Singularly perturbed Neumann problem for fractional Schr\"odinger equations

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 $\mathbf R^n$.