The Brezis-Nirenberg type critical problem for the nonlinear Choquard equation

We establish some existence results for the Brezis-Nirenberg type problem of the nonlinear Choquard equation −Δu=(∫Ω|u(y)|2μ*|x−y|μdy)|u|2μ*−2u+λuinΩ,\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$ - \Delta u = \left( {\int_\Omega {\frac{{{{\left| {u\left( y \right)} \right|}^{2_\mu ^*}}}}{{{{\left| {x - y} \right|}^\mu }}}dy} } \right){\left| u \right|^{2_\mu ^* - 2}}u + \lambda uin\Omega ,$$\end{document}, where Ω is a bounded domain of RN with Lipschitz boundary, λ is a real parameter, N ≥ 3, 2μ*=(2N−μ)/(N−2)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$2_\mu ^* = \left( {2N - \mu } \right)/\left( {N - 2} \right)$$\end{document} is the critical exponent in the sense of the Hardy-Littlewood-Sobolev inequality.