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This paper deals with the following fractional Schr$ \ddot{\textrm{o}}$dingerequations with Choquard-type nonlinearities \begin{equation*}\left\{\begin{array}{r@{\ \ }c@{\ \ }ll}  (-\Delta)^{\frac{\alpha}{2}}u + u - C_{n,-\beta} \,(|x|^{\beta-n}\astu^{p})\, u^{p-1}& = &0 & \mbox{in}\ \ \mathbb{R}^{n}\,, \\[0.05cm]  u & > & 0 & \mbox{on}\ \ \mathbb{R}^{n}, \end{array}\right. \end{equation*}where $ 0< \alpha,\beta < 2, 1\leq p <\infty \,\,and\,\, n\geq 2. $ First weconstruct a decay result at infinity and a narrow region principle for relatedequations. Then we establish the radial symmetry of positive solutions for theabove equation with the generalized direct method of moving planes.

Symmetry of Positive Solutions for the Fractional Schr$ \ddot{\textrm{o}}$dinger Equations with Choquard-type Nonlinearities