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 − Δ u = ( ∫ Ω | u ( y ) | 2 μ * | x − y | μ d y ) | u | 2 μ * − 2 u + λ u i n Ω , , where Ω is a bounded domain of R N with Lipschitz boundary, λ is a real parameter, N ≥ 3, 2 μ * = ( 2 N − μ ) / ( N − 2 ) is the critical exponent in the sense of the Hardy-Littlewood-Sobolev inequality.