We investigate the nonnegative solutions of the system involving the fractional Laplacian: $$\{\begin{array}{l}{(-\Delta )}^{\alpha}{{\displaystyle u}}_{i}(x)={{\displaystyle f}}_{i}(u),\hspace{1em}x\in {?}^{n},i=\mathrm{1},\mathrm{2},\mathrm{...},m,\\ u(x)=({u}_{\mathrm{1}}(x),{u}_{\mathrm{2}}(x),\mathrm{...},{u}_{m}(x)),\end{array}$$

Where $\mathrm{0}\u27e8\alpha \u27e8\mathrm{1},n\u27e9\mathrm{2},{{\displaystyle 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 ${{\displaystyle p}}_{i}=(n+\mathrm{2}\alpha )/(n-\mathrm{2}\alpha )$ for each 1≤*i*≤*m*; and the only nonnegative solution of this system is *u* ≡ 0 if $\mathrm{1}\u27e8{{\displaystyle p}}_{i}\u27e8(n+\mathrm{2}\alpha )/(n-\mathrm{2}\alpha )$ for all 1≤*i*≤*m*.

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