Annals of Global Analysis and Geometry

, Volume 54, Issue 2, pp 237–256 | Cite as

Improved Moser–Trudinger inequality of Tintarev type in dimension n and the existence of its extremal functions

  • Van Hoang NguyenEmail author


Let \(\Omega \) be a smooth bounded domain in \(\mathbb R^n\) with \(n\ge 2\), \(W^{1,n}_0(\Omega )\) be the usual Sobolev space on \(\Omega \) and define \(\lambda _1(\Omega ) = \inf \nolimits _{u\in W^{1,n}_0(\Omega )\setminus \{0\}}\frac{\int _\Omega |\nabla u|^n \mathrm{d}x}{\int _\Omega |u|^n \mathrm{d}x}\). Based on the blow-up analysis method, we shall establish the following improved Moser–Trudinger inequality of Tintarev type
$$\begin{aligned} \sup _{u\in W^{1,n}_0(\Omega ), \int _\Omega |\nabla u|^n \mathrm{{d}}x-\alpha \int _\Omega |u|^n \mathrm{{d}}x \le 1} \int _\Omega \exp (\alpha _{n} |u|^{\frac{n}{n-1}}) \mathrm{{d}}x < \infty , \end{aligned}$$
for any \(0 \le \alpha < \lambda _1(\Omega )\), where \(\alpha _{n} = n \omega _{n-1}^{\frac{1}{n-1}}\) with \(\omega _{n-1}\) being the surface area of the unit sphere in \(\mathbb R^n\). This inequality is stronger than the improved Moser–Trudinger inequality obtained by Adimurthi and Druet (Differ Equ 29:295–322, 2004) in dimension 2 and by Yang (J Funct Anal 239:100–126, 2006) in higher dimension and extends a result of Tintarev (J Funct Anal 266:55–66, 2014) in dimension 2 to higher dimension. We also prove that the supremum above is attained for any \(0< \alpha < \lambda _{1}(\Omega )\). (The case \(\alpha =0\) corresponding to the Moser–Trudinger inequality is well known.)


Improved Moser–Trudinger inequality Blow-up analysis Extremal functions Elliptic regularity theory 

