Liouville type theorems for general weighted integral system with negative exponents

Abstract

In this paper, we consider the weighted integral system with negative exponents on the upper half space $ \mathbb{R}^{n+1}_+ $ as follows$ \begin{equation*} \begin{cases} u(X) = \displaystyle{\int}_{\mathbb{R}^{n+1}_+}\frac{f(u, v)(Y)}{t^\alpha z^\beta|X-Y|^\lambda}dY, &X\in\mathbb{R}^{n+1}_+, \\ v(X) = \displaystyle{\int}_{\mathbb{R}^{n+1}_+}\frac{g(u, v)(Y)}{ t^\beta z^\alpha|X-Y|^\lambda}dY, &X\in\mathbb{R}^{n+1}_+, \end{cases} \end{equation*} $where $ \alpha, \beta\le0 $, $ \lambda<0 $ and $ X = (x, t), \, Y = (y, z). $ Under the natural conditions on $ f $ and $ g $, we obtain the classification and symmetry of positive solutions by the method of moving spheres in integral forms. Moreover, we generalize our results to integral system on $ \mathbb{R}^{n+m} $.