In this paper, we consider the Choquard-type equation with the fractional Laplacian $\left\{\begin{array}{ll}(-\mathrm{\Delta }{)}^{s}u(x)+au(x)=\left(\frac{1}{|x{|}^{n-\alpha }}\ast u^p\right)u^{p-1},& x\in {\mathbb{R}}^n,\\ u\left(x\right)>0,& x\in {\mathbb{R}}^n,\end{array}\right.$ where 0<s<1, $0<\alpha <n$, $1\le p<\mathrm{\infty }$, $a\ge 0$ is a constant and $n\ge 2$. We will give a variant decay at infinity and a variant narrow region principle, then combining the direct method of moving planes to prove the solutions of the above equation must be radially symmetric and monotone decreasing about some point in ${\mathbb{R}}^n$.

在本文中，我们考虑分数拉普拉斯算子的Choquard型方程 $\left\{\begin{array}{ll}（-\mathrm{\Delta }{）}^sü（X）+\mathrm{一种}ü（X）=\left(\frac{\mathrm{1个}}{|X{|}^{ñ-\alpha }}\ast {ü}^p\right){ü}^{p-\mathrm{1个}}，& X\in {\mathbb{[R}}^{ñ}，\\ ü（X）>0，& X\in {\mathbb{[R}}^{ñ}，\end{array}\right.$0 < s <1，$0<\alpha <ñ$， $\mathrm{1个}\le p<\mathrm{\infty }$， $\mathrm{一种}\ge 0$ 是一个常数， $ñ\ge 2$。我们将给出无穷大处的变体衰减和窄区域变体原理，然后结合移动平面的直接方法来证明上述方程的解必须是径向对称的，并且单调在点处减小${\mathbb{[R}}^{ñ}$。