Fourth order weighted elliptic problem under exponential nonlinear growth

We deal with nonlinear weighted biharmonic problem in the unit ball of ${\mathbb{R}}^4$. The weight is of logarithm type. The nonlinearity is critical in view of Adam’s inequalities in the weighted Sobolev space $W_0^{2,2}(B,w)$. 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.