Weighted Trudinger–Moser inequalities and applications

Trudinger–Moser inequalities provide continuous embeddings in the borderline cases of the standard Sobolev embeddings, in which the embeddings into Lebesgue $L^p$ spaces break down. One is led to consider their natural generalization, which are embeddings into Orlicz spaces with corresponding maximal growth functions which are of exponential type. Some parameters come up in the description of these growth functions. The parameter ranges for which embeddings exist increase by the use of weights in the Sobolev norm, and one is led to consider weighted TM inequalities. Some interesting cases are presented for special weights in dimension two, with applications to mean field equations of Liouville type.