  We deal with a weighted biharmonic problem in the unit ball of
$\mathbb{R}^{4}$.
  The non-linearity is assumed to have critical exponential growth in view of
Adam's type inequalities. The weight $w(x)$ is of logarithm type and the
potential $V$ is a positive continuous function on $\overline{B}$. It is proved
that there is a nontrivial positive weak solution to this problem by the
mountain Pass Theorem. We avoid the loss of compactness by proving a
concentration compactness result and by a suitable asymptotic condition.
