Qualitative analysis to an eigenvalue problem of the Hartree type Br\'ezis-Nirenberg problem

In this paper, we are concerned with the critical Hartree equation {equation*} {cases} -Δ u=({{∫Ω}}{u^{2*μ}(y)}{|x-y|μ}dy)u^{2*μ-1}+ε u, u>0, &in Ω,\\ u=0, &on ∂Ω, {cases} {equation*} where Ω⊂ RN (N≥ 5) is a smooth bounded domain, μ∈ (0,4) and 2*μ=2N-μ/N-2 is the upper critical exponent in the sense of the Hardy-Littlewood-Sobolev inequality. Under a non-degeneracy condition on the critical point x₀∈Ω of the Robin function $R(x)$, we perform that for ε>0 sufficiently small, the Morse index of the blow-up solutions u_ε concentrating at x₀ can be computed in terms of the negative eigenvalues of the Hessian matrix D²R(x) at x₀. Compared with the usual local cases, our problem is non-local due to the nonlinearity with Hartree-type, and several difficulties arise and new estimates of the eigenpairs \(λi,ε,vi,ε)\ to the associated linearized problem at uε should be introduced. To our knowledge, this seems to be the first paper to consider the qualitative analysis of a Hartree type Br\'ezis-Nirenberg problem and our results extend the works established by M. Grossi et al in {GP} and F. Takahashi in {Ta3} to the non-local case.