Axisymmetry of locally bounded solutions to an Euler-Lagrangesystem of the weighted Hardy-Littlewood-Sobolev inequality

This paper is concerned with the symmetry results for the $2k$-order singular Lane-Emden type partial differential system $$ \left\{\begin{array}{ll} (-\Delta)^k(|x|^{\alpha}u(x)) =|x|^{-\beta} v^{q}(x), \\ (-\Delta)^k(|x|^{\beta}v(x)) =|x|^{-\alpha} u^p(x), \end{array} \right. $$ and the weighted Hardy-Littlewood-Sobolev type integral system $$ \left \{ \begin{array}{l} u(x) = \frac{1}{|x|^{\alpha}}\int_{R^{n}} \frac{v^q(y)}{|y|^{\beta}|x-y|^{\lambda}} dy\\ v(x) = \frac{1}{|x|^{\beta}}\int_{R^{n}} \frac{u^p(y)}{|y|^{\alpha}|x-y|^{\lambda}} dy. \end{array} \right. $$ Here $x \in R^n \setminus \{0\}$. We first establish the equivalence of this integral system and an fractional order partial differential system, which includes the $2k$-order PDE system above. For the integral system, we prove that the positive locally bounded solutions are symmetric and decreasing about some axis by means of the method of moving planes in integral forms introduced by Chen-Li-Ou. In addition, we also show that the integrable solutions are locally bounded. Thus, the equivalence implies the positive solutions of the PDE system, particularly including the higher integer-order PDE system, also have the corresponding properties.