Liouville theorems for an integral system with Poisson kernel on the upper half space

Abstract

We investigate the Liouville theorem for an integral system with Poisson kernel on the upper half space R ⁿ₊ ,

$$\left\{ {\begin{array}{*{20}{c}} {u\left( x \right) = \frac{2}{{n{w_n}}}\int_{\partial \mathbb{R}_ + ^n} {\frac{{{x_n}f\left( {v\left( y \right)} \right)}}{{{{\left| {x - y} \right|}^n}}}dy,\;x \in \mathbb{R}_ + ^n,} } \\ {v\left( y \right) = \frac{2}{{n{w_n}}}\int_{\mathbb{R}_ + ^n} {\frac{{{x_n}g\left( {u\left( x \right)} \right)}}{{{{\left| {x - y} \right|}^n}}}dx,\;y \in \partial \mathbb{R}_ + ^n,} } \end{array}} \right.$$

, where n ≥ 3, ω _n is the volume of the unit ball in Rⁿ. This integral system arises from the Euler-Lagrange equation corresponding to an integral inequality on the upper half space established by Hang et al. (2008). With natural structure conditions on f and g, we classify the positive solutions of the above system based on the method of moving spheres in integral form and the inequality mentioned above.