Symmetry and nonexistence of positive solutions to an integral system with weighted functions

Abstract

Consider the system of integral equations with weighted functions in ℝⁿ,

$$ \left\{ \begin{gathered} u\left( x \right) = \int_{\mathbb{R}^n } {\left| {x - y} \right|^{\alpha - n} Q\left( y \right)v\left( y \right)^q dy,} \hfill \\ v\left( x \right) = \int_{\mathbb{R}^n } {\left| {x - y} \right|^{\alpha - n} K\left( y \right)u\left( y \right)^p dy,} \hfill \\ \end{gathered} \right. $$

where 0 < α < n, $\frac{1} {{p + 1}} + \frac{1} {{q + 1}} \geqslant \frac{{n - \alpha }} {n},\frac{\alpha } {{n - \alpha }}$, Q(x) and K(x) satisfy some suitable conditions. It is shown that every positive regular solution (u(x), v(x)) is symmetric about some plane by developing the moving plane method in an integral form. Moreover, regularity of the solution is studied. Finally, the nonexistence of positive solutions to the system in the case 0 < p,q < $\frac{{n + \alpha }} {{n - \alpha }}$ is also discussed.