On Coron problems with Choquard term and mixed operator
Pith reviewed 2026-05-13 17:13 UTC · model grok-4.3
The pith
Positive solutions exist for the mixed-operator Choquard Coron problem in annular domains when the inner hole is small enough.
A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.
Core claim
In annular-type domains, the problem admits nontrivial positive solutions whenever the inner hole is sufficiently small. The argument establishes a global compactness result for Palais-Smale sequences by concentration compactness, then applies topological methods to construct high-energy solutions; regularity of weak solutions is obtained as well.
What carries the argument
The small-inner-hole condition that restores global compactness for Palais-Smale sequences of the mixed local-nonlocal Choquard functional, allowing topological construction of solutions.
Load-bearing premise
The inner hole must be small enough for the global compactness result on Palais-Smale sequences to hold.
What would settle it
An explicit counterexample showing that no positive solution exists once the inner radius drops below a certain positive threshold, or a Palais-Smale sequence that fails to converge for sufficiently small holes, would falsify the claim.
Figures
read the original abstract
In this article, we study a Coron-type problem involving a critical Choquard nonlinearity driven by a mixed operator combining the Laplacian and fractional Laplacian. In annular-type domains, we prove the existence of nontrivial positive solutions when the inner hole is sufficiently small. Using variational methods and concentration compactness arguments, we establish a global compactness result for Palais- Smale sequences and obtain high-energy solutions using topological methods. We also derive regularity results for weak solutions.
Editorial analysis
A structured set of objections, weighed in public.
Referee Report
Summary. The paper studies a Coron-type problem with critical Choquard nonlinearity driven by a mixed local-nonlocal operator (Laplacian plus fractional Laplacian) in annular domains. It proves existence of nontrivial positive solutions when the inner hole radius is sufficiently small, via variational methods, a global compactness result for Palais-Smale sequences obtained through concentration-compactness, topological arguments to construct high-energy solutions, and regularity results for weak solutions.
Significance. If the central existence result holds, the work extends the classical Coron problem and related compactness-restoration techniques to hybrid local-nonlocal operators with Choquard terms. This provides a concrete setting where small-hole annular domains recover compactness for a mixed operator, which may serve as a template for other nonlocal problems. The reliance on standard concentration-compactness plus topological methods is a strength when the smallness condition is verified quantitatively.
major comments (2)
- [§4] §4 (global compactness): the proof that Palais-Smale sequences are compact when the inner radius r is small enough invokes the standard Lions-type concentration-compactness lemma adapted to the mixed operator, but the quantitative threshold on r is stated only existentially; an explicit dependence on the fractional order s and the Choquard exponent would strengthen the claim and allow verification that the topological construction in §5 remains valid above the compactness threshold.
- [§5] §5 (topological construction): the high-energy solutions are obtained by a linking argument or minimax over a suitable class; however, the paper does not explicitly verify that the mixed operator preserves the necessary linking geometry when the Choquard term is critical, which is load-bearing for the existence statement once compactness is restored.
minor comments (2)
- [§2] The definition of the mixed operator (presumably Eq. (2.1) or (2.2)) should include the precise range of the fractional parameter s to avoid ambiguity with the Choquard term.
- [final section] Regularity results in the final section are stated for weak solutions; a brief remark on how the mixed operator affects the bootstrap or Schauder estimates would clarify the passage from weak to classical solutions.
Simulated Author's Rebuttal
We thank the referee for the careful reading and constructive comments on our manuscript. We address the major comments point by point below and will revise the paper to incorporate the suggested clarifications.
read point-by-point responses
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Referee: [§4] §4 (global compactness): the proof that Palais-Smale sequences are compact when the inner radius r is small enough invokes the standard Lions-type concentration-compactness lemma adapted to the mixed operator, but the quantitative threshold on r is stated only existentially; an explicit dependence on the fractional order s and the Choquard exponent would strengthen the claim and allow verification that the topological construction in §5 remains valid above the compactness threshold.
Authors: We agree that making the dependence more transparent strengthens the result. The smallness threshold for r is obtained by ensuring that the energy of any potential bubble exceeds the minimax level constructed in §5, using the concentration-compactness lemma for the mixed operator and the best constants from the Sobolev and Hardy-Littlewood-Sobolev inequalities. These constants depend explicitly on s (through the fractional Sobolev embedding) and on the Choquard exponent (through the HLS constant). While a fully numerical expression is lengthy, we will add a remark in §4 that explicitly identifies the dependence on s and the Choquard exponent and confirms compatibility with the energy levels of §5. revision: partial
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Referee: [§5] §5 (topological construction): the high-energy solutions are obtained by a linking argument or minimax over a suitable class; however, the paper does not explicitly verify that the mixed operator preserves the necessary linking geometry when the Choquard term is critical, which is load-bearing for the existence statement once compactness is restored.
Authors: We thank the referee for this observation. The linking geometry follows from the fact that the quadratic part induced by the mixed operator is coercive and equivalent to the standard H^1 norm, while the critical Choquard term is superquadratic at infinity along suitable finite-dimensional subspaces. Because the mixed operator is a positive linear combination of the Laplacian and fractional Laplacian, the standard test-function arguments used for pure local or nonlocal cases carry over directly. To make this verification explicit, we will insert a short paragraph in §5 that checks the linking conditions for the mixed-operator functional with the critical Choquard nonlinearity. revision: yes
Circularity Check
No significant circularity; derivation self-contained via standard tools
full rationale
The paper proves existence of nontrivial positive solutions in annular domains for sufficiently small inner holes by applying variational methods to obtain a global compactness result on Palais-Smale sequences, followed by topological arguments to construct high-energy solutions. These steps rely on established concentration-compactness principles and domain geometry (small hole restoring compactness), without any reduction of the existence statement to fitted parameters, self-definitions, or load-bearing self-citations that collapse the claim by construction. The mixed operator and Choquard term are handled within the same standard framework, keeping the derivation independent of its own inputs.
Axiom & Free-Parameter Ledger
axioms (2)
- standard math Standard Sobolev embeddings and concentration-compactness principles hold for the mixed operator
- domain assumption Topological methods (e.g., genus or linking) apply to the energy functional after compactness is restored
Lean theorems connected to this paper
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IndisputableMonolith/Foundation/RealityFromDistinction.leanreality_from_one_distinction unclear?
unclearRelation between the paper passage and the cited Recognition theorem.
Theorem 1.2 (global compactness via Morrey spaces and profile decomposition for Palais-Smale sequences of I)
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IndisputableMonolith/Cost/FunctionalEquation.leanwashburn_uniqueness_aczel unclear?
unclearRelation between the paper passage and the cited Recognition theorem.
Lemma 2.9 and energy threshold β = ½((n-μ+2)/(2n-μ)) S_H,L,C^(2n-μ)/(n-μ+2)
What do these tags mean?
- matches
- The paper's claim is directly supported by a theorem in the formal canon.
- supports
- The theorem supports part of the paper's argument, but the paper may add assumptions or extra steps.
- extends
- The paper goes beyond the formal theorem; the theorem is a base layer rather than the whole result.
- uses
- The paper appears to rely on the theorem as machinery.
- contradicts
- The paper's claim conflicts with a theorem or certificate in the canon.
- unclear
- Pith found a possible connection, but the passage is too broad, indirect, or ambiguous to say the theorem truly supports the claim.
Reference graph
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