Blowing-up solutions for the Choquard type Brezis-Nirenberg problem in dimension three

In this paper, we are interested in the existence of solutions for the following Choquard type Brezis-Nirenberg problem {align*} \{ {array}{ll} -Δ u=(∫Ω{u6-α(y)}{|x-y|^α}dy)u5-α+λ u, \ \ &in\ Ω, u=0, \ \ &on\ ∂ Ω, {array} . {align*} where Ω is a smooth bounded domain in R³, α∈ (0,3), 6-α is the upper critical exponent in the sense of the Hardy-Littlewood-Sobolev inequality, and λ is a real positive parameter. By applying the reduction argument, we find and characterize a positive value λ₀ such that if λ-λ₀>0 is small enough, then the above problem admits a solution, which blows up and concentrates at the critical point of the Robin function as λ→ λ₀. Moreover, we consider the above problem under zero Neumann boundary condition.