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The Brezis-Nirenberg type critical problem for the 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 RN 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.

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Acknowledgements

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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  1. Department of Mathematics, Zhejiang Normal University, Jinhua, 321004, China

    Fashun Gao & Minbo Yang

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  1. Fashun Gao
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  2. Minbo Yang
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Correspondence to Minbo Yang.

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Gao, F., Yang, M. The Brezis-Nirenberg type critical problem for the nonlinear Choquard equation. Sci. China Math. 61, 1219–1242 (2018). https://doi.org/10.1007/s11425-016-9067-5

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