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 $u_{ \varepsilon} \in L^{ \infty}(\Omega)$.
Categories: math.AP