination of mixed local nonlocal operators, given by \begin{equation*} \left\{\begin{aligned} &-\Delta u + (-\Delta)^s u - \lambda u = |u|^{2^*-2}u \;\;\mbox{ in } \Omega, &\quad u=0\,\mbox{ in }\mathbb{R}^N\setminus \Omega. \end{aligned} \right. \end{equation*} where $\Omega\subseteq \mathbb{R}^{N}$ is a smooth bounded domain with $N \geq 3$, $s\in (0,1),\,\lambda\in\mathbb{R}$ is a real parameter and $2^* = \frac{2N}{N - 2} $ denotes the critical Sobolev exponent. As an application of the derived global compactness result, we further study the existence of positive solution of the corresponding Coron-type problem (C. R. Acad. Sci. Paris S\'{e}r I Math, 299(7):209-212, 1984) when $\lambda=0$.

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