Existence of positive solution for a class of nonlocal elliptic problems in the half space with a hole

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.