Singularly perturbed Neumann problem for fractional Schrödinger equations

This paper is concerned with a Neumann type problem for singularly perturbed fractional nonlinear Schrödinger equations with subcritical exponent. For some smooth bounded domain Ω ⊂ Rn, our boundary condition is given by ∫u(x)−u(y)|x−y|n+2sdy=0forx∈ℝn∖Ω¯\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\int {\frac{{u\left( x \right) - u\left( y \right)}}{{{{\left| {x - y} \right|}^{n + 2s}}}}} dy = 0forx \in {\mathbb{R}^n} \setminus \overline \Omega $$\end{document}. We establish existence of non-negative small energy solutions, and also investigate the integrability of the solutions on Rn.