Abstract
Let Ω be a smooth bounded domain in ℝ2, W1,20 (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 ∈ W1,20 (Ω) with ∥∇v∥22 ≤ 4π.
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Supported by the Fundamental Research Funds for the Central Universities, and the Research Funds of Renmin University of China (Grant No. 20XHN105)
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Wang, Y.M. Energy Estimates on Existence of Extremals for Trudinger–Moser Inequalities. Acta. Math. Sin.-English Ser. 36, 829–841 (2020). https://doi.org/10.1007/s10114-020-9528-5
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DOI: https://doi.org/10.1007/s10114-020-9528-5