  In this article we are interested in the following non-linear Schr\"odinger
equation with non-local regional diffusion $$
(-\Delta)_{\rho_\epsilon}^{\alpha}u + u = f(u) \hbox{ in } \mathbb{R}^n, \quad
u \in H^\alpha(\mathbb{R}^n), \qquad\qquad(P_\epsilon) $$ where $\epsilon >0$,
$0< \alpha < 1$, $(-\Delta)_{\rho_\epsilon}^{\alpha}$ is a variational version
of the regional laplacian, whose range of scope is a ball with radius
$\rho_\epsilon(x)=\rho(\epsilon x)>0$, where $\rho$ is a continuous function.
We give general conditions on $\rho$ and $f$ which assure the existence and
multiplicity of solution for $(P_\epsilon)$.
