  Let $\lambda^*>0$ denote the largest possible value of $\lambda$ such that
\begin{align*} \left\{\begin{aligned} \Delta^2 u & = \la e^u && \text{in $B $}
u &= \pd{u}{n} = 0 && \text{on $ \pa B $} \end{aligned} \right. \end{align*}
has a solution, where $B$ is the unit ball in $\R^N$ and $n$ is the exterior
unit normal vector. We show that for $\lambda=\lambda^*$ this problem possesses
a unique {\em weak} solution $u^*$. We prove that $u^*$ is smooth if $N\le 12$
and singular when $N\ge 13$, in which case $ u^*(r) = - 4 \log r + \log
(8(N-2)(N-4) / \lambda^*) + o(1)$ as $r\to 0$. We also consider the problem
with general constant Dirichlet boundary conditions.
