On the Wolff-type Integral System with Negative Exponents

In this paper, we are concerned with the positive continuous entire
solutions of the Wolff-type integral system
\begin{equation*}
\left\{
\begin{array}{ll} &u(x)
=C_{1}(x)W_{\beta,\gamma}
(v^{-q})(x), \\[3mm] &v(x)
=C_{2}(x)W_{\beta,\gamma}
(u^{-p})(x), \end{array} \right.
\end{equation*} where $n\geq1$,
$\min\{p,q\}>0$,
$\gamma>1$,
$\beta>0$ and
$\beta\gamma\neq n$. In
addition, $C_{i}(x) \ (i=1,2)$ are some double
bounded functions. If
$\beta\gamma\in (0,n)$,
the Serrin-type condition is critical for existence of the positive
solutions for some double bounded functions $C_{i}(x)$ $(i=1,2)$.
Such an integral equation system is related to the study of the
$\gamma$-Laplace system and $k$-Hessian system with
negative exponents. Estimated by the integral of the Wolff type
potential, we obtain the asymptotic rates and the integrability of
positive solutions, and studied whether the radial solutions exist.