A new log-weighted Trudinger-Moser inequality in $ \mathbb{R}^N,\ N \geq 2, $ and applications to some mean field equation

In this paper, we establish some new Trudinger-Moser's inequalities in the whole space $ \mathbb{R}^N, \ N \geq 2. $ These inequalities are concerned with functions lying in weighted Sobolev spaces. The weights are of logarithmic type. Our inequalities are completely new and they extend some previous recent works due to M. Calanchi, E. Massa and B. Ruf [12], and S. Deng [18]. The novelty in the present work mainly lies in the fact that we are dealing with the whole euclidean space $ \mathbb{R}^N, \ N \geq 2. $ To the very best knowledge of the author, this is the first attempt to prove such a result, and some sophisticated technical arguments are used in the proof. Taking profit of these inequalities, we were able to treat, in the last part of this work, some nonlinear elliptic mean field equation of Liouville type.