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

We are interested in the existence and asymptotic behavior of the solutions of the following critical Hartree equation with small parameter epsilon> 0, { - Delta u = (integral(Omega) u(2*) (mu)(y) / | x - y|(mu) dy) u(2*) (mu -1) + epsilon u, in Omega, u = 0, on. partial derivative Omega, (0.1) where N >= 5, mu is an element of(0, 4], Omega is a bounded smooth domain in R-N, and 2(*) (mu)= 2N- mu/N-2is 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 ueconcentrating around the critical point of Robin function under some suitable assumptions, if epsilon -> 0, N >= 5, mu is an element of(0, 4] sufficiently close to 0, 4or = 4. (c) 2022 Elsevier Inc. All rights reserved.