A critical nonlinear elliptic equation with non local regional diffusion

In this article we are interested in the nonlocal regional Schr"odinger equation with critical exponent \begin{eqnarray*} &\epsilon^{2\alpha} (-\Delta){\rho}^{\alpha}u + u = \lambda u^q + u^{2{\alpha}^{}-1} \mbox{ in } $\mathbb{R}$^{N}, \ & u \in H^{\alpha}($\mathbb{R}$^{N}), \end{eqnarray} where $ϵ$ is a small positive parameter, $\alpha \in (0,1)$, $q\in (1,2_{\alpha}^{}-1)$,$2_{\alpha}^{} = \frac{2N}{N-2\alpha}$ is the critical Sobolev exponent, $\lambda >0$ is a parameter and $(-\mathrm{\Delta }{)}_{\rho }^{\alpha }$ is a variational version of the regional laplacian, whose range of scope is a ball with radius $\rho (x)>0$. We study the existence of a ground state and we analyze the behavior of semi-classical solutions as $\epsilon \to 0$.