Liouville theorem and qualitative properties of solutions for an integral system

In this paper, we are concerned with an integral system $$ \left\{ \begin{aligned} &u(x)= W_{\beta,\gamma}(u^{p-1}v)(x), \ u>0 \ \text{in} \ R^{n},\\ &v(x)=I_{\alpha}(u^{p})(x), \ v>0 \ \text{in} \ R^{n}, \end{aligned} \right. $$ where $p>0,$ $0<\alpha, \beta\gamma1$. Base on the integrability of positive solutions, we obtain some Liouville theorems and the decay rates of positive solutions at infinity. In addition, we use the properties of the contraction map and the shrinking map to prove that $u$ is Lipschitz continuous. In particular, the Serrin type condition is established, which plays an important role to classify the positive solutions.