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A Brezis-Oswald approach for mixed local and nonlocal operators

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arxiv 2103.11382 v3 pith:PCXO5FCU submitted 2021-03-21 math.AP

Stefano Biagi , Dimitri Mugnai , Eugenio Vecchi This is my paper · ORCID
classification math.AP
keywords brezis-oswalddeltalocalnonlocalresultadditionapproachcelebrated
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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., $$\mathcal{L}_{p,s} = -\Delta_p + (-\Delta)^s_p.$$ Our main result is resemblant to the celebrated work by Brezis-Oswald [10]. In addition, we prove a regularity result of independent interest.

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  1. On an eigenvalue problem associated with mixed operators under mixed boundary conditions

    math.AP 2024-11 conditional novelty 6.0 of 10

    The paper establishes the principal eigenvalue theory and bifurcation from zero and infinity for a mixed local-nonlocal elliptic operator under mixed Dirichlet-Neumann boundary conditions.

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