A generalized Trudinger–Moser inequality on a compact Riemannian surface. (April 2018)

• Record Type: Journal Article
• Authors: Zhu, Xiaobao
• Is Part Of: Nonlinear analysis. Volume 169(2018)
• Page Start: 38
• Page End: 58
• Publication Date: 2018-04
• Subjects: 58J05; Trudinger–Moser inequality -- Blow-up analysis -- Kazdan–Warner equation; Mathematical analysis -- Periodicals; Functional analysis -- Periodicals; Nonlinear theories -- Periodicals; Analyse mathématique -- Périodiques; Analyse fonctionnelle -- Périodiques; Théories non linéaires -- Périodiques; Functional analysis; Mathematical analysis; Nonlinear theories; Periodicals; Electronic journals; 515.7248
• DOI: 10.1016/j.na.2017.12.001
• Languages: English
• ISSNs: 0362-546X

Abstract

Abstract: Let ( Σ, g ) be a compact Riemannian surface. Let ψ, h be two smooth functions on Σ with ∫ Σ ψ d v g ≠ 0 and h ≥ 0, h ≢ 0 . In this paper, using a method of blow-up analysis, we prove that the functional (1) J ψ, h ( u ) = 1 2 ∫ Σ | ∇ g u | 2 d v g + 8 π 1 ∫ Σ ψ d v g ∫ Σ ψ u d v g − 8 π log ∫ Σ h e u d v g is bounded from below in W 1, 2 ( Σ, g ) . Moreover, we obtain a sufficient condition under which J ψ, h attains its infimum in W 1, 2 ( Σ, g ) . These results generalize the main results in Ding–Jost–Li–Wang (1997) and Yang–Zhu (2017).