Existence of nonnegative solutions for fractional Schrödinger equations with Neumann condition

In this paper we study a Neumann problem for the fractional Laplacian, namely \begin{equation}\left{ \begin{array}{rcll} \varepsilon^{2s}(- Δ)^{s}u + u &=& f(u) \ \ &\mbox{in} \ \ Ω\ $\mathcal{N}$_{s}u &=& 0 , ,, &$\text{in}$ ,, $\mathbb{R}$^{N}\backslash Ω\end{array}\right. \end{equation} where $\mathrm{\Omega }\subset {\mathbb{R}}^N$ is a smooth bounded domain, $N>2s$, $s\in (0,1)$, $\epsilon >0$ 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 $u_{\epsilon }$, and we use the Moser-Nash iteration procedure to show that $u_{\epsilon }\in L^{\mathrm{\infty }}(\mathrm{\Omega })$.