  We mainly consider the focusing biharmonic Schr\"odinger equation with a
large radial repulsive potential $V(x)$: \begin{equation*} \left\{
\begin{aligned}
  iu_{t}+(\Delta^2+V)u-|u|^{p-1}u=0,\;\;(t,x) \in {{\bf{R}}\times{\bf{R}}^{N}},
  u(0, x)=u_{0}(x)\in H^{2}({\bf{R}}^{N}),
  \end{aligned}\right.
  \end{equation*} If $N>8$, \ $1+\frac{8}{N}<p<1+\frac{8}{N-4}$ (i.e. the
$L^{2}$-supercritical and $\dot{H}^{2}$-subcritical case ), and $\langle
x\rangle^\beta \big(|V(x)|+|\nabla V(x)|\big)\in L^\infty$ for some
$\beta>N+4$, then we firstly prove a global well-posedness and scattering
result for the radial data $u_0\in H^2({\bf R}^N)$ which satisfies that $$
M(u_0)^{\frac{2-s_c}{s_c}}E(u_0)<M(Q)^{\frac{2-s_c}{s_c}}E_{0}(Q) \ \
{\rm{and}}\ \ \|u_{0}\|^{\frac{2-s_c}{s_c}}_{L^{2}}\|H^{\frac{1}{2}}
u_{0}\|_{L^{2}}<\|Q\|^{\frac{2-s_c}{s_c}}_{L^{2}}\|\Delta Q\|_{L^{2}}, $$ where
$s_c=\frac{N}{2}-\frac{4}{p-1}\in(0,2)$, $H=\Delta^2+V$ and $Q$ is the ground
state of $\Delta^2Q+(2-s_c)Q-|Q|^{p-1}Q=0$.
  We crucially establish full Strichartz estimates and smoothing estimates of
linear flow with a large poetential $V$, which are fundamental to our
scattering results.
  Finally, based on the method introduced in \cite[T. Boulenger, E. Lenzmann,
Blow up for biharmonic NLS, Ann. Sci. $\acute{E}$c. Norm. Sup$\acute{e}$r.,
50(2017), 503-544]{B-Lenzmann}, we also prove a blow-up result for a class of
potential $V$ and the radial data $u_0\in H^2({\bf R}^N)$ satisfying that $$
M(u_0)^{\frac{2-s_c}{s_c}}E(u_0)<M(Q)^{\frac{2-s_c}{s_c}}E_{0}(Q) \ \
{\rm{and}}\ \ \|u_{0}\|^{\frac{2-s_c}{s_c}}_{L^{2}}\|H^{\frac{1}{2}}
u_{0}\|_{L^{2}}>\|Q\|^{\frac{2-s_c}{s_c}}_{L^{2}}\|\Delta Q\|_{L^{2}}. $$
