标题： 带有垂直权重的上半空间上的逆Hardy-Littlewood-Sobolev不等式 显示英文标题

标题： Reversed Hardy-Littlewood-Sobolev inequalities with vertical weights on the upper half space

摘要： 在本文中，我们获得了上半空间上带有垂直权重的逆Hardy-Littlewood-Sobolev不等式，并讨论了极值函数。 我们通过引入归一化方法证明了该不等式中的最佳常数是可以达到的。 通过移动球面的方法讨论了相应极值函数的分类。 此外，我们通过使用弱意义下的Pohozaev恒等式证明了Euler-Lagrange方程正解存在的充分必要条件。 这种归一化方法是无需重排的，也可以用于证明扩展核和其他类似不等式的最佳（逆）Hardy-Littlewood-Sobolev不等式的极值函数的存在性。 显示英文摘要

摘要： In this paper, we obtain the reversed Hardy-Littlewood-Sobolev inequality with vertical weights on the upper half space and discuss the extremal functions. We show that the sharp constants in this inequality are attained by introducing a renormalization method. The classification of corresponding extremal functions is discussed via the method of moving spheres. Moreover, we prove the sufficient and necessary conditions of existence for positive solutions to the Euler-Lagrange equations by using Pohozaev identities in weak sense. This renormalization method is rearrangement free, which can be also applied to prove the existence of extremal functions for sharp (reversed) Hardy-Littlewood-Sobolev inequality with extended kernels and other similar inequalities.