  In this article, we study the following problem $$\Delta(w(x)\Delta u) = \
f(x,u) \quad\mbox{ in }\quad B, \quad u=\frac{\partial u}{\partial n}=0
\quad\mbox{ on } \quad\partial B,$$
  where $B$ is the unit ball of $\mathbb{R}^{4}$ and $ w(x)$ a singular weight
of logarithm type. The reaction source $f(x,u)$ is a radial function with
respect to $x$ and it is critical in view of exponential inequality of Adams'
type. The existence result is proved by using the constrained minimization in
Nehari set coupled with the quantitative deformation lemma and degree theory
results.
