Skip to main content
Journal of Inequalities and Applications
Download PDF
  • Research Article
  • Open access
  • Published:

A Hardy Inequality with Remainder Terms in the Heisenberg Group and the Weighted Eigenvalue Problem

Journal of Inequalities and Applications volume 2007, Article number: 032585 (2007) Cite this article

Abstract

Based on properties of vector fields, we prove Hardy inequalities with remainder terms in the Heisenberg group and a compact embedding in weighted Sobolev spaces. The best constants in Hardy inequalities are determined. Then we discuss the existence of solutions for the nonlinear eigenvalue problems in the Heisenberg group with weights for the-sub-Laplacian. The asymptotic behaviour, simplicity, and isolation of the first eigenvalue are also considered.

[1234567891011121314151617181920]

References

  1. Capogna L, Danielli D, Garofalo N: An embedding theorem and the Harnack inequality for nonlinear subelliptic equations. Communications in Partial Differential Equations 1993,18(9–10):1765–1794.

    Article  MathSciNet  MATH  Google Scholar 

  2. Domokos A, Manfredi JJ: -regularity for-harmonic functions in the Heisenberg group fornear 2. In The p-Harmonic Equation and Recent Advances in Analysis, Contemporary Mathematics. Volume 370. American Mathematical Society, Providence, RI, USA; 2005:17–23.

    Google Scholar 

  3. Allegretto W: Principal eigenvalues for indefinite-weight elliptic problems in . Proceedings of the American Mathematical Society 1992,116(3):701–706.

    MathSciNet  MATH  Google Scholar 

  4. Adimurthi , Chaudhuri N, Ramaswamy M: An improved Hardy-Sobolev inequality and its application. Proceedings of the American Mathematical Society 2002,130(2):489–505. 10.1090/S0002-9939-01-06132-9

    Article  MathSciNet  MATH  Google Scholar 

  5. Chaudhuri N, Ramaswamy M: Existence of positive solutions of some semilinear elliptic equations with singular coefficients. Proceedings of the Royal Society of Edinburgh: Section A 2001,131(6):1275–1295. 10.1017/S0308210500001396

    Article  MathSciNet  MATH  Google Scholar 

  6. Adimurthi , Esteban MJ: An improved Hardy-Sobolev inequality in and its application to Schrödinger operators. Nonlinear Differential Equations and Applications 2005,12(2):243–263. 10.1007/s00030-005-0009-4

    Article  MathSciNet  MATH  Google Scholar 

  7. García Azorero JP, Peral Alonso I: Hardy inequalities and some critical elliptic and parabolic problems. Journal of Differential Equations 1998,144(2):441–476. 10.1006/jdeq.1997.3375

    Article  MathSciNet  MATH  Google Scholar 

  8. Lê A: Eigenvalue problems for the -Laplacian. Nonlinear Analysis: Theory, Methods & Applications 2006,64(5):1057–1099. 10.1016/j.na.2005.05.056

    Article  MathSciNet  MATH  Google Scholar 

  9. Vázquez JL, Zuazua E: The Hardy inequality and the asymptotic behaviour of the heat equation with an inverse-square potential. Journal of Functional Analysis 2000,173(1):103–153. 10.1006/jfan.1999.3556

    Article  MathSciNet  MATH  Google Scholar 

  10. De Napoli PL, Pinasco JP: Eigenvalues of the -Laplacian and disconjugacy criteria. Journal of Inequalities and Applications 2006, 2006: 8 pages.

    Article  MathSciNet  MATH  Google Scholar 

  11. Jaroš J, Takaŝi K, Yoshida N: Picone-type inequalities for nonlinear elliptic equations and their applications. Journal of Inequalities and Applications 2001,6(4):387–404. 10.1155/S1025583401000236

    MathSciNet  MATH  Google Scholar 

  12. Yaotian S, Zhihui C: General Hardy inequalities with optimal constants and remainder terms. Journal of Inequalities and Applications 2005, (3):207–219.

    Google Scholar 

  13. Barbatis G, Filippas S, Tertikas A: A unified approach to improved Hardy inequalities with best constants. Transactions of the American Mathematical Society 2004,356(6):2169–2196. 10.1090/S0002-9947-03-03389-0

    Article  MathSciNet  MATH  Google Scholar 

  14. Barbatis G, Filippas S, Tertikas A: Series expansion for Hardy inequalities. Indiana University Mathematics Journal 2003,52(1):171–190. 10.1512/iumj.2003.52.2207

    Article  MathSciNet  MATH  Google Scholar 

  15. Niu P, Zhang H, Wang Y: Hardy type and Rellich type inequalities on the Heisenberg group. Proceedings of the American Mathematical Society 2001,129(12):3623–3630. 10.1090/S0002-9939-01-06011-7

    Article  MathSciNet  MATH  Google Scholar 

  16. D'Ambrosio L: Critical degenerate inequalities on the Heisenberg group. Manuscripta Mathematica 2001,106(4):519–536. 10.1007/s229-001-8031-2

    Article  MathSciNet  MATH  Google Scholar 

  17. D'Ambrosio L: Hardy-type inequalities related to degenerate elliptic differential operators. Annali della Scuola Normale Superiore di Pisa 2005,4(3):451–486.

    MathSciNet  MATH  Google Scholar 

  18. Garofalo N, Nhieu D-M: Isoperimetric and Sobolev inequalities for Carnot-Carathéodory spaces and the existence of mimimal surfaces. Communications on Pure and Applied Mathematics 1996,49(10):1081–1144. 10.1002/(SICI)1097-0312(199610)49:10<1081::AID-CPA3>3.0.CO;2-A

    Article  MathSciNet  MATH  Google Scholar 

  19. Struwe M: Variational Methods: Applications to Nonlinear Partial Differential Equations and Hamiltonian Systems, A Series of Modern Surveys in Mathematics. Volume 34. 3rd edition. Springer, Berlin, Germany; 2000:xviii+274.

    Book  MATH  Google Scholar 

  20. Boccardo L, Murat F: Almost everywhere convergence of the gradients of solutions to elliptic and parabolic equations. Nonlinear Analysis: Theory, Methods & Applications 1992,19(6):581–597. 10.1016/0362-546X(92)90023-8

    Article  MathSciNet  MATH  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Department of Applied Mathematics, Northwestern Polytechnical University, Xi'an, Shaanxi, 710072, China

    Jingbo Dou, Pengcheng Niu & Zixia Yuan

Authors
  1. Jingbo Dou
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Pengcheng Niu
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Zixia Yuan
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Pengcheng Niu.

Rights and permissions

Open Access This article is distributed under the terms of the Creative Commons Attribution 2.0 International License ( https://creativecommons.org/licenses/by/2.0 ), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Reprints and permissions

About this article

Cite this article

Dou, J., Niu, P. & Yuan, Z. A Hardy Inequality with Remainder Terms in the Heisenberg Group and the Weighted Eigenvalue Problem. J Inequal Appl 2007, 032585 (2007). https://doi.org/10.1155/2007/32585

Download citation

  • Received:

  • Revised:

  • Accepted:

  • Published:

  • DOI: https://doi.org/10.1155/2007/32585

Keywords