  In this paper, we are interested in some problems related to the following
biharmonic hartree equation
  \begin{equation*}
  \Delta^{2} u=(|x|^{-\alpha}\ast |u|^{p})u^{p-1},\sp \text{in}\quad\R^N.
  \end{equation*}
  where $p=\frac{2N-\alpha}{N-4}$, $N\geq 9$ and $0<\alpha<N$. First, by using
the spherical harmonic decomposition and the Funk-Heck formula of the spherical
harmonic functions, we prove the nondegeneracy of the positive solutions of the
above biharmonic equation. As applications, we investigate the stability of a
version of nonlocal Sobolev inequality
  \begin{equation}
  \int_{\R^N}|\Delta u|^{2} \geq S^{*}\left( \int_{\R^N}\Big(|x|^{-\alpha}\ast
|u|^{p}\Big)u^{p} dx\right)^{\frac{1}{p}},
  \end{equation}
  and give a gradient form remainder. Moreover, by applying a finite dimension
reduction and local Poho$\check{z}$aev identity, we can also construct
multi-bubble solutions for the following equation with potential
  \begin{equation*}
  \Delta^2 u+V(|x'|, x'')u
  =\Big(|x|^{-\alpha}\ast |u|^{p}\Big)u^{p-1}\hspace{4.14mm}x\in \mathbb{R}^N.
  \end{equation*}
  where $N\geq9$, $(x',x'')\in \mathbb{R}^2\times\mathbb{R}^{N-2}$ and $V(|x'|,
x'')$ is a bounded and nonnegative function. We will show what is the role fo
the dimension and the order of the Riesz potential in proving the existence
result. In fact, the existence result is restricted to the range
$6-\frac{12}{N-4}\leq\alpha<N$.
