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

Authors:

Institution:

College of Mathematics and Information Science, Henan Normal University, People's Republic of China ; Department of Applied Mathematics, University of Colorado, Boulder, Colorado 80309

Abstract:

We prove the uniqueness for the solutions of the singular nonlinear PDE system:

$\displaystyle \left\{\begin{array}{ll} - \lap ( \vert x\vert^{\alpha} u(x) ) = ... ...rt^{\beta} v(x) ) = \dfrac{u^p (x)}{\vert x\vert^{\alpha}}. \end{array} \right.$

(1)

In the special case when $\alpha = \beta$ and , we classify all the solutions and thus obtain the best constant in the corresponding weighted Hardy-Littlewood-Sobolev inequality.

Keywords:

Weighted Hardy-Littlewood-Sobolev inequality best constants system of singular PDEs uniqueness radial symmetry classifications

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