Brezis-Nirenberg type critical problem for nonlinear Choquard equation



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.


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


35J25 35J60 35A15 


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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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© 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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