A nonlocal concave-convex problem with nonlocal mixed boundary data

The aim of this paper is to study the following problem....... In this setting, Nsu can be seen as a Neumann condition of nonlocal type that is compatible with the probabilistic interpretation of the fractional Laplacian, as introduced in [20], and Bsu is a mixed Dirichlet-Neumann exterior datum. The main purpose of this work is to prove existence, nonexistence and multiplicity of positive energy solutions to problem (P) for suitable ranges of and p and to understand the interaction between the concave-convex nonlinearity and the Dirichlet-Neumann data.