Mathematics Subject Classification

26D10 46E35 


  1. 1.
    Adams, D.R.: A sharp inequality of J. Moser for higher order derivatives. Ann. Math. 128(2), 385–398 (1988)MathSciNetCrossRefzbMATHGoogle Scholar
  2. 2.
    Adimurthi, A., Druet, O.: Blow-up analysis in dimension \(2\) and a sharp form of Trudinger-Moser inequality. Commun. Partial Differ. Equ. 29, 295–322 (2004)MathSciNetCrossRefzbMATHGoogle Scholar
  3. 3.
    Adimurthi, A., Yang Y.: An interpolation of Hardy inequality and Trudinger–Moser inequality in \(\mathbb{R}^n\) and its applications. Int. Math. Res. Not. 13, 2394–2426 (2010)Google Scholar
  4. 4.
    Balogh, J., Manfredi, J., Tyson, J.: Fundamental solution for the \(Q-\)Laplacian and sharp Moser–Trudinger inequality in Carnot groups. J. Funct. Anal. 204, 35–49 (2003)MathSciNetCrossRefzbMATHGoogle Scholar
  5. 5.
    Carleson, L., Chang, S.Y.A.: On the existence of an extremal function for an inequality of J. Moser. Bull. Sci. Math. 110, 113–127 (1986)MathSciNetzbMATHGoogle Scholar
  6. 6.
    Černy, R., Cianchi, A., Hencl, S.: Concentration-compactness principles for Moser-Trudinger inequalities: new results and proofs. Ann. Math. Pura Appl. (4) 192(2), 225–243 (2013)Google Scholar
  7. 7.
    Chang, S.A., Yang, P.: Conformal deformation of metric on \(\mathbb{S}^2\). J. Differ. Geom. 27, 259–296 (1988)CrossRefGoogle Scholar
  8. 8.
    Cianchi, A.: Moser–Trudinger inequalities without boundary conditions and isoperimetric problems. Indiana Univ. Math. J. 54, 669–705 (2005)MathSciNetCrossRefzbMATHGoogle Scholar
  9. 9.
    Cohn, W.S., Lu, G.: Best constants for Moser–Trudinger inequalities on the Heisenberg group. Indiana Univ. Math. J. 50, 1567–1591 (2001)MathSciNetCrossRefzbMATHGoogle Scholar
  10. 10.
    Csato, G., Roy, P.: Extremal functions for the singular Moser–Trudinger inequality in \(2\) dimensions. Calc. Var. Partial Differ. Equ. 54, 2341–2366 (2015)MathSciNetCrossRefzbMATHGoogle Scholar
  11. 11.
    Csato, G., Roy, P.: Singular Moser–Trudinger inequality on simply connected domains. Commun. Partial Differ. Equ. 41, 838–847 (2016)MathSciNetCrossRefzbMATHGoogle Scholar
  12. 12.
    de Figueiredo, D.G., do Ó, J.M.: On an inequality by N. Trudinger and J. Moser and related elliptic equations. Commun. Pure Appl. Math. 55, 135–152 (2002)MathSciNetCrossRefzbMATHGoogle Scholar
  13. 13.
    do Ó, J.M., de Souza, M.: A sharp inequality of Trudinger–Moser type and extremal functions in \(H^{1, n}(\mathbb{R}^n)\). J. Differ. Equ. 258, 4062–4101 (2015)CrossRefzbMATHGoogle Scholar
  14. 14.
    do Ó, J.M., de Souza, M.: Trudinger–Moser inequality on the whole plane and extremal functions. Commun. Contemp. Math. 18(5), 1550054 (2016)MathSciNetCrossRefzbMATHGoogle Scholar
  15. 15.
    Druet, O., Hebey, E., Robert, F.: Blow-Up Theory for Elliptic PDEs in Riemannian Geometry. Mathematical notes, vol. 45. Princeton University Press, Princeton (2004)zbMATHGoogle Scholar
  16. 16.
    Esposito P.: A classification result for the quasi-linear Liouville equation. Ann. Inst. H. Poincaré Anal. Nonlinéaire (to appear)Google Scholar
  17. 17.
    Flucher, M.: Extremal functions for the Trudinger–Moser inequality in \(2\) dimensions. Comment. Math. Helv. 67, 471–497 (1992)MathSciNetCrossRefzbMATHGoogle Scholar
  18. 18.
    Lam, N., Lu, G.: Sharp Moser–Trudinger inequality on the Heisenberg group at the critical case and applications. Adv. Math. 231, 3259–3287 (2012)MathSciNetCrossRefzbMATHGoogle Scholar
  19. 19.
    Lam, N., Lu, G.: A new approach to sharp Moser–Trudinger and Adams type inequalities: a rearrangement-free argument. J. Differ. Equ. 255, 298–325 (2013)MathSciNetCrossRefzbMATHGoogle Scholar
  20. 20.
    Li, Y.: Moser–Trudinger inequality on compact Riemannian manifolds of dimension two. J. Partial Differ. Equa. 14, 163–192 (2001)MathSciNetzbMATHGoogle Scholar
  21. 21.
    Li, Y.: Extremal functions for the Moser–Trudinger inequalities on compact Riemannian manifolds. Sci. China Ser. A 48, 618–648 (2005)MathSciNetCrossRefzbMATHGoogle Scholar
  22. 22.
    Li, Y., Ruf, B.: A sharp Trudinger–Moser type inequality for unbounded domains in \(\mathbb{R}^n\). Indiana Univ. Math. J. 57, 451–480 (2008)MathSciNetCrossRefzbMATHGoogle Scholar
  23. 23.
    Lin, K.: Extremal functions for Moser’s inequality. Trans. Am. Math. Soc. 348, 2663–2671 (1996)MathSciNetCrossRefzbMATHGoogle Scholar
  24. 24.
