A fractional critical problem in the halfspace with Neumann conditions

Given $s\in (0,1)$ and $n>2s$, we construct nontrivial solutions of the problem \begin{equation*} \begin{cases} (-\Delta)^s u=u^{2^s-1} &{\mbox{ in }}$\mathbb{R}$^n+,\ {$\mathcal{N}$}s u=0&{\mbox{ in }}$\mathbb{R}$^n-,\end{cases} \end{equation}where ${\mathcal{N}}_s$ represents the nonlocal Neumann condition of exterior type.