In this article, we are interested in the non-linear Schrödinger equation with non-local regional diffussion (Formula presented.) where f is a super-linear sub-critical function, (-Δ)αρ is a variational version of the regional Laplacian, whose range of scope is a ball with radius ρ(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.