Existence of extremals for critical Trudinger-Moser inequalities via the method of energy estimate

We reprove the existence of extremals for the critical Trudinger-Moser inequality on a smooth bounded domain of R 2 via the method of energy estimate, which was recently developed by Malchiodi-Martinazzi [11] , Mancini-Martinazzi [12] and Mancini-Thizy [13] . For this purpose, unlike [12] , [13] , it suffices to calculate local energy of maximizers for subcritical Trudinger-Moser functionals near the blow-up point.