Necessary condition in a Brezis–Oswald-type problem for mixed local and nonlocal operators

Biagi, Stefano

doi:10.1016/j.aml.2022.108177

Cited by 10 publications

(6 citation statements)

References 14 publications

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“…1 , as claimed. Moreover, by letting t → 0 + in (2.3) and recalling the extended definition of f , we have for a.e.x ∈ Ω and all t ∈ R −…”

mentioning

confidence: 60%

“…(a) The boundedness condition H (i) is obviously satisfied in the autonomous case, i.e., f ∈ C(R + ). (b) The growth condition H (ii) acts on the reaction from above only, as is the case in [6,9] but differently from previous results in the nonlocal setting such as [1,2,25]. (c) The strict monotonicity condition H (iii) classifies our reaction as a (p − 1)-sublinear one.…”

Section: Version Of June 22 2022 1 Introductionmentioning

confidence: 80%

“…In the nonlocal framework, we recall the related results for the fractional Laplacian in R N [27], for the spectral fractional Laplacian in a bounded domain [23], for the fractional p-Laplacian with nonlocal Robin conditions [25], and a mixed local-nonlocal operator with Dirichlet conditions [1,2]. We remark again that the last three contributions present partial results regarding the necessary condition for existence, and they both employ bilateral growth conditions on the reaction.…”

Section: Version Of June 22 2022 1 Introductionmentioning

confidence: 99%

“…This is a nonlinear, nonlocal operator which in special case reduces, up to a multiplicative constant, to the fractional Laplacian (p = 2, s ∈ (0, 1)), to the p-Laplacian (p > 1, s = 1), and in particular to the Laplacian (p = 2, s = 1). The fractional p-Laplacian is degenerate if p > 2, singular if p ∈ (1,2). An introduction to this operator and related problems can be found in [24,26].…”

Section: Version Of June 22 2022 1 Introductionmentioning

confidence: 99%

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Optimal solvability for the fractional p-Laplacian with Dirichlet conditions

Iannizzotto¹,

Mugnai²

2022

Preprint

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We study a nonlinear, nonlocal Dirichlet problem driven by the fractional p-Laplacian, involving a (p − 1)-sublinear reaction. By means of a weak comparison principle we prove uniqueness of the solution. Also, comparing the problem to 'asymptotic' weighted eigenvalue problems for the same operator, we prove a necessary and sufficient condition for the existence of a solution. Our work extends classical results due to Brezis-Oswald [6] and Diaz-Saa [9] to the quasilinear nonlocal framework.

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“…1 , as claimed. Moreover, by letting t → 0 + in (2.3) and recalling the extended definition of f , we have for a.e.x ∈ Ω and all t ∈ R −…”

mentioning

confidence: 60%

Section: Version Of June 22 2022 1 Introductionmentioning

confidence: 80%

Section: Version Of June 22 2022 1 Introductionmentioning

confidence: 99%

Section: Version Of June 22 2022 1 Introductionmentioning

confidence: 99%

See 2 more Smart Citations

Optimal solvability for the fractional p-Laplacian with Dirichlet conditions

Iannizzotto¹,

Mugnai²

2022

Preprint

Self Cite

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“…[2,9,13,17,24,25]), boundary Harnack principle [14], boundary regularity and overdetermined problems [10,36], qualitative properties of solutions [3], existence of solutions and asymptotics (see e.g. [6,7,8,12,15,20,21,26,33,35]) and shape optimization problems [4,5,27].…”

Section: Introductionmentioning

confidence: 99%

Variational methods for nonpositive mixed local-nonlocal operators

Maione¹,

Mugnai²,

Vecchi³

2022

Preprint

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We prove the existence of a weak solution for boundary value problems driven by a mixed local-nonlocal operator. The main novelty is that such an operator is allowed to be nonpositive definite. 2020 Mathematics Subject Classification. 35A15, 35J62, 35R11. Key words and phrases. Operators of mixed order, variational methods, Saddle Point Theorem, Linking Theorem, asymptotically linear growth, superlinear and subcritical growth.A. Maione is supported by the DFG SPP 2256 project "Variational Methods for Predicting Complex Phenomena in Engineering Structures and Materials", by the University of Freiburg and by INdAM-GNAMPA Project "Equazioni differenziali alle derivate parziali in fenomeni non lineari". D. Mugnai is supported by the FFABR "Fondo per il finanziamento delle attività base di ricerca" 2017 and by the INdAM-GNAMPA Project "PDE ellittiche a diffusione mista". E. Vecchi is supported by INdAM-GNAMPA Project "PDE ellittiche a diffusione mista".

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Klein–Gordon–Maxwell Equations Driven by Mixed Local–Nonlocal Operators

Cangiotti,

Caponi,

Maione

et al. 2023

Milan J. Math.

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Classical results concerning Klein–Gordon–Maxwell type systems are shortly reviewed and generalized to the setting of mixed local–nonlocal operators, where the nonlocal one is allowed to be nonpositive definite according to a real parameter. In this paper, we provide a range of parameter values to ensure the existence of solitary (standing) waves, obtained as Mountain Pass critical points for the associated energy functionals in two different settings, by considering two different classes of potentials: constant potentials and continuous, bounded from below, and coercive potentials.

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Necessary condition in a Brezis–Oswald-type problem for mixed local and nonlocal operators

Cited by 10 publications

References 14 publications

Optimal solvability for the fractional p-Laplacian with Dirichlet conditions

Optimal solvability for the fractional p-Laplacian with Dirichlet conditions

Variational methods for nonpositive mixed local-nonlocal operators

Klein–Gordon–Maxwell Equations Driven by Mixed Local–Nonlocal Operators

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