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

Abstract

We consider the fractional nonlinear Schrodinger 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 Schrodinger system.