Symmetric ground state solution for a non-linear Schrodinger equation with non-local regional diffusion

In this article, we are interested in the non-linear Schrodinger equation with non-local regional diffussion (-Delta)(rho)(alpha)u + u = f(x, u) in R-n, u is an element of H-alpha(R-n), where f is a super-linear sub-critical function, (-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 solution, furthermore we prove that the ground state level is achieved by a radially symmetric solution. The proof is carried out by using variational methods jointly with rearrangement arguments.