A Log-Weighted Sharp Trudinger-Moser Inequalities Involving Singular and Vanishing Radial Potentials and Applications

The main goal of this paper is to investigate some Trudinger-Moser inequality with a logarithmic weight involving unbounded or decaying radial weights in weighted Sobolev spaces. After that, we provide a further improvement of these inequalities. Next, we establish a version of the concentration-compactness principle, attributed to Lions, providing some new improvements of the Trudinger-Moser inequalities established in the second part of this work. In the light of this last result, we treat some elliptic quasilinear problems involving new type of doubly exponential growth condition at infinity.