  This work concerns with the existence of solutions for the following class of
nonlocal elliptic problems \begin{eqnarray}\label{eq:0.1}
&&\left\{\begin{array}{l} (-\Delta)^{s} u+u=|u|^{p-2} u \text { in } \Omega_{r}
\\ u \geq 0 \quad \text { in }\Omega_{r} \text { and } u \neq 0 \\ u=0 \quad
\mathbb{R}^{N} \backslash \Omega_{r} \end{array}\right., \end{eqnarray}
involving the fractional Laplacian operator $(-\Delta)^{s},$ where $s \in(0,1),
N>2 s$, $\Omega_{r}$ is the half space with a hole in $\mathbb{R}^N$ and $p
\in\left(2,2_{s}^{*}\right) .$ The main technical approach is based on
variational and topological methods.
