Existence solutions for a weighted equation of p-biharmonic type in the unit ball of ${\mathbb{R}}^N$ with critical exponential growth

We study a weighted $\frac{N}{2}$ biharmonic equation involving a positive continuous potential in $\bar{B}$. The non-linearity is assumed to have critical exponential growth in view of logarithmic weighted Adams’ type inequalities in the unit ball of ${\mathbb{R}}^N$. 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.