A fractional critical problem in the halfspace with Neumann conditions

Abstract

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.