We investigate the nonnegative solutions of the system involving the fractional Laplacian:
$$\left\{ {\begin{array}{*{20}c} {\begin{array}{*{20}c} {( - \Delta )^\alpha u_i (x) = f_i (u),} & {x \in \mathbb{R}^n , i = 1,2, \ldots ,m,} \\ \end{array} } \\ {u(x) = (u_1 (x),u_2 (x), \ldots ,u_m (x)),} \\ \end{array} } \right.$$
where 0 <
α < 1, n > 2,
f i (
u), 1 ≤
i ≤
m, are real-valued nonnegative functions of homogeneous degree
p i ≥ 0 and nondecreasing with respect to the independent variables
u 1,
u 2,...,
u m . By the method of moving planes, we show that under the above conditions, all the positive solutions are radially symmetric and monotone decreasing about some point
x 0 if
p i = (
n + 2
α)/(
n ? 2
α) for each 1 ≤
i ≤
m; and the only nonnegative solution of this system is
u ≡ 0 if 1 <
p i < (
n + 2
α)/(
n ? 2
α) for all 1 ≤
i ≤
m.