Classifications of positive solutions to an integral system involving the multilinear fractional integral inequality

In this paper, we classify the positive solutions to the following integral system

$ \begin{eqnarray*} \begin{cases} u_s(x_s) = \int_{\mathbb{R}^{N(k-1)}}\frac{\prod\limits_{j\neq s}u^{p_j}_j(x_j)}{\prod_{1\leq i<j\leq k}|x_i-x_j|^{N-h_{ij}}}dX_{\widehat{s}}, \\ u_s\geq0, \; \rm{in}\quad\mathbb{R}^N, \quad s = 1, 2, \cdots, k, \end{cases} \end{eqnarray*} $

where $ N\ge1, p_j>1, 0<h_{ij}<N $ for all $ i, j\in\{1, 2, \cdots, k\} $. Up to a positive constant multiplier, this system is the Euler-Lagrangian equations associated to the multilinear fractional integral inequality established by Beckner. Employing the method of moving spheres, we give the explicit form of positive solutions to the above system with $ p_j = \frac{\sum_{1\leq i<j\leq k} \ \ h_{ij}-(k-3)N}{(k-1)N-\sum_{1\leq i<j\leq k} \ \ h_{ij}} $ satisfying

$ \sum\limits^k_{j = 1}\frac{1}{p_j+1} = \sum\limits_{1\leq i<j\leq k}\frac{N-h_{ij}}{N}, $

and show the nonexistence of positive solutions for $ p_j>1 $ with

$ \sum\limits^k_{j = 1}\frac{1}{p_j+1}>\sum\limits_{1\leq i<j\leq k}\frac{N-h_{ij}}{N}. $