Abstract: In this paper, we study an integral system involving $m$ equations $$\left\{ \begin{aligned} u_i(x) &= \int_{\mathbb{R}^n}\frac{u_{i+1}^{p_{i+1}}(y)}{|x-y|^{n-\alpha}}{\rm d}y, \quad i=1,2,\cdots,m-1, \\ u_m(x) &= \int_{\mathbb{R}^n}\frac{u_{1}^{p_{1}}(y)}{|x-y|^{n-\alpha}}{\rm d}y, \quad m\geq3,\ n\geq1, \end{aligned} \right. $$ where $u_i>0$ in $\mathbb{R}^n$, $0<\alpha<n$, and $p_i>1 \ (i=1,2,\cdots,m).$ Based on the optimal integrability intervals, we estimate the decay rates of the positive solutions of the system at infinity. But estimating these rates is difficult because the relation between $p_i \ (i=1,2,\cdots,m)$ is uncertain. To overcome this difficulty, we obtain the asymptotic behavior of all cases by discussing them separately. In addition, we also get the radial symmetry of positive solutions under some integrability condition.

Key words: integral equation, Riesz potentials, radial symmetry, asymptotic behavior