The best constant in a weighted Hardy-Littlewood-Sobolev inequality

Abstract

We prove the uniqueness for the solutions of the singular nonlinear PDE system: (1) { − δ ( | x | α u ( x ) ) = v q ( x ) | x | β , − δ ( | x | β v ( x ) ) = u p ( x ) | x | α . \begin{equation}\tag {1} \begin {cases} - \delta ( |x|^{\alpha } u(x) ) = \dfrac {v^q (x)}{|x|^{\beta }} ,\\ - \delta ( |x|^{\beta } v(x) ) = \dfrac {u^p (x)}{|x|^{\alpha }}. \end{cases} \end{equation} In the special case when α = β \alpha = \beta and p = q p = q , we classify all the solutions and thus obtain the best constant in the corresponding weighted Hardy-Littlewood-Sobolev inequality.