A Brezis-Oswald approach for mixed local and nonlocal operators

Abstract

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.

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

“…This fact shows that, when Ω is sufficiently regular, X 1,p 0 (Ω) coincides with the space X p (Ω) introduced in [6] (for p = 2) and in [9] (for a general p > 1).…”

“…To begin with, we recall the following result proved in [9] which establishes the existence of the smallest eigenvalue and detects its basic properties. Proposition 2.6 ([9, Proposition 5.1]).…”

“…In this paper we consider a nonlinear operator arising from the superposition of a classical p-Laplace operator and a fractional p-Laplace operator, of the form Given a bounded open set Ω ⊆ R n , we consider the eigenvalue problem for the operator L p,s with homogeneous Dirichlet boundary conditions (i.e., the eigenfunctions are prescribed to vanish in the complement of Ω). In particular, we define λ 1 (Ω) to be the smallest of such eigenvalues and λ 2 (Ω) to be the second smallest one (in the sense made precise in [9,41]).…”

“…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].…”

Combined effects in mixed local-nonlocal stationary problems

“…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.…”