Qualitative properties of single blow-up solutions for nonlinear Hartree equation with slightly subcritical exponent

In this paper, we study the qualitative properties of single blow-up solutions to the nonlocal equations with slightly subcritical exponents \begin{equation*} -\Delta u=(|x|^{-(n-2)}\ast u^{p-\epsilon})u^{p-1-\epsilon}\quad \mbox{in}~~\Omega,~~ u=0\quad \mbox{on}~~\partial\Omega, \end{equation*} where $\Omega$ is a smooth bounded domain in $\mathbb{R}^n$ for $n=3,4,5$, $\ast$ denotes the standard convolution, $\epsilon>0$ is a small parameter and $p=\frac{n+2}{n-2}$ is $\mathcal{D}^{1,2}$ energy-critical exponent. By exploiting various local Pohozaev identities and blow-up analysis, we provide a number of estimates on the first $(n+2)$-eigenvalues and their corresponding eigenfunctions, and examine the qualitative behavior of the eigenpairs $(\lambda_{i,\epsilon}, v_{i,\epsilon})$ to the linearied problem of the above nonlocal equations for $i=1,\cdots,n+2$. As a corollary, we derive the Morse index of a single-bubble solution in a nondegenerate setting.