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 \begin{equation*} \begin{cases} -\Delta u=\left(\displaystyle{\displaystyle{\int_{\Omega}}}\frac{u^{2^{*}_{\mu}}(y)}{|x-y|^{\mu}}dy\right)u^{2^{*}_{\mu}-1}+\varepsilon u,\quad u>0,\quad &\text{in $\Omega$,}\\ u=0,\quad &\text{on $\partial\Omega$,} \end{cases} \end{equation*} where $\Omega\subset \mathbb{R}^N$ ($N\geq 5$) is a smooth bounded domain, $\mu\in (0,4)$ and $2^{*}_{\mu}=\frac{2N-\mu}{N-2}$ is the upper critical exponent in the sense of the Hardy-Littlewood-Sobolev inequality. Under a non-degeneracy condition on the critical point $x_0\in\Omega$ of the Robin function $R(x)$, we perform that for $\varepsilon>0$ sufficiently small, the Morse index of the blow-up solutions $u_\varepsilon$ concentrating at $x_0$ can be computed in terms of the negative eigenvalues of the Hessian matrix $D^{2}R(x)$ at $x_0$. 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 $\{\left(\lambda_{i,\varepsilon},v_{i,\varepsilon}\right)\}$ to the associated linearized problem at $u_{\varepsilon}$ 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 \cite{GP} and F. Takahashi in \cite{Ta3} to the non-local case.