Non-linear Schrödinger equation with non-local regional diffusion

In this article we are interested in the nonlinear Schrödinger equation with non-local regional difussion ϵ2α(-Δ)ραu+u=f(u)inRn,u∈Hα(Rn),\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned}&\epsilon ^{2\alpha } (-\Delta )_{\rho }^{\alpha }u + u = f(u) \hbox { in } \mathbb {R}^{n}, \\&u \in H^{\alpha }(\mathbb {R}^{n}), \end{aligned}$$\end{document}where f\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$f$$\end{document} is a super-linear sub-critical function and (-Δ)ρα\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$(-\Delta )_{\rho }^{\alpha }$$\end{document} is a variational version of the regional laplacian, whose range of scope is a ball with radius ρ(x)>0\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\rho (x)>0$$\end{document}. We study the existence of a ground state and we analyze the behavior of semi-classical solutions as ε→0\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\varepsilon \rightarrow 0$$\end{document}.