Existence and multiplicity of solutions for a nonlinear Schr\"odinger equation with non-local regional diffusion

TL;DR

This paper investigates the existence and multiplicity of solutions for a nonlinear Schrödinger equation involving a non-local regional diffusion operator, under general conditions on the diffusion scope and nonlinearity.

Contribution

It establishes new conditions on the regional diffusion scope and nonlinearity that guarantee solutions and multiple solutions for the non-local Schrödinger equation.

Findings

Existence of solutions under broad conditions.

Multiple solutions are proven to exist.

Results depend on the properties of the function  and the scope function .

Abstract

In this article we are interested in the following non-linear Schr\"odinger equation with non-local regional diffusion $$ (-\Delta)_{\rho_\epsilon}^{\alpha}u + u = f(u) \hbox{ in } \mathbb{R}^n, \quad u \in H^\alpha(\mathbb{R}^n), \qquad\qquad(P_\epsilon) $$ where $\epsilon >0$, $0< \alpha < 1$, $(-\Delta)_{\rho_\epsilon}^{\alpha}$ is a variational version of the regional laplacian, whose range of scope is a ball with radius $\rho_\epsilon(x)=\rho(\epsilon x)>0$, where $\rho$ is a continuous function. We give general conditions on $\rho$ and $f$ which assure the existence and multiplicity of solution for $(P_\epsilon)$.