Sharp Subcritical and Critical Trudinger-Moser Inequalities on and their Extremal Functions

In this paper, we study on some new types of the sharp subcritical and critical Trudinger-Moser inequality that have close connections to the study of the optimizers for the classical Trudinger-Moser inequalities. For instance, one of our results can be read as follows: Let 0 a beta < 2, p 0, alpha 0. Then if and only if alpha < 4 pi or alpha = 4 pi, p a 2. The attainability and inattainability of these sharp inequalties will be also investigated using a new approach, namely the relations between the supremums of the sharp subcritical and critical ones. This new method will enable us to compute explicitly the supremums of the subcritical Trudinger-Moser inequalities in some special cases. Also, a version of Concentration-compactness principle in the spirit of Lions (Lions, I. Rev. Mat. Iberoam. 1(1) 145-01 1985) will also be studied.