Existence and symmetry of positive solutions of an integral equation system (Q614321)

The main result of this paper is the following existence and uniqueness theorem. Assume \((u,v)\) is a pair of positive \({C}^1\) solutions of the system of integral equations \[ u(x) = \int_{\mathbb{R}^n} |x-y|^{\alpha -n} v^p(y) dy,\quad v(x) =\int_{\mathbb{R}^n} |x-y|^{\beta -n} u^q(y) dy, \] where \( 0<\alpha, \beta < n,\) \(1<p\leq \frac{n+\alpha}{n-\beta},\) \(1<q < \frac{n+\beta}{n-\alpha},\) then this system of integral equations has no positive solution unless \(p=\frac{n+\alpha}{n-\beta}\) and \(q=\frac{n+\beta}{n-\alpha}.\) Furthermore, when \(p=\frac{n+\alpha}{n-\beta}\) and \(q=\frac{n+\beta}{n-\alpha},\) \(u(x), v(x)\) are radially symmetric and monotonic about some points with the form \[ u(x) = \biggl( \frac{a_1}{d_1+|x-x_0|^2} \biggr )^{\frac{n-\alpha}{2}},\quad v(x) = \biggl( \frac{a_2}{d_2+|x-x_0|^2} \biggr )^{\frac{n-\beta}{2}}, \] where \(a_1, d_1, a_2, d_2 >0\) are constants, and \(x_0\in\mathbb{R}^n\).