Improved Singular Moser–Trudinger Inequalities and Their Extremal Functions


In this paper, we prove several improvements for the sharp singular Moser–Trudinger inequality. We first establish an improved singular Moser–Trudinger inequality in the spirit of Tintarev (J. Funct. Anal. 266, 55–66, 2014) and prove the existence of extremal functions for this improved singular Moser–Trudinger inequality. Our proof is based on the blow-up analysis method. Due to appearance of the singularity of weights, our proof is more complicated and difficult in dealing with analyzing the asymptotic behavior of the sequence of maximizers in the subcritical case near the blow-up point when comparing with the previous works (see, e.g., Nguyen (Ann. Global Anal. Geom. 54(2), 237–256, 2018), Yang (J. Funct. Anal. 239(1), 100–126, 2006)). To overcome these difficulties, we shall prove a classification result for a singular quasi-linear Liouville equation which maybe is of independent interest. Finally, we derive another improvement of the singular Moser–Trudinger inequality in the sprit of Adimurthi and Druet (Comm. Partial Differential Equations 29, 295–322, 2004). Our results extend many well-known Moser–Trudinger type inequalities to more general setting (e.g., singular weights, higher dimension, any domain, etc).

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Correspondence to Van Hoang Nguyen.

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Nguyen, V.H. Improved Singular Moser–Trudinger Inequalities and Their Extremal Functions. Potential Anal 53, 55–88 (2020).

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  • Singular Moser–Trudinger inequality
  • Improvement
  • Extremal functions
  • Blow-up analysis
  • Elliptic regularity theory
  • Green function
  • Singular quasi-linear Liouville equation
  • Classification

Mathematics Subject Classification (2010)

  • 26D10
  • 46E35