Brezis-Nirenberg type critical problem for nonlinear Choquard equation

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Abstract

We establish some existence results for the Brezis-Nirenberg type problem of the nonlinear Choquard equation
$$ - \Delta u = \left( {\int_\Omega {\frac{{{{\left| {u\left( y \right)} \right|}^{2_\mu ^*}}}}{{{{\left| {x - y} \right|}^\mu }}}dy} } \right){\left| u \right|^{2_\mu ^* - 2}}u + \lambda uin\Omega ,$$
, where Ω is a bounded domain of R N with Lipschitz boundary, λ is a real parameter, N ≥ 3, \(2_\mu ^* = \left( {2N - \mu } \right)/\left( {N - 2} \right)\) is the critical exponent in the sense of the Hardy-Littlewood-Sobolev inequality.

Keywords

Brezis-Nirenberg problem Choquard equation Hardy-Littlewood-Sobolev inequality critical exponent 

MSC(2010)

35J25 35J60 35A15 

Notes

Acknowledgments

This work was supported by National Natural Science Foundation of China (Grant Nos. 11571317 and 11671364) and Natural Science Foundation of Zhejiang (Grant No. LY15A010010). The authors thank the anonymous referees for their useful comments and suggestions which helped to improve the presentation of the paper greatly.

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Copyright information

© Science China Press and Springer-Verlag GmbH Germany, part of Springer Nature 2018

Authors and Affiliations

  1. 1.Department of MathematicsZhejiang Normal UniversityJinhuaChina

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