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 (2014) 893-908) and Mancini-Martinazzi (Calc. Var. (2017) 56:94). 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. Amer. Math. Soc. 359 (2007) 5761-5776; J. Differential Equations 258 (2015) 3161-3193) and Lu-Yang (Discrete Contin. Dyn. Syst. 25 (2009) 963-979) by using another method.