Existence of nonnegative solutions for fractional Schr\"odinger equations with Neumann condition

TL;DR

This paper proves the existence of nonnegative solutions for a fractional Schrödinger equation with Neumann boundary conditions, using variational methods and Moser-Nash iteration to establish boundedness.

Contribution

It introduces new existence results for nonnegative solutions to fractional Schrödinger equations with Neumann conditions, employing a novel approach with the nonlocal normal derivative.

Findings

Existence of nonnegative, non-constant small energy solutions.

Solutions are shown to be bounded in L-infinity norm.

Methodology involves variational techniques and Moser-Nash iteration.

Abstract

In this paper we study a Neumann problem for the fractional Laplacian, namely \begin{equation}\left\{ \begin{array}{rcll} \varepsilon^{2s}(- \Delta)^{s}u + u &=& f(u) \ \ &\mbox{in} \ \ \Omega \\ \mathcal{N}_{s}u &=& 0 , \,\, &\text{in} \,\, \mathbb{R}^{N}\backslash \Omega \end{array}\right. \end{equation} where $\Omega \subset \mathbb{R}^{N}$ is a smooth bounded domain, $N>2s$, $s \in (0,1)$, $\varepsilon > 0$ is a parameter and $\mathcal{N}_{s}$ is the nonlocal normal derivative introduced by Dipierro, Ros-Oton, and Valdinoci. We establish the existence of a nonnegative, non-constant small energy solution $u_{\varepsilon}$, and we use the Moser-Nash iteration procedure to show that $u_{\varepsilon} \in L^{\infty}(\Omega)$.