    Lions, P.L.: The concentration–compactness principle in the calculus of variations. The limit case. I. Rev. Mat. Iberoam. 1, 145–201 (1985)MathSciNetCrossRefzbMATHGoogle Scholar
  25. 25.
    Lu G., Zhu M.: A sharp Moser–Trudinger type inequality involving \(L^n\) norm in the entire space \(\mathbb{R}^n\). arXiv:1703.00901
  26. 26.
    Mancini, G., Sandeep, K.: Moser–Trudinger inequality on conformal discs. Commun. Contemp. Math. 12, 1055–1068 (2010)MathSciNetCrossRefzbMATHGoogle Scholar
  27. 27.
    Mancini, G., Sandeep, K., Tintarev, C.: Trudinger–Moser inequality in the hyperbolic space \(\mathbb{H}^n\). Adv. Nonlinear Anal. 2, 309–324 (2013)MathSciNetzbMATHGoogle Scholar
  28. 28.
    Moser J.: A sharp form of an inequality by N. Trudinger. Indiana Univ. Math. J. 20:1077–1092 (1970/71)Google Scholar
  29. 29.
    Nguyen V.H.: Extremal functions for the Moser–Trudinger inequality of Adimurthi–Druet type in \(W^{1,n}(\mathbb{R}^n)\). arXiv:1702.07970
  30. 30.
    Nguyen, V.H.: Improved Moser–Trudinger inequality for functions with mean value zero in \(\mathbb{R}^n\) and its extremal functions. Nonlinear Anal. 163, 127–145 (2017)MathSciNetCrossRefzbMATHGoogle Scholar
  31. 31.
    Pohožaev, S.I.: On the eigenfunctions of the equation \(\Delta u + \lambda f(u) = 0\). Dokl. Akad. Nauk. SSSR 165, 36–39 (1965). (Russian) MathSciNetGoogle Scholar
  32. 32.
    Ruf, B.: A sharp Trudinger–Moser type inequality for unbounded domains in \(\mathbb{R}^2\). J. Funct. Anal. 219, 340–367 (2005)MathSciNetCrossRefzbMATHGoogle Scholar
  33. 33.
    Serrin, J.: Local behavior of solutions of quasi-linear equations. Acta. Math. 111, 248–302 (1964)MathSciNetCrossRefzbMATHGoogle Scholar
  34. 34.
    Struwe, M.: Critical points of embeddings of \(H^{1, n}_0\) into Orlicz spaces. Ann. Inst. H. Poincaré Anal. Nonlinéaire 5, 425–464 (1988)CrossRefzbMATHGoogle Scholar
  35. 35.
    Tintarev, C.: Trudinger–Moser inequality with remainder terms. J. Funct. Anal. 266, 55–66 (2014)MathSciNetCrossRefzbMATHGoogle Scholar
  36. 36.
    Trudinger, N.S.: On imbedding into Orlicz spaces and some applications. J. Math. Mech. 17, 473–483 (1967)MathSciNetzbMATHGoogle Scholar
  37. 37.
    Tolksdorf, P.: Regularity for a more general class of quasilinear elliptic equations. J. Differ. Equ. 51, 126–150 (1984)MathSciNetCrossRefzbMATHGoogle Scholar
  38. 38.
    Yang, Y.: A sharp form of Moser–Trudinger inequality in high dimension. J. Funct. Anal. 239(1), 100–126 (2006)MathSciNetCrossRefzbMATHGoogle Scholar
  39. 39.
    Yang, Y.: Extremal functions for a sharp Moser–Trudinger inequality. Int. J. Math. 17, 331–338 (2006)MathSciNetCrossRefzbMATHGoogle Scholar
  40. 40.
    Yang, Y.: A sharp form of the Moser–Trudinger inequality on a compact Riemannian surface. Trans. Am. Math. Soc. 359, 5761–5776 (2007)MathSciNetCrossRefzbMATHGoogle Scholar
  41. 41.
    Yang Y: Corrigendum to: a sharp form of Moser–Trudinger inequality in high dimension [J. Funct. Anal. 239, 100–126 (2006); MR2258218]. J. Funct. Anal. 242, 669–671 (2007)Google Scholar
  42. 42.
    Yang, Y.: Trudinger–Moser inequalities on complete noncompact Riemannian manifolds. J. Funct. Anal. 263, 1894–1938 (2012)MathSciNetCrossRefzbMATHGoogle Scholar
  43. 43.
    Yang, Y.: Extremal functions for Trudinger–Moser inequalities of Adimurthi–Druet type in dimension two. J. Differ. Equ. 258, 3161–3193 (2015)MathSciNetCrossRefzbMATHGoogle Scholar
  44. 44.
    Yang, Q., Su, D., Kong, Y.: Sharp Moser–Trudinger inequalities on Riemannian manifolds with negative curvature. Ann. Mat. Pura Appl. 195, 459–471 (2016)MathSciNetCrossRefzbMATHGoogle Scholar
  45. 45.
    Yang, Y., Zhu, X.: Blow-up analysis concerning singular Trudinger–Moser inequalities in dimension two. J. Funct. Anal. 272, 3347–3374 (2017)MathSciNetCrossRefzbMATHGoogle Scholar
  46. 46.
    Yuan, A., Zhu, X.: An improved singular Trudinger–Moser inequality in unit ball. J. Math. Anal. Appl. 435, 244–252 (2016)MathSciNetCrossRefzbMATHGoogle Scholar
  47. 47.
    Yudovič, V.I.: Some estimates connected with integral operators and with solutions of elliptic equations. Dokl. Akad. Nauk. SSSR 138, 805–808 (1961). (Russian) MathSciNetGoogle Scholar

Copyright information

© Springer Science+Business Media B.V., part of Springer Nature 2018

Authors and Affiliations

  1. 1.Institute of Research and DevelopmentDuy Tan UniversityDa NangVietnam

Personalised recommendations