A nonlocal concave-convex problem with nonlocal mixed boundary data

The aim of this paper is to study the following problem (P) ≡{ (-Δ)^su = u^q + u^p in ; u > 0 inΩ; Bsu = 0 in R^N/Ω with 0 < q < 1 < p, N > 2s, > 0,Ω⊂ R^N is a smooth bounded domain, (-Δ)^su(x) = aN;s P:V: Z R^N u(x) -u(y)/x-y/^N+2s dy; aN;s is a normalizing constant, and B_su = uX∑₁ + NsuX∑₂ : Here, ∑₁ and ∑₂ are open sets in R^N/Ω such that∑₂ ∩ ∑₂ = ; and ∑₁ ∩ ∑₂ = R^N/Ω: 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.