Symmetry of integral equation systems on bounded domains (Q624602)

The authors investigate the symmetry of domains and solutions of integral equation systems on bounded domains. They consider the following integral equation systems on bounded domain \(\Omega \subset\mathbb R^n\) with \(\partial \Omega \in C^1\) \[ \begin{aligned} u(x) &= A\int_{\Omega} |x-y|^{\alpha-n}(v(y))^p dy + C, \quad x \in \Omega\\ v(x) &= B\int_{\Omega} |x-y|^{\beta-n}(u(y))^q dy + D, \quad x \in \Omega,\end{aligned} \] \(p, q > 1,\) \(0 < \alpha, \beta < n\). Under some natural integrability conditions, they prove that the domains are balls, all positive solutions of the systems are radially symmetric and monotone decreasing with respect to the radius.