What does this research mean for the field?

A nontrivial solution to a Sobolev critical problem can be produced by using a mixed-order operator where a fractional Laplacian with the 'wrong sign' acts as a nonlocal perturbation to a purely local Laplacian. Novelty: ClaimNovelty.NOVEL_FINDING. Consensus alignment: ConsensusAlignment.NEUTRAL.

What question did this study set out to answer?

Investigate a Sobolev critical problem governed by a mixed-order operator formed by superposing a standard Laplacian with a fractional Laplacian bearing an opposite sign.

August 22, 2026Bulletin of the London Mathematical SocietyOpen Access

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

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Investigate a Sobolev critical problem governed by a mixed-order operator formed by superposing a standard Laplacian with a fractional Laplacian bearing an opposite sign.
Formulated a mixed local-nonlocal differential operator modeling the fractional Laplacian as a negative nonlocal perturbation of the classical local operator.
Applied variational and analytical techniques for critical Sobolev growth equations to examine the existence of non-zero solutions.
Proved that assigning the 'wrong sign' to the fractional Laplacian serves as a crucial nonlocal perturbation to overcome non-existence barriers.
Demonstrated the existence of nontrivial solutions for the critical mixed-order differential equation.

Implication

Theoretical analysis demonstrates nontrivial solutions for mixed-order critical problems with a wrong-sign fractional Laplacian, highlighting the role of nonlocal perturbations.

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 , 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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Cite This Study

Biagi et al. (2026) studied this question.

synapsesocial.com/papers/6a895e6bca7ade938187c8cb https://doi.org/10.1112/blms.70477

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Implication

Theoretical analysis demonstrates nontrivial solutions for mixed-order critical problems with a wrong-sign fractional Laplacian, highlighting the role of nonlocal perturbations.

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

Key Points

Implication

Abstract

Discussion

Cite This Study

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

Key Points

Implication

Abstract

Discussion

Cite This Study

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