Existence of concentrating solutions of the Hartree type Brezis-Nirenberg problem

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We are interested in the existence and asymptotic behavior of the solutions of the following critical Hartree equation with small parameter ε>0,(0.1){−Δu=(∫Ωu2μ⁎(y)|x−y|μdy)u2μ⁎−1+εu,inΩ,u=0,on∂Ω, where N≥5, μ∈(0,4], Ω is a bounded smooth domain in RN, and 2μ⁎=2N−μN−2 is the upper critical exponent in the sense of the Hardy-Littlewood-Sobolev inequality. By applying the reduction arguments, we prove that equation (0.1) has a family of solutions uε concentrating around the critical point of Robin function under some suitable assumptions, if ε→0, N≥5, μ∈(0,4] sufficiently close to 0, 4 or =4.