Existence and multiplicity of solutions for a nonlinear Schr\"odinger equation with non-local regional diffusion

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 $(P_\epsilon)$.