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

• Weiming Liu
• Miaomiao Niu
• Yanfang Peng
Article

## 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.

## Keywords

Nonlinear fractional Schrödinger system Reduction method Vector solutions

## Mathematics Subject Classification

35B99 35J20 35J65

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