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Achievability of a supremum for the Hardy-Littlewood-Sobolev inequality with supercritical exponent

  • Xiaoming An
  • Shuangjie PengEmail author
  • Chaodong Xie
Article

Abstract

In this paper, we prove that the supremum
$$\sup \left\{{\int_B {\int_B {{{{{\left| {u(y)} \right|}^{p(\left| y \right|)}}{{\left| {u(x)} \right|}^{p(\left| x \right|)}}} \over {{{\left| {x - y} \right|}^\mu}}}}} dxdy:u \in H_{0,{\rm{rad}}}^1(B),\;\;{{\left\| {\nabla u} \right\|}_{{L^2}(B)}} = 1} \right\}$$
is attained, where B denotes the unit ball in ℝN (N ⩾ 3), μ ∈ (0, N), p(r) = 2 μ * + rt, t ∈ (0, min{N/2 − μ/4, N − 2}) and 2 μ * = (2Nμ)/(N − 2) is the critical exponent for the Hardy-Littlewood-Sobolev inequality.

Keywords

Hardy-Littlewood-Sobolev inequality achievability of a supremum supercritical exponent 

MSC(2010)

35J20 35J25 35J30 

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Notes

Acknowledgements

This work was supported by National Natural Science Foundation of China (Grant Nos. 11831009 and 11571130).

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Copyright information

© Science China Press and Springer-Verlag GmbH Germany, part of Springer Nature 2019

Authors and Affiliations

  1. 1.School of Mathematics and Statistics, Hubei Key Laboratory of Mathematical SciencesCentral China Normal UniversityWuhanChina
  2. 2.School of Economics ManagementGuizhou University for Ethinic MinoritiesGuiyangChina

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