Skip to main content
SpringerLink
Log in

Energy Estimates on Existence of Extremals for Trudinger–Moser Inequalities

  • Published:
Acta Mathematica Sinica, English Series Aims and scope Submit manuscript

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

$$\mathop {\sup}\limits_{v \in W_0^{1,2}\left({\rm{\Omega}} \right),\left\| {{\nabla _u}} \right\|_2^2 \le 4{\rm{\pi}}} \int_{\rm{\Omega}} {\left({{{\rm{e}}^{{v^2}}} - \alpha {v^2}} \right)dx}$$

can be achieved by some function v ∈ W1,20 (Ω) with ∥∇v∥22 ≤ 4π.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

  1. Adams, D.: A sharp inequality of J. Moser for higher order derivatives. Ann. Math., 128, 385–398 (1988)

    Article  MathSciNet  Google Scholar 

  2. Adimurthi, A., Struwe, M.: Global compactness properties of semilinear elliptic equation with critical exponential growth. J. Funct. Anal., 175, 125–167 (2000)

    Article  MathSciNet  Google Scholar 

  3. Carleson, L., Chang, A.: On the existence of an extremal function for an inequality of J. Moser. Bull. Sci. Math., 110, 113–127 (1986)

    MathSciNet  MATH  Google Scholar 

  4. Ding, W. Y., Jost, J., Li, J. Y., et al.: The differential equation − Δu = 8π − 8πheu on a compact Riemann Surface. Asian J. Math., 1, 230–248 (1997)

    Article  MathSciNet  Google Scholar 

  5. Druet, O.: Multibumps analysis in dimension 2, quantification of blow-up levels. Duke Math. J., 132, 217–269 (2006)

    Article  MathSciNet  Google Scholar 

  6. Druet, O., Thizy, P. D.: Multi-bumps analysis for Trudinger–Moser nonlinearities i-quantification and location of concentration points. arXiv:1710.08811

  7. Flucher, M.: Extremal functions for Trudinger–Moser inequality in 2 dimensions. Comment. Math. Helv., 67, 471–497 (1992)

    Article  MathSciNet  Google Scholar 

  8. Fontana, L.: Sharp borderline Sobolev inequalities on compact Riemannian manifolds. Comment. Math. Helv., 68, 415–454 (1993)

    Article  MathSciNet  Google Scholar 

  9. Gidas, B., Ni, W. M., Nirenberg, L.: Symmetry and related properties via the maximum principle. Comm. Math. Phys., 68, 209–243 (1979)

    Article  MathSciNet  Google Scholar 

  10. Li, Y. X.: Moser–Trudinger inequality on compact Riemannian manifolds of dimension two. J. Partial Differ. Equ., 14, 163–192 (2001)

    MathSciNet  MATH  Google Scholar 

  11. Li, Y. X.: Remarks on the extremal functions for the Moser–Trudinger inequality. Acta Math. Sin. Engl. Ser., 22, 545–550 (2006)

    Article  MathSciNet  Google Scholar 

  12. Lin, K. C.: Extremal functions for Moser’s inequality. Trans. Amer. Math. Soc., 348, 2663–2671 (1996)

    Article  MathSciNet  Google Scholar 

  13. Lions, P. L.: The concentration-compactness principle in the calculus of variation, the limit case, part I. Rev. Mat. Iberoamericana, 1, 145–201 (1985)

    Article  MathSciNet  Google Scholar 

  14. Lu, G. Z., Yang, Y. Y.: The sharp constant and extremal functions for Moser–Trudinger inequalities involving Lp norms. Discrete and Continuous Dynamical Systems, 25, 963–979 (2009)

    Article  MathSciNet  Google Scholar 

  15. Malchiodi, A., Martinazzi, L.: Critical points of the Moser–Trudinger functional on a disk. J. Eur. Math. Soc., 16, 893–908 (2014)

    Article  MathSciNet  Google Scholar 

  16. Mancini, G., Martinazzi, L.: The Moser–Trudinger inequality and its extremals on a disk via energy estimates. Calc. Var. Partial Differential Equations, 56, 94 (2017)

    Article  MathSciNet  Google Scholar 

  17. Mancini, G., Thizy, P. D.: Non-existence of extremals for the Adimurthi–Druet inequality. J. Differential Equations, 266, 1051–1072 (2019)

