Skip to main content
SpringerLink
Log in

Maximizers for the variational problems associated with Sobolev type inequalities under constraints

  • Published:
Mathematische Annalen Aims and scope Submit manuscript

Abstract

We propose a new approach to study the existence and non-existence of maximizers for the variational problems associated with Sobolev type inequalities both in the subcritical case and critical case under the equivalent constraints. The method is based on an useful link between the attainability of the supremum in our variational problems and the attainablity of the supremum of some special functions defined on \((0,\infty )\). Our approach can be applied to the same problems related to the fractional Laplacian operators. Our main results are new in the critical case and in the setting of the fractional Laplacian operator which was left open in the work of Ishiwata and Wadade (Math Ann 364:1043–1068, 2016) and Ishiwata (On variational problems associated with Sobolev type inequalities and related topics. http://www.rism.it/doc/Ishiwata.pdf, 2015). In the subcritical case, our approach provides a new and elementary proof of the results of Ishiwata and Wadade (2016) and Ishiwata (2015).

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. Adachi, S., Tanaka, K.: Trudinger type inequalities in \({\mathbb{R}}^N\) and their best constant. Proc. Am. Math. Soc. 128, 2051–2057 (2000)

    Article  Google Scholar 

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

    Article  MathSciNet  Google Scholar 

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

    Article  MathSciNet  Google Scholar 

  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. Aubin, T.: Problèmes isopérimétriques et espaces de Sobolev. J. Differ. Geom. 11, 573–598 (1976)

    Article  Google Scholar 

  6. Brasco, L., Mosconi, S., Squassina, M.: Optimal decay of extremals for the fractional Sobolev inequality. Calc. Var. Partial Differ. Equ. 55, 32 (2016)

    Article  MathSciNet  Google Scholar 

  7. Brezis, H., Lieb, E.H.: A relation between pointwise convergence of functions and convergence of functionals. Proc. Am. Math. Soc. 88, 486–490 (1983)

    Article  MathSciNet  Google Scholar 

  8. Brothers, J.E., Ziemer, W.P.: Minimal rearrangements of Sobolev functions. J. Reine. Angew. Math. 348, 153–179 (1988)

    MathSciNet  MATH  Google Scholar 

  9. Cao, D.: Nontrivial solution of semilinear elliptic equations with critical exponent in \({\mathbb{R}}^2\). Comm. Partial Differ. Equ. 17, 407–435 (1992)

    Article  Google Scholar 

  10. 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 

  11. Cordero-Erausquin, D., Nazaret, B., Villani, C.: A mass-transportation approach to sharp Sobolev and Gagliardo-Nirenberg inequalities. Adv. Math. 182, 307–332 (2004)

    Article  MathSciNet  Google Scholar 

  12. Di Nezza, E., Palatucci, G., Valdinoci, E.: Hitchhiker’s guide to the fractional Sobolev spaces. Bull. Sci. Math. 136, 521–573 (2012)

    Article  MathSciNet  Google Scholar 

  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)

    Article  Google Scholar 

  14. do Ó, J.M., Sani, F., Tarsi, C.: Vanishing–concentration–compactness alternative for the Trudinger–Moser inequality in \({\mathbb{R}}^{N}\). Commun. Contemp. Math 20, 1650036 (2018). (pp. 27)

    Article  MathSciNet  Google Scholar 

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

    Article  MathSciNet  Google Scholar 

  16. Ishiwara, M.: Existence and nonexistence of maximizers for variational problems associated with Trudinger-Moser inequalities in \(\mathbb{R}^N\). Math. Ann. 351, 781–804 (2011)

    Article  MathSciNet  Google Scholar 

  17. Ishiwata, M., Wadade, H.: On the effect of equivalent constrain on a maximizing problem associated with the Sobolev type embedding in \({\mathbb{R}}^N\). Math. Ann. 364, 1043–1068 (2016)

    Article  MathSciNet  Google Scholar 

  18. Ishiwata, M.: On variational problems associated with Sobolev type inequalities and related topics. In: Nonlinear phenomena in mathematics and economics, 14–18 Sep 2015. http://www.rism.it/doc/Ishiwata.pdf (2015)

  19. Lam, N., Lu, G., Zhang, L.: Equivalence of critical and subcritical sharp Trudinger-Moser-Adams inequalities. Rev. Mat. Iberoam. 33, 1219–1246 (2017)

    Article  MathSciNet  Google Scholar 

  20. 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)

    Article  MathSciNet  Google Scholar 

  21. Lieb, E.H.: Sharp constants in the Hardy-Littlewood-Sobolev inequalities and related inequalities. Ann. Math. 118, 349–374 (1983)

    Article  MathSciNet  Google Scholar 

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

    Article  MathSciNet  Google Scholar 

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

    Article  Google Scholar 

  24. Lions, P.L.: The concentration-compactness principle in the calculus of variations, the limit case, part \(2\). Rev. Mat. Iberoamericane 1, 45–121 (1985)

    Article  Google Scholar 

  25. Maz’ya, V., Shaposhnikova, T.: On the Bourgain, Brezis, and Mironescu theorem concerning limiting embeddings of fractional Sobolev spaces. J. Funct. Anal. 195, 230–238 (2002)

    Article  MathSciNet  Google Scholar 

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

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

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

    MathSciNet  Google Scholar 

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

    Article  MathSciNet  Google Scholar 

  30. Struwe, M.: Critical points of embeddings of \(H_0^{1,n}\) into Orlicz spaces. Ann. Inst. H. Poincaré, Anal. Non Linéaire 5, 425–464 (1988)

    Article  MathSciNet  Google Scholar 

  31. Talenti, G.: Best constants in Sobolev inequality. Ann. Mat. Pura Appl. 110, 353–372 (1976)

    Article  MathSciNet  Google Scholar 

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

    MathSciNet  MATH  Google Scholar 

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

    MathSciNet  Google Scholar 

Download references

Acknowledgements

This work was supported by the CIMI’s postdoctoral research fellowship.

Author information

Authors and Affiliations

  1. Institute of Research and Development, Duy Tan University, Da Nang, Vietnam

    Van Hoang Nguyen

Authors
  1. Van Hoang Nguyen
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Van Hoang Nguyen.

Additional information

Communicated by Loukas Grafakos.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Nguyen, V.H. Maximizers for the variational problems associated with Sobolev type inequalities under constraints. Math. Ann. 372, 229–255 (2018). https://doi.org/10.1007/s00208-018-1683-y

Download citation

  • Received:

  • Revised:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s00208-018-1683-y

Mathematics Subject Classification

Search

Navigation