A Brezis-Oswald approach for mixed local and nonlocal operators

TL;DR

This paper establishes conditions for the existence and uniqueness of positive solutions to mixed local and nonlocal sublinear Dirichlet problems, extending Brezis-Oswald type results and including a new regularity theorem.

Contribution

It provides necessary and sufficient conditions for solutions to a combined local and nonlocal operator problem, extending classical Brezis-Oswald results.

Findings

Existence and uniqueness criteria for solutions

Extension of Brezis-Oswald framework to mixed operators

New regularity results for solutions

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.