Improved Singular Moser–Trudinger Inequalities and Their Extremal Functions

Abstract

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).

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

References

  1. 1.

    Adams, D.R.: A sharp inequality of J. Moser for higher order derivatives. Ann. Math. 128(2), 385–398 (1988)

    MathSciNet  MATH  Google Scholar 

  2. 2.

    Adimurthi, Druet, O.: Blow–up analysis in dimension 2 and a sharp form of Trudinger–Moser inequality. Comm. Partial Differential Equations 29, 295–322 (2004)

  3. 3.

    Adimurthi, Sandeep, K.: A singular Moser–Trudinger embedding and its applications. Nonlinear Differ. Equ. Appl. 13, 585–603 (2007)

  4. 4.

    Adimurthi, Yang, Y.: An interpolation of Hardy inequality and Trudinger–Moser inequality in \(\mathbb {R}^{n}\) and its applications. Int. Math. Res. Not. IMRN 13, 2394–2426 (2010)

    MATH  Google Scholar 

  5. 5.

    Alvino, A., Brock, F., Chiacchio, F., Mercaldo, A., Posteraro, M.R.: Some isoperimetric inequalities on rN with respect to weights |x|a. J. Math. Anal. Appl. 451(1), 280–318 (2017)

    MathSciNet  MATH  Google Scholar 

  6. 6.

    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)

    MathSciNet  MATH  Google Scholar 

  7. 7.

    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)

    MathSciNet  MATH  Google Scholar 

  8. 8.

    Černy, R., Cianchi, A., Hencl, S.: Concentration–compactness principles for Moser–Trudinger inequalities: new results and proofs. Ann. Mat. Pura Appl. 192(4), 225–243 (2013)

    MathSciNet  MATH  Google Scholar 

  9. 9.

    Chang, S.A., Yang, P.: Conformal deformation of metric on s2. J. Differential Geom. 27, 259–296 (1988)

    MathSciNet  Google Scholar 

  10. 10.

    Cianchi, A.: MoserTrudinger inequalities without boundary conditions and isoperimetric problems. Indiana Univ. Math. J. 54, 669–705 (2005)

    MathSciNet  MATH  Google Scholar 

  11. 11.

    Cohn, W.S., Lu, G.: Best constants for Moser–Trudinger inequalities on the Heisenberg group. Indiana Univ. Math. J 50, 1567–1591 (2001)

    MathSciNet  MATH  Google Scholar 

  12. 12.

    Cohn, W.S., Lu, G.: Sharp constants for Moser–Trudinger inequalities on spheres in complex space cn. Comm. Pure Appl. Math. 57, 1458–1493 (2004)

    MathSciNet  MATH  Google Scholar 

  13. 13.

    Csató, G., Roy, P.: Extremal functions for the singular Moser–Trudinger inequality in 2 dimensions. Calc. Var. Partial Differential Equations 54, 2341–2366 (2015)

    MathSciNet  MATH  Google Scholar 

  14. 14.

    Csató, G., Roy, P.: Singular Moser–Trudinger inequality on simply connected domains, Comm. Partial Differential Equations 41, 838–847 (2016)

    MathSciNet  MATH  Google Scholar 

  15. 15.

    Csató, G., Roy, P., Nguyen, V.H.: Extremals for the singular Moser-Trudinger inequality via n-harmonic transplantation. arXiv:1801.03932v3

  16. 16.

    de Figueiredo, D.G., do Ó, J.M., Ruf, B.: On an inequality by N. Trudinger and J. Moser and related elliptic equations. Comm. Pure Appl. Math. 55, 135–152 (2002)

    MathSciNet  MATH  Google Scholar 

  17. 17.

    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. Differential Equations 258, 4062–4101 (2015)

    MathSciNet  MATH  Google Scholar 

  18. 18.

    Druet, O., Hebey, E., Robert, F.: Blow-Up Theory for Elliptic PDEs in Riemannian Geometry, Math Notes, vol. 45. Princeton University press, Princeton (2004)

    Google Scholar 

  19. 19.

    Esposito, P.: A classification result for the quasi-linear Liouville equation. Ann. Inst. H. Poincaré, Anal. Non Linéaire 35(3), 781–801 (2018)

    MathSciNet  MATH  Google Scholar 

  20. 20.

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

    MathSciNet  MATH  Google Scholar 

  21. 21.

    Heinonen, J., Kilpelaine, T., Martio, O.: Non-Linear Potential Theory of Degenerate Elliptic Equations. Clarendon Press, Oxford (1993)

    Google Scholar 

  22. 22.

    Iula, S., Mancini, G.: Extremal functions for singular Moser–Trudinger embeddings. Nonlinear Anal. 156, 215–248 (2017)

    MathSciNet  MATH  Google Scholar 

  23. 23.

    Lam, N., Lu, G.: Sharp Moser–Trudinger inequality on the Heisenberg group at the critical case and applications. Adv. Math. 231, 3259–3287 (2012)

    MathSciNet  MATH  Google Scholar 

  24. 24.

    Li, Y.: Moser–trudinger inequaity on compact Riemannian manifolds of dimension two. J. Partial Differ. Equa. 14, 163–192 (2001)

    MATH  Google Scholar 

  25. 25.

