A remark on energy estimates concerning extremals for Trudinger–Moser inequalities on a disc

In this short note, we generalized an energy estimate due to Malchiodi–Martinazzi (J Eur Math Soc 16:893–908, 2014 ) and Mancini–Martinazzi (Calc Var 56:94, 2017 ). As an application, we used it to reprove existence of extremals for Trudinger–Moser inequalities of Adimurthi–Druet type on the unit disc. Such existence problems in general cases had been considered by Yang (Trans Am Math Soc 359:5761–5776, 2007 ; J Differ Equ 258:3161–3193, 2015 ) and Lu–Yang (Discrete Contin Dyn Syst 25:963–979, 2009 ) by using another method.