Abstract
We study a weighted fourth order equation involving a positive continuous potential in B. The non-linearity is assumed to have critical exponential growth in view of Adams' type inequalities. The weight w(x) is of logarithm type. It is proved that there is a nontrivial 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.