Optimal integrability for some integral system of Wolff type

Abstract

In the paper, we obtain the optimal integrability for positive solutions of the following integral system involving Wolff potential:
$$
\left\{
\begin{array}{ll}
&u(x)=W_{\beta,\gamma}(v^{q})(x), \quad x\in R^{n},\\[3mm]
&v(x)=W_{\beta,\gamma}(u^{p})(x), \quad x\in R^{n},
\end{array}
\right.
$$
where $p,q>0$, $\beta>0$, $\gamma>1$ and $0<\beta\gamma<n.$ Ma, Chen and Li [\emph{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.