Multiplicity and symmetry results for a nonlinear Schrödinger equation with non‐local regional diffusion

Abstract

In this paper, we are interested in the nonlinear Schrödinger equation with non-local regional diffusionwhere 0 <˛< 1 and . /˛ is a variational version of the regional Laplacian, whose range of scope is a ball with radius .x/ > 0. The novelty of this paper is that, assuming f is of subquadratic growth as juj ! C1, we show that (1) possesses infinitely many solutions via the genus properties in critical point theory. Furthermore, if f.x, u/ D a.x/juj 1 , where.R n , R C / is a nonincreasing radially symmetric function, then the solution of (1) is radially symmetric.