Liouville Type Theorems for General Integral System with Negative Exponents

Abstract

In this paper, we establish a Liouville type theorem for the following integral system with negative exponents \[ \begin{cases} u(x) = \int_{\mathbb{R}^n} |x-y|^{\nu} f(u,v)(y) \, dy, & x \in \mathbb{R}^n, \\ v(x) = \int_{\mathbb{R}^n} |x-y|^{\nu} g(u,v)(y) \, dy, & x \in \mathbb{R}^n, \end{cases} \] where $n \geq 1$, $\nu \gt 0$, and $f$, $g$ are continuous functions defined on $\mathbb{R}_{+} \times \mathbb{R}_{+}$. Under nature structure conditions on $f$ and $g$, we classify each pair of positive solutions for above integral system by using the method of moving sphere in integral forms. Moreover, some other Liouville theorems are established for similar integral systems.