Concentration phenomenon for fractional nonlinear Schrödinger equations

Abstract

We study the concentration phenomenon for solutions of the fractional nonlinear Schrödinger equation, which is nonlocal. We mainly use the Lyapunov-Schmidt reduction method. Precisely, consider the nonlinear equation \begin{equation}\label{e:abstract} (-\varepsilon^2Δ)^sv+Vv-|v|^αv=0\quad\mbox{in}\quad\mathbf R^n, \end{equation} where $n=1,2,3$, $\max \{\frac{1}{2},\frac{n}{4}\}<s<1$, $1\le \alpha <{\alpha }_{\ast }(s,n)$, $V\in C_b^3({\mathbf{R}}^n)$. Here the exponent ${\alpha }_{\ast }(s,n)=\frac{4s}{n-2s}$ for $0<s<\frac{n}{2}$ and ${\alpha }_{\ast }(s,n)=\infty$ for $s\ge \frac{n}{2}$. 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 $\epsilon \to 0$.

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