In this paper, we are interested in the nonlinear Schrödinger equation with non-local regional diffusion 1 (Formula presented.) where 0 < α < 1 and (Formula presented.) is a variational version of the regional Laplacian, whose range of scope is a ball with radius ρ(x) > 0. The novelty of this paper is that, assuming f is of subquadratic growth as |u|→+∞, we show that possesses infinitely many solutions via the genus properties in critical point theory. Furthermore, if f(x,u) = γa(x)|u|γ − 1, where (Formula presented.) is a nonincreasing radially symmetric function, then the solution of is radially symmetric. Copyright © 2015 John Wiley & Sons, Ltd.