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

Chen, Wenxiong; Li, Congming

doi:10.1090/S0002-9939-07-09232-5

The best constant in a weighted Hardy-Littlewood-Sobolev inequality
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Proc. Amer. Math. Soc. 136 (2008), 955-962 Request permission

Abstract:

We prove the uniqueness for the solutions of the singular nonlinear PDE system: \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 $\alpha = \beta$ and $p = q$, we classify all the solutions and thus obtain the best constant in the corresponding weighted Hardy-Littlewood-Sobolev inequality.

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