Existence and multiplicity of solutions for a nonlinear Schrödinger equation with non-local regional diffusion

Abstract

In this article we are interested in the following non-linear Schrödinger equation with non-local regional diffusion $$ (-Δ)_{ρ_ε}^αu + u = f(u) \hbox{ in } \mathbb{R}^n, \quad u \in H^α(\mathbb{R}^n), \qquad\qquad(P_ε) $$ where $ε>0$, $0< α< 1$, $(-Δ)_{ρ_ε}^α$ is a variational version of the regional laplacian, whose range of scope is a ball with radius $ρ_ε(x)=ρ(εx)>0$, where $ρ$ is a continuous function. We give general conditions on $ρ$ and $f$ which assure the existence and multiplicity of solution for $(P_ε)$.