  We let $\Omega$ be a smooth bounded domain of $\mathbb{R}^4$ and a sequence
of fonctions $(V_k)_{k\in\mathbb{N}}\in C^0(\Omega)$ such that $\lim_{k\to
+\infty}V_k=1$ in $C^0_{loc}(\Omega)$. We consider a sequence of functions
$(u_k)_{k\in\mathbb{N}}\in C^4(\Omega)$ such that $$\Delta^2 u_k=V_k e^{4u_k}$$
in $\Omega$ for all $k\in\mathbb{N}$. We address in this paper the question of
the asymptotic behaviour of the $(u_k)'s$ when $k\to +\infty$. The
corresponding problem in dimension 2 was considered by Br\'ezis-Merle and
Li-Shafrir (among others), where a blow-up phenomenon was described and where a
quantization of this blow-up was proved. Surprisingly, as shown by Adimurthi,
Struwe and the author, a similar quantization phenomenon does not hold for this
fourth order problem. Assuming that the $u_k$'s are radially symmetrical, we
push further the previous analysis. We prove that there are exactly three types
of blow-up and we describe each type in a very detailed way.
