Optimal integrability for some integral system of Wolff type

In the paper, we obtain the optimal integrability for positive solutions of the following integral system involving Wolff potentials: {u(x) = W?,?(vq)(x); x ?, Rn, v(x) = W?,?(up)(x), x ? Rn, where p,q > 0,? > 0, ? > 1 and 0 < ?? < n. Ma, Chen and Li [Advances in Mathematics, 226(2011), 2676-2699] developed the regularity lifting method and obtained the optimal integrability for p > 1, q > 1. Here, based on some new observations, we overcome the difficulty there, and derive the optimal integrability for the case of p > 0,q > 0 and pq > 1. This integrability plays a key role in estimating the asymptotic behavior of positive solutions.