Multiplicity of solutions for a class of fractional elliptic problems with critical exponential growth and nonlocal Neumann condition

In this paper we consider the existence and multiplicity of weak solutions for the following class of fractional elliptic problem \begin{document}$ \begin{equation} \begin{cases} (-\Delta)^{\frac{1}{2}}u + u = Q(x)f(u)\;\;\mbox{in}\;\;\mathbb{R} \setminus (a, b)\\ \mathcal{N}_{1/2}u(x) = 0\;\;\qquad \qquad \quad \mbox{in}\;\;(a, b), \end{cases} \end{equation} \ \ \ \ \ \ \ \ (0.1) $\end{document} where \begin{document}$ a, b\in \mathbb{R} $\end{document} with \begin{document}$ a , \begin{document}$ (-\Delta)^{\frac{1}{2}} $\end{document} denotes the fractional Laplacian operator and \begin{document}$ \mathcal{N}_{1/2} $\end{document} is the nonlocal operator that describes the Neumann boundary condition, which is given by \begin{document}$ \mathcal{N}_{1/2}u(x) = \frac{1}{\pi} \int_{\mathbb{R}\setminus (a, b)} \frac{u(x) - u(y)}{|x-y|^{2}}dy, \;\;x\in [a, b]. $\end{document}