Vector solutions with clustered peaks for nonlinear fractional Schrödinger systems in $\mathbb {R}^{N}$

Abstract

We consider the fractional nonlinear Schrödinger system

$$\begin{aligned} \left\{ \begin{array}{ll} \epsilon ^{2s}(-\Delta )^s u +P_1( x)u=\mu _1 |u|^{2p-2}u+\beta |v|^p|u|^{p-2}u, \quad x\in \mathbb {R}^N,\\ \epsilon ^{2s}(-\Delta )^s v +P_2( x)v=\mu _2 |v|^{2p-2}v+\beta |u|^p|v|^{p-2}v, \quad \; x\in \mathbb {R}^N,\\ \end{array} \right. \end{aligned}$$

where $\epsilon >0$ is a small parameter, $0<s<1,$ $P_1$ and $P_2$ are positive potentials, $\mu _1>0,~\mu _2>0$, and $\beta \in \mathbb {R}$ is a coupling constant. To construct solutions to this system, we use the Lyapunov–Schmidt reduction that takes advantage of the variational structure of the problem. For any positive integer $k\ge 2$, we construct k interacting spikes concentrating near the local maximum point $x_{0}$ of $P_1$ and $P_2$ when $P_{1}(x_{0})=P_{2}(x_{0})$ in the attractive case. For any two positive integers $k,m\ge 2$, we construct k interacting spikes for u near the local maximum point $x_{1,0}$ of $P_1$ and m interacting spikes for v near the local maximum point $x_{2,0}$ of $P_2$, respectively, when $x_{1,0}\ne x_{2,0}$. For $s = 1$, this corresponds to the system studied by Peng and Pi (Discrete Contin Dyn Syst 36:2205–2227, 2016) for the classical nonlinear Schrödinger system.