We study the concentration phenomenon for solutions of the fractional nonlinear Schr\"{ o}dinger equation, which is nonlocal. We mainly use the Lyapunov-Schmidt reduction method. Precisely, consider the nonlinear equation \begin{ equation}\label{ e:abstract} (-\varepsilon^2\Delta)^sv+Vv-|v|^{ \alpha}v=0\quad\mbox{ in}\quad\mathbf R^n, \end{ equation} where $n =1, 2, 3$, $\max\{ \frac{ 1}{ 2}, \frac{ n}{ 4}\}< s < 1$, $1 \leq \alpha < \alpha_*(s,n)$, $V\in C^3_{ b}(\mathbf{ R}^n)$. Here the exponent $\alpha_*(s,n)=\frac{ 4s}{ n-2s}$ for $0 < s < \frac{ n}{ 2}$ and $\alpha_*(s,n)=\infty$ for $s \geq\frac{ n}{ 2}$. Then for each non-degenerate critical point $z_0$ of $V$, there is a nontrivial solution of equation (\ref{ e:abstract}) concentrating to $z_0$ as $\varepsilon\to 0$.
Categories: math.AP math.FA