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Alves","submitted_at":"2016-11-07T14:03:42Z","abstract_excerpt":"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. 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