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., $${\mathcal{L}}_{p,s}=-{\Delta }_p+(-\Delta {)}_p^s.$$ Our main result is resemblant to the celebrated work by Brezis-Oswald [10]. In addition, we prove a regularity result of independent interest.

arXiv commentThe title of the manuscript has been changed after the revision