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

Abstract

<p style='text-indent:20px;'>In this paper, we classify the positive solutions to the following integral system</p><p style='text-indent:20px;'><disp-formula> <label/> <tex-math id="FE1"> \begin{document}$ \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&lt;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*} $\end{document} </tex-math></disp-formula></p><p style='text-indent:20px;'>where <inline-formula><tex-math id="M1">\begin{document}$ N\ge1, p_j&gt;1, 0&lt;h_{ij}&lt;N $\end{document}</tex-math></inline-formula> for all <inline-formula><tex-math id="M2">\begin{document}$ i, j\in\{1, 2, \cdots, k\} $\end{document}</tex-math></inline-formula>. Up to a positive constant multiplier, this system is the Euler-Lagrangian equations associated to the multilinear fractional integral inequality established by Beckner. Employing the method of moving spheres, we give the explicit form of positive solutions to the above system with <inline-formula><tex-math id="M3">\begin{document}$ p_j = \frac{\sum_{1\leq i&lt;j\leq k} \ \ h_{ij}-(k-3)N}{(k-1)N-\sum_{1\leq i&lt;j\leq k} \ \ h_{ij}} $\end{document}</tex-math></inline-formula> satisfying</p><p style='text-indent:20px;'><disp-formula> <label/> <tex-math id="FE2"> \begin{document}$ \sum\limits^k_{j = 1}\frac{1}{p_j+1} = \sum\limits_{1\leq i&lt;j\leq k}\frac{N-h_{ij}}{N}, $\end{document} </tex-math></disp-formula></p><p style='text-indent:20px;'>and show the nonexistence of positive solutions for <inline-formula><tex-math id="M4">\begin{document}$ p_j&gt;1 $\end{document}</tex-math></inline-formula> with</p><p style='text-indent:20px;'><disp-formula> <label/> <tex-math id="FE3"> \begin{document}$ \sum\limits^k_{j = 1}\frac{1}{p_j+1}&gt;\sum\limits_{1\leq i&lt;j\leq k}\frac{N-h_{ij}}{N}. $\end{document} </tex-math></disp-formula></p>