    Li, Y.: The existence of the extremal function of Moser–Trudinger inequality on compact Riemannian manifolds. Sci. China Ser. A 48, 618–648 (2005)

    MathSciNet  MATH  Google Scholar 

  26. 26.

    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)

    MathSciNet  MATH  Google Scholar 

  27. 27.

    Li, X., Yang, Y.: Extremal functions for singular Trudinger–Moser inequalities in the entire Euclidean space. J. Differential Equations 264, 4901–4943 (2018)

    MathSciNet  MATH  Google Scholar 

  28. 28.

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

    MathSciNet  MATH  Google Scholar 

  29. 29.

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

    MathSciNet  MATH  Google Scholar 

  30. 30.

    Lu, G., Zhu, M.: A sharp Moser–Trudinger type inequality involving Ln, norm in the entire space \(\mathbb {R}^{n}\). arXiv:1703.00901

  31. 31.

    Mancini, G., Sandeep, K., Tintarev, C.: Trudinger–moser inequality in the hyperbolic space \(\mathbb {H}^{n}\). Adv. Nonlinear Anal. 2, 309–324 (2013)

    MathSciNet  MATH  Google Scholar 

  32. 32.

    Mancini, G., Thizy, P.D.: Non–existence of extremals for the Adimurthi–Druet inequality. J. Differential Equations 266(2-3), 1051–1072 (2019)

    MathSciNet  MATH  Google Scholar 

  33. 33.

    Moser, J.: A sharp form of an inequality by N. Trudinger. Indiana Univ. Math. J. 20(71), 1077–1092 (1970)

    MathSciNet  MATH  Google Scholar 

  34. 34.

    Nguyen, V.H.: Extremal functions for the Moser-Trudinger inequality of Adimurthi–Druet type in \(W^{1,n}(\mathbb {R}^{n})\). Commun. Contemp. Math., in press. https://doi.org/10.1142/S0219199718500232

  35. 35.

    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)

    MathSciNet  MATH  Google Scholar 

  36. 36.

    Nguyen, V.H.: Improved Moser–Trudinger inequality of Tintarev type in dimension n and the existence of its extremal functions. Ann. Global Anal. Geom. 54(2), 237–256 (2018)

    MathSciNet  MATH  Google Scholar 

  37. 37.

    Pohožaev, S.I.: On the eigenfunctions of the equation Δu + f(u) = 0. (Russian), Dokl. Akad. Nauk. SSSR 165, 36–39 (1965)

    Google Scholar 

  38. 38.

    Ruf, B.: A sharp Trudinger–Moser type inequality for unbounded domains in \(\mathbb {R}^{2}\). J. Funct. Anal. 219, 340–367 (2005)

    MathSciNet  MATH  Google Scholar 

  39. 39.

    Struwe, M.: Critical points of embeddings of \(h^{1,n}_{0}\) into Orlicz spaces. Ann. Inst. H. Poincaré, Anal. Non Linéaire 5, 425–464 (1988)

    MathSciNet  MATH  Google Scholar 

  40. 40.

    Tintarev, C.: Trudinger–moser inequality with remainder terms. J. Funct. Anal. 266, 55–66 (2014)

    MathSciNet  MATH  Google Scholar 

  41. 41.

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

    MathSciNet  MATH  Google Scholar 

  42. 42.

    Yang, Y.: A sharp form of Moser–Trudinger inequality in high dimension. J. Funct. Anal. 239(1), 100–126 (2006)

    MathSciNet  MATH  Google Scholar 

  43. 43.

    Yang, Y.: Extremal functions for a sharp Moser–Trudinger inequality. Internat J. Math. 17, 331–338 (2006)

    MathSciNet  MATH  Google Scholar 

  44. 44.

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

    MathSciNet  MATH  Google Scholar 

  45. 45.

    Yang, Y.: Corrigendum to: a sharp form of Moser–Trudinger inequality in high dimension [J Funct. Anal., 239 (2006) 100–126; MR2258218]. J. Funct. Anal. 242, 669–671 (2007)

    MathSciNet  Google Scholar 

  46. 46.

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

    MathSciNet  MATH  Google Scholar 

  47. 47.

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

    MathSciNet  MATH  Google Scholar 

  48. 48.

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

    MathSciNet  MATH  Google Scholar 

  49. 49.

    Yudovič, V. I.: Some estimates connected with integral operators and with solutions of elliptic equations, (Russian). Dokl. Akad. Nauk SSSR 138, 805–808 (1961)

    MathSciNet  Google Scholar 

Download references

Author information

Affiliations

Authors

Corresponding author

Correspondence to Van Hoang Nguyen.

Additional information

Publisher’s Note

Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

Rights and permissions

Reprints and Permissions

About this article

Verify currency and authenticity via CrossMark

Cite this article

Nguyen, V.H. Improved Singular Moser–Trudinger Inequalities and Their Extremal Functions. Potential Anal 53, 55–88 (2020). https://doi.org/10.1007/s11118-018-09759-3

Download citation

Keywords

  • 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