Semilinear elliptic equations involving mixed local and nonlocal operators

Biagi, Stefano; Vecchi, Eugenio; Dipierro, Serena; Valdinoci, Enrico

doi:10.1017/prm.2020.75

Cited by 73 publications

(34 citation statements)

References 81 publications

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“…Here, ∆ p u = div(|∇u| p−2 ∇u) is the classical p−Laplacian operator and, for fixed s ∈ (0, 1) and up to a multiplicative positive constant, the fractional p−Laplacian is defined as (−∆) s p u(x) := 2 P.V. Problems driven by operators like L p,s have raised a certain interest in the last few years, both for the mathematical complications that the combination of two so different operators imply and for the wide range of applications, see for instance [5,4,6,11,12,13,14] and the references therein. A common feature of the aforementioned papers is to deal with weak solutions, in contrast with other results existing in the literature where viscosity solutions have been considered, see e.g.…”

Section: Introductionmentioning

confidence: 99%

A Brezis-Oswald approach for mixed local and nonlocal operators

Biagi¹,

Mugnai²,

Vecchi³

2021

Preprint

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In this paper we provide necessary and sufficient conditions for the existence of a unique positive weak solution for some sublinear Dirichlet problems driven by the sum of a quasilinear local and a nonlocal operator, i.e., Lp,s = −∆p + (−∆) s p . Our main result is resemblant to the celebrated work by Brezis-Oswald [10]. In addition, we prove a regularity result of independent interest. 2010 Mathematics Subject Classification. 35A01, 35R11. Key words and phrases. Operators of mixed order, p−sublinear Dirichlet problems, existence and uniqueness of solutions, strong maximum principle, L ∞ −estimate. The authors are members of the Gruppo Nazionale per l'Analisi Matematica, la Probabilità e le loro Applicazioni (GNAMPA) of the Istituto Nazionale di Alta Matematica (INdAM). S. Biagi is partially supported by the INdAM-GNAMPA project Metodi topologici per problemi al contorno associati a certe classi di equazioni alle derivate parziali. D. Mugnai is supported by the FFABR "Fondo per il finanziamento delle attività base di ricerca" 2017 and by the INdAM-GNAMPA Project Equazioni alle derivate parziali: problemi e modelli. E. Vecchi is partially supported by the INdAM-GNAMPA project Convergenze variazionali per funzionali e operatori dipendenti da campi vettoriali.

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Section: Introductionmentioning

confidence: 99%

A Brezis-Oswald approach for mixed local and nonlocal operators

Biagi¹,

Mugnai²,

Vecchi³

2021

Preprint

Self Cite

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“…Very recently a great amount of attention has been paid in studying elliptic problems involving mixed type of operator having both local and nonlocal behaviours. Some questions related to structural results like existence, maximum principle and interior Sobolev and Lipschitz regularity [1,11,14], symmetry results [13], Faber-Krahn type inequality [12], Neumann problems [42], Green functions estimates [26,27] has been answered.…”

Section: Introductionmentioning

confidence: 99%

Combined effects in mixed local-nonlocal stationary problems

Arora¹,

Rădulescu²

2021

Preprint

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In this work, we study an elliptic problem involving an operator of mixed order with both local and nonlocal aspects, and in either the presence or the absence of a singular nonlinearity. We investigate existence or non-existence properties, power and exponential type Sobolev regularity results, and the boundary behavior of the weak solution, in the light of the interplay between the summability of the datum and the power exponent in singular nonlinearities.

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“…The case of mixed operators. The study of mixed local/nonlocal operators has been recently received an increasing level of attention, both in view of their intriguing mathematical structure, which combines the classical setting and the features typical of nonlocal operators in a framework that is not scale-invariant [40,45,46,5,32,10,21,4,20,24,23,22,39,7,1,18,30,27,28,35,36,37,38,19,9,6,54], and of their importance in practical applications such as the animal foraging hypothesis [29,51].…”

Section: Introductionmentioning

confidence: 99%

A Hong-Krahn-Szegö inequality for mixed local and nonlocal operators

Biagi¹,

Dipierro²,

Valdinoci³

et al. 2021

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Given a bounded open set Ω ⊆ R n , we consider the eigenvalue problem of a nonlinear mixed local/nonlocal operator with vanishing conditions in the complement of Ω.We prove that the second eigenvalue λ2(Ω) is always strictly larger than the first eigenvalue λ1(B) of a ball B with volume half of that of Ω.This bound is proven to be sharp, by comparing to the limit case in which Ω consists of two equal balls far from each other. More precisely, differently from the local case, an optimal shape for the second eigenvalue problem does not exist, but a minimizing sequence is given by the union of two disjoint balls of half volume whose mutual distance tends to infinity.

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Semilinear elliptic equations involving mixed local and nonlocal operators

Cited by 73 publications

References 81 publications

A Brezis-Oswald approach for mixed local and nonlocal operators

A Brezis-Oswald approach for mixed local and nonlocal operators

Combined effects in mixed local-nonlocal stationary problems

A Hong-Krahn-Szegö inequality for mixed local and nonlocal operators

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