In the present paperwe study the existence of nontrivial solutions of a class of static coupled nonlinear fractional Hartree type system. First, we use the direct moving plane methods to establish the maximum principle(Decay at infinity and Narrow region principle) and prove the symmetry and nonexistence of positive solution of this nonlocal system. Second, we make complete classification of positive solutions of the system in the critical case when some parameters are equal. Finally, we prove the existence of multiple nontrivial solutions in the critical case according to the different parameters ranges by using variational methods. To accomplish our results we establish the maximum principle for the fractional nonlocal system.

中文翻译：

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具有非局部相互作用的非线性分数Hartree型系统的定性分析

本文研究了一类静耦合非线性分数Hartree型系统的非平凡解的存在性。首先，我们利用直动平面法建立了极大值原理（无限衰减和窄域原理），证明了这个非局域系统正解的对称性和不存在性。其次，在某些参数相等的情况下，我们对系统的正解进行了完整分类。最后，我们通过变分方法证明了在临界情况下，根据不同的参数范围，存在多个非平凡解。为了完成我们的结果，我们建立了分数非局部系统的最大原理。