Liouville type theorems for integral equations and integral systems (Q1937824)

The author establishes some Liouville type theorems for positive solutions of integral equations and integral systems in \(\mathbb R^N\) such as \[ u(x) = \int\limits_{\mathbb R^N} \frac{1}{|x-y|^{N-\alpha}} f(u(y)) dy, \, x \in \mathbb R^N, \] and \[ \begin{aligned} u(x) = \int\limits_{\mathbb R^N} \frac{1}{|x-y|^{N-\alpha}} f(u(y), v(y)) dy, \, x \in \mathbb R^N,\\ v(x) = \int\limits_{R^N} \frac{1}{|x-y|^{N-\alpha}} g(u(y), v(y)) dy, \, x \in \mathbb R^N, \end{aligned} \] where \(N \geq 2\), \(0 < \alpha < N\). The method of moving planes is used as the main technique.