Non-linear Schrodinger equation with non-local regional diffusion

Abstract

In this article we are interested in the nonlinear Schrodinger equation with non-local regional difussion

epsilon(2 alpha)(-Delta)(rho)(alpha)u + u = f (u) in R-n,

u epsilon H-alpha(R-n),

where f is a super-linear sub-critical function and (-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 --> 0.