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

Abstract

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.

On nonexistence of extremals for the Trudinger-Moser functionals involving <inline-formula><tex-math id="M1">\begin{document}$ L^p $\end{document}</tex-math></inline-formula> norms

“…We improve the results in [16] to the case of all p > 1 on the general smooth bounded domain Ω. Similar method was also used by Yang [30,31]. Different from Lu-Yang [13], we give more precise error term to estimate the energy on the whole domain.…”

A modified singular Trudinger–Moser inequality

“…Recently, using a method of energy estimates in [19], Mancini-Martinazzi [20] reproved Carleson-Chang's result. For applications of this method, we refer the reader to Yang [29]. Using the same idea, they proved that the supremum sup u∈W 1,2 0 (B),…”

“…We are aim to prove two main results: One is to explain the new supremum is finite; the other is to discuss the existence of extremals for such functionals. In our proof, unlike the previous energy estimate procedure in [19,20,29], we mainly employ the method of blow-up analysis as in [11,14,15,18] to prove the supremum in the following (9) can be achieved. Based on Mancini-Martinazzi [20] (see pages 3 and 4), we assume the function g in (9) satisfies…”

Nonexistence of extremals for an inequality of Adimurthi-Druet on a closed Riemann surface

“…This is different from ([28], Step 3.4). Before ending this introduction, we mention related works such as de Souza-do Ó [9,14], Ishiwata [20], Martinazzi [29], Martinazzi-Struwe [30], Lamm-Robert-Struwe [21], Adimurthi-Yang [3], del Pino-Musso-Ruf [10,11], Yang [42] and Figueroa-Musso [17]. The remaining part of this paper will be organized as follows: In Section 2, we prove Theorem 1 by using an energy estimate (Proposition 3); In Section 3, we prove Proposition 3 by using blow-up analysis.…”