Abstract: We shall give a new approach to the Moser–Trudinger inequality and the existence of its extremals on the unit disk of R².  Subtly estimating the energy of critical points of subcritical inequalities, we will show that a suitable sequence of such critical points does not blow up and in fact converges to an extremal of the critical Moser–Trudinger inequality. This approach allows to prove existence of critical points for small perturbations of the  Moser-Trudinger functional. Several open questions will be discussed.
This talk is based on a joint work with Luca Martinazzi.