  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.
