A remark on energy estimates concerning extremals for Trudinger–Moser inequalities on a disc



In this short note, we generalized an energy estimate due to Malchiodi–Martinazzi (J Eur Math Soc 16:893–908, 2014) and Mancini–Martinazzi (Calc Var 56:94, 2017). As an application, we used it to reprove existence of extremals for Trudinger–Moser inequalities of Adimurthi–Druet type on the unit disc. Such existence problems in general cases had been considered by Yang  (Trans Am Math Soc 359:5761–5776, 2007; J Differ Equ 258:3161–3193, 2015) and Lu–Yang (Discrete Contin Dyn Syst 25:963–979, 2009) by using another method.


Trudinger–Moser inequality Extremal function Energy estimate Blow-up analysis 

Mathematics Subject Classification

35A01 35B09 


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  1. 1.
    A. Adimurthi and O. Druet, Blow-up analysis in dimension 2 and a sharp form of Moser-Trudinger inequality, Comm. Partial Differential Equations 29 (2004), 295–322.MathSciNetCrossRefMATHGoogle Scholar
  2. 2.
    Adimurthi and S. Prashanth, Failure of Palais-Smale condition and blow-up analysis for the critical exponent problem in \(\mathbb{R}^2\), Proc. Indian Acad. Sci. Math. Sci. 107 (1997), 283–317.Google Scholar
  3. 3.
    A. Adimurthi and M. Struwe, Global compactness properties of semilinear elliptic equations with critical exponential growth, J. Funct. Anal. 175 (2000), 125–167.MathSciNetCrossRefMATHGoogle Scholar
  4. 4.
    A. Adimurthi and Y. Yang, Multibubble analysis on \(N\)-Laplace equation in \(\mathbb{R}^N\), Calc. Var. 40 (2011), 1–14.MathSciNetCrossRefMATHGoogle Scholar
  5. 5.
    L. Carleson and A. Chang, On the existence of an extremal function for an inequality of J. Moser, Bull. Sci. Math. 110 (1986), 113–127.MathSciNetMATHGoogle Scholar
  6. 6.
    M. del Pino, M. Musso, and B. Ruf, New solutions for Trudinger-Moser critical equations in \(\mathbb{R}^2\), J. Funct. Anal. 258 (2010), 421–457.MathSciNetCrossRefMATHGoogle Scholar
  7. 7.
    M. del Pino, M. Musso, and B. Ruf, Byond the Trudinger-Moser supremum, Calc. Var. 44 (2012), 543–567.CrossRefMATHGoogle Scholar
  8. 8.
    O. Druet, Multibumps analysis in dimension 2, quantification of blow-up levels, Duke Math. J. 132 (2006), 217–269.MathSciNetCrossRefMATHGoogle Scholar
  9. 9.
    T. Lamm, F. Robert, and M. Struwe, The heat flow with a critical exponential nonlinearity, J. Funct. Anal. 257 (2009), 2951–2998.MathSciNetCrossRefMATHGoogle Scholar
  10. 10.
    G. Lu and Y. Yang, The sharp constant and extremal functions for Moser-Trudinger inequalities involving \(L^{p}\) norms, Discrete Contin. Dyn. Syst. 25 (2009), 963–979.MathSciNetCrossRefMATHGoogle Scholar
  11. 11.
    A. Malchiodi and L. Martinazzi, Critical points of the Moser-Trudinger functional on a disk, J. Eur. Math. Soc. 16 (2014), 893–908.MathSciNetCrossRefMATHGoogle Scholar
  12. 12.
    G. Mancini and L. Martinazzi, The Moser-Trudinger inequality and its extremals on a disk via energy estimates, Calc. Var. (2017) 56:94,  https://doi.org/10.1007/s00526-017-1184-y.
  13. 13.
    G. Mancini and P. Thizy, Non-existence of extremals for the Adimurthi-Druet inequality, arXiv:1711.05022.
  14. 14.
    L. Martinazzi, A threshold phenomenon for embeddings of \(H_0^m\) into Orlicz spaces, Calc. Var. 36 (2009), 493–506.MathSciNetCrossRefMATHGoogle Scholar
  15. 15.
    L. Martinazzi and M. Struwe, Quantization for an elliptic equation of order \(2m\) with critical exponential non-linearity, Math Z. 270 (2012), 453–486.MathSciNetCrossRefMATHGoogle Scholar
  16. 16.
    J. Moser, A sharp form of an inequality by N. Trudinger, Indiana Univ. Math. J. 20 (1971), 1077–1092.MathSciNetCrossRefMATHGoogle Scholar
  17. 17.
    C. Tintarev, Trudinger-Moser inequality with remainder terms, J. Funct. Anal. 266 (2014), 55–66.MathSciNetCrossRefMATHGoogle Scholar
  18. 18.
    Y. Yang, A sharp form of the Moser-Trudinger inequality on a compact Riemannian surface, Trans. Amer. Math. Soc. 359 (2007), 5761–5776.MathSciNetCrossRefMATHGoogle Scholar
  19. 19.
    Y. Yang, Extremal functions for Trudinger-Moser inequalities of Adimurthi-Druet type in dimension two, J. Differential Equations 258 (2015), 3161–3193.MathSciNetCrossRefMATHGoogle Scholar
  20. 20.
    Y. Yang, Quantization for an elliptic equation with critical exponential growth on compact Riemannian surface without boundary, Calc. Var. 53 (2015), 901–941.MathSciNetCrossRefMATHGoogle Scholar

Copyright information

© Springer International Publishing AG, part of Springer Nature 2018

Authors and Affiliations

  1. 1.Department of MathematicsRenmin University of ChinaBeijingPeople’s Republic of China

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