    Article  MathSciNet  Google Scholar 

  18. Moser, J.: A sharp form of an inequality by N. Trudinger. Indiana Univ. Math. J., 20, 1077–1091 (1971)

    Article  MathSciNet  Google Scholar 

  19. Peetre, J.: Espaces d’interpolation et theoreme de Soboleff. Ann. Inst. Fourier (Grenoble), 16, 279–317 (1966)

    Article  MathSciNet  Google Scholar 

  20. Pohozaev, S.: The Sobolev embedding in the special case pl = n. Proceedings of the technical scientific conference on advances of scientific reseach 1964–1965, Mathematics sections, 158–170, Moscov. Energet. Inst., Moscow, 1965

    Google Scholar 

  21. Struwe, M.: Critical points of embedding of H 10 into Orlicz spaces. Ann. Inst. H. Poincaré Anal. Non Linéaire, 5, 425–464 (1988)

    Article  MathSciNet  Google Scholar 

  22. Su, X. F.: A Trudinger–Moser type inequality and its extremal functions in dimension two. J. Math. Inequal., in press

  23. Trudinger, N. S.: On embeddings into Orlicz space and some applications. J. Math. Mech., 17, 473–483 (1967)

    MathSciNet  MATH  Google Scholar 

  24. Yang, Y. Y.: A sharp form of Moser–Trudinger inequality in high dimension. J. Funct. Anal., 239, 100–126 (2006)

    Article  MathSciNet  Google Scholar 

  25. Yang, Y. Y.: Corrigendum to: “A sharp form of Moser–Trudinger inequality in high dimension”. J. Funct. Anal., 242, 669–671 (2007)

    Article  MathSciNet  Google Scholar 

  26. Yang, Y. Y.: A sharp form of the Moser–Trudinger inequality on a compact Riemannian surface. Trans. Amer. Math. Soc., 359, 5761–5776 (2007)

    Article  MathSciNet  Google Scholar 

  27. Yang, Y. Y.: Extremal functions for Trudinger–Moser inequalities of Adimurthi–Druet type in dimension two. J. Differential Equations, 258, 3161–3193 (2015)

    Article  MathSciNet  Google Scholar 

  28. Yang, Y. Y.: Nonexistence of extremals for an inequality of Adimurthi–Druet on a closed Riemann surface. Sci. China Math., 63, (2020)

  29. Yang, Y. Y.: Energy estimate related to Hardy–Trudinger–Moser inequality. J. Partial Differ. Equ., 32, 342–351 (2019)

    Article  Google Scholar 

  30. Yang, Y. Y., Zhu, X. B.: Blow-up analysis concerning singular Trudinger–Moser inequalities in dimension two. J. Funct. Anal., 272, 3347–3374 (2017)

    Article  MathSciNet  Google Scholar 

  31. Yang, Y. Y.: A remark on energy estimates concerning extremals for Trudinger–Moser inequalities on a disc. Arch. Math. (Basel), 111, 215–223 (2018)

    Article  MathSciNet  Google Scholar 

  32. Yang, Y. Y.: Existence of extremals for critical Trudinger–Moser inequalities via the method of energy estimate. J. Math. Anal. Appl., 479, 1281–1291 (2019)

    Article  MathSciNet  Google Scholar 

  33. Yuan, A. F., Zhu, X. B.: An improved singular Trudinger–Moser inequality in unit ball. J. Math. Anal. Appl., 435, 244–252 (2016)

    Article  MathSciNet  Google Scholar 

  34. Yudovich, V. I.: Some estimates connected with integral operators and with solutions of elliptic equations. Sov. Math. Docl., 2, 746–749 (1961)

    MATH  Google Scholar 

  35. Zhu, J. Y.: Improved Moser–Trudinger inequality involving Lp norm in n dimensions. Advanced Nonlinear Studies, 14, 273–293 (2014)

    Article  MathSciNet  Google Scholar 

Download references

Acknowledgements

We thank the referees for their time and comments.

Author information

Authors and Affiliations

  1. School of Mathematics, Renmin University of China, Beijing, 100872, P. R. China

    Ya Min Wang

Authors
  1. Ya Min Wang
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Ya Min Wang.

Additional information

Supported by the Fundamental Research Funds for the Central Universities, and the Research Funds of Renmin University of China (Grant No. 20XHN105)

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

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

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s10114-020-9528-5

Keywords

MR(2010) Subject Classification

Search

Navigation