We study a weighted N 2 biharmonic equation involving a positive continuous potential in B. The non-linearity is assumed to have critical exponential growth in view of logarithmic weighted Adams' type inequalities in the unit ball of R N . 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.

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We deal with the nonlinear weighted elliptic problemwhere B is the unit ball of R 4 and w(x) = log e |x| β , β ∈ (0, 1) a singular logarithm weight. The nonlinearity is critical in view of Adam's inequalities in the weighted Sobolev space W 2,2 0 (B, w). We prove the existence of non trivial solutions via the critical point theory. The main difficulty is the loss of compactness due to the critical exponential growth of the nonlinear term f . We give a new growth condition and we point out its importance for checking the Palais-Smale compactness condition.