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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 nonlocalnormal derivative introduced by Dipierro, Ros-Oton, and Valdinoci. We establishthe existence of a nonnegative, non-constant small energy solution$u_{\varepsilon}$, and we use the Moser-Nash iteration procedure to show that$u_{\varepsilon} \in L^{\infty}(\Omega)$.

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