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

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## 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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## Copyright information

© Springer International Publishing AG, part of Springer Nature 2017

## Authors and Affiliations

1. 1.School of Mathematics and StatisticsHubei Normal UniversityHuangshiPeople’s Republic of China
2. 2.College of Applied ScienceBeijing University of TechnologyBeijingPeople’s Republic of China
3. 3.Department of Mathematics and Computer ScienceGuizhou Normal UniversityGuiyangPeople’s Republic of China