In this paper, we study a fractional NLS system with trapping potentials in \({\mathbb {R}}^{2}\). By constructing a constrained minimization problem, we show that minimizers exist for the minimization problem if and only if the attractive interaction strength \(a_{i}<a^{*}{:}{=}\Vert Q\Vert _{2}^{2s}\), where \(i=1, 2\) and Q is the unique positive radial solution of \((-\Delta )^{s}u+su-|u|^{2s}u=0\) in \({\mathbb {R}}^{2}\), \(s\in (0, 1)\). Moreover, by analyzing some precise energy estimates, we obtain the concentration and blow-up behavior for the minimizers of the minimization problem as \((a_{1}, a_{2})\nearrow (a^{*}, a^{*})\). Comparing to the NLS system and fractional NLS equation, we encounter some new difficulties because of the nonlocal nature of the fractional Laplace. One of the main difficulties is that the energy functional is changed, we have to develop a suitable trial function to do some precise integral computation for the energy of minimization problem. Another difficulty is given by the fact that the Q(x) is polynomially decay at infinity, which is in contrast to the fact that the ground state exponentially decays at infinity in \(s=1\), we need to give a more detailed proof to establish the best estimate of the trial function. The last major difficulty lies in the decay estimates of the sequences of solution to the nonlocal problem at infinity are different from those in the case of the classical local problem, we must build decay estimates for nonlocal operators.

在本文中，我们研究了在\({\mathbb {R}}^{2}\)中具有捕获势的分数 NLS 系统。通过构造一个受约束的最小化问题，我们表明当且仅当有吸引力的相互作用强度\(a_{i}<a^{*}{:}{=}\Vert Q\Vert _{2 }^{2s}\)，其中\(i=1, 2\)和Q是\((-\Delta )^{s}u+su-|u|^{2s}u的唯一正径向解=0\)在\({\mathbb {R}}^{2}\)，\(s\in (0, 1)\)。此外，通过分析一些精确的能量估计，我们获得了最小化问题的最小值的集中和爆炸行为\((a_{1}, a_{2})\nearrow (a^{*}, a^ {*})\). 与 NLS 系统和分数阶 NLS 方程相比，由于分数阶拉普拉斯的非局部性质，我们遇到了一些新的困难。主要困难之一是能量泛函改变了，我们必须开发一个合适的试函数来对能量最小化问题进行一些精确的积分计算。另一个困难是Q ( x ) 在无穷大处呈多项式衰减，这与基态在\(s=1\)中在无穷大呈指数衰减的事实形成对比, 我们需要给出更详细的证明来建立试验函数的最佳估计。最后一个主要困难在于无穷远非局部问题的解序列的衰减估计与经典局部问题的情况不同，我们必须为非局部算子建立衰减估计。