Energy Estimates on Existence of Extremals for Trudinger–Moser Inequalities

Let Ω be a smooth bounded domain in R2, W01,2（Ω） be the standard Sobolev space. By the method of energy estimate developed by Malchiodi–Martinazzi(J. Eur. Math. Soc., 16, 893–908（2014）), Mancini–Martinazzi(Calc. Var. Partial Differential Equations, 56, 94（2017）) and Mancini–Thizy(J. Differential Equations, 266, 1051–1072（2019）), we reprove the results of Carleson–Chang(Bull. Sci. Math., 110, 113–127（1986）), Flucher(Comment. Math. Helv., 67, 471–497（1992）), Li(Acta Math. Sin. Engl. Ser., 22, 545–550（2006）) and Su（J. Math. Inequal., in press）. Namely, for any real number α≤ 1, the supremum ■ can be achieved by some function v ∈ W01,2（Ω） with ║?v║2～2≤ 4π.