Energy Estimates on Existence of Extremals for Trudinger-Moser Inequalities

Let Omega be a smooth bounded domain in R2, W 1,2 0 (Omega) 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 ManciniThizy (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 alpha >= 1, the supremum sup(v is an element of W01,2(Omega), parallel to del v parallel to 22 <= 4 pi) integral(Omega)(e(v2) - alpha(v2))dx can be achieved by some function v is an element of W-0(1,2)(Omega) with parallel to del v parallel to(2)(2) = 4 pi.