A nonlocal concave-convex problem with nonlocal mixed boundary data

The aim of this paper is to study the following problem

$(P_{\lambda}) \equiv\left\{\begin{array}{rcll}(-\Delta)^s u& = &\lambda u^{q}+u^{p}&{\text{ in }}\Omega,\\ u&>&0 &{\text{ in }} \Omega, \\ \mathcal{B}_{s}u& = &0 &{\text{ in }} \mathbb{R}^{N}\backslash \Omega,\end{array}\right.$

with $0<q<1<p$, $N>2s$, $λ> 0$, $Ω \subset \mathbb{R}^{N}$ is a smooth bounded domain,

$(-Δ)^su(x) = a_{N,s}\;P.V.∈t_{\mathbb{R}^{N}}\frac{u(x)-u(y)}{|x-y|^{N+2s}}\,dy,$

$a_{N,s}$ is a normalizing constant, and $\mathcal{B}_{s}u = uχ_{Σ_{1}}+\mathcal{N}_{s}uχ_{Σ_{2}}.$ Here, $Σ_{1}$ and $Σ_{2}$ are open sets in $\mathbb{R}^{N}\backslash Ω$ such that $Σ_{1} \cap Σ_{2} = \emptyset$ and $\overline{Σ}_{1}\cup \overline{Σ}_{2} = \mathbb{R}^{N}\backslash Ω.$

In this setting, $\mathcal{N}_{s}u$ 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 $\mathcal{B}_{s}u$ 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 ($P_{λ}$) for suitable ranges of $λ$ and $p$ and to understand the interaction between the concave-convex nonlinearity and the Dirichlet-Neumann data.