Singularly perturbed Neumann problem for fractional Schr\"odinger equations

TL;DR

This paper studies a Neumann boundary problem for fractional Schrödinger equations with singular perturbations, establishing existence of small energy solutions and analyzing their integrability properties.

Contribution

It introduces a new Neumann boundary condition for fractional Schrödinger equations and proves the existence of small energy solutions in this setting.

Findings

Existence of nonnegative small energy solutions.

Solutions are integrable on ^n.

Boundary condition involves a nonlocal integral constraint.

Abstract

This paper is concerned with a Neumann type problem for singularly perturbed fractional nonlinear Schr\"odinger equations with subcritical exponent. For some smooth bounded domain $\Omega\subset \mathbf R^n$, our boundary condition is given by \begin{equation*} \int_{\Omega}\frac{u(x)-u(y)}{|x-y|^{n+2s}}dy=0\quad\mbox{for }x\in \mathbf R^n\setminus\bar\Omega. \end{equation*} We establish existence of nonnegative small energy solutions, and also investigate the integrability of the solutions on $\mathbf R^n$.