Log in

Asymptotic estimates and blow-up theory for critical equations involving the p-Laplacian

  • Original Article
  • Published:
Calculus of Variations and Partial Differential Equations Aims and scope Submit manuscript

Abstract

We prove the SH1p—theory for critical equations involving the p-Laplace operator on compact manifolds. We also prove pointwise estimates for these equations.

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

Access this article

Subscribe and save

Springer+ Basic
EUR 32.99 /Month
  • Get 10 units per month
  • Download Article/Chapter or Ebook
  • 1 Unit = 1 Article or 1 Chapter
  • Cancel anytime
Subscribe now

Buy Now

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

Instant access to the full article PDF.

Similar content being viewed by others

References

  1. Alves, C.O.: Existence of positive solutions for a problem with lack of compactness involving the p-Laplacian. Nonlinear Anal. 51(7), 1187–1206 (2002)

    Article  MATH  MathSciNet  Google Scholar 

  2. Brezis, H., Coron, J.M.: Convergence of solutions of H-systems or how to blow bubbles. Arch. Rational Mech. Anal. 89(1), 21–56 (1985)

    Article  MathSciNet  Google Scholar 

  3. Brezis, H., Lieb, E.: A relation between pointwise convergence of functions and convergence of functionals. Proc. Amer. Math. Soc. 88(3), 486–490 (1983)

    MathSciNet  Google Scholar 

  4. Brezis, H., Nirenberg, L.: Positive solutions of nonlinear elliptic equations involving critical Sobolev exponents. Comm. Pure Appl. Math. 36(4), 437–477 (1983)

    MathSciNet  Google Scholar 

  5. Damascelli, L., Pacella, F.: Monotonicity and symmetry results for p-Laplace equations and applications. Adv. Differential Equations 5, 1179–1200 (2000)

    MathSciNet  Google Scholar 

  6. Damascelli, L., Pacella, F., Ramaswamy, M.: Symmetry of ground states of p-Laplace equations via the moving plane method. Arch. Rational Mech. Anal. 148, 291–308 (1999)

    Article  MathSciNet  Google Scholar 

  7. Demengel, F., Hebey, E.: On some nonlinear equations involving the p-Laplacian with critical Sobolev growth. Adv. Differential Equations 3(4), 533–574 (1998)

    MathSciNet  Google Scholar 

  8. Druet, O.: Generalized scalar curvature type equations on compact Riemannian manifolds. Proc. Roy. Soc. Edinburgh Sect. A 130(4), 767–788 (2000)

    MATH  MathSciNet  Google Scholar 

  9. Druet, O.: Isoperimetric inequalities on compact manifolds. Geom. Dedicata 90, 217–236 (2002)

    Article  MATH  MathSciNet  Google Scholar 

  10. Druet, O.: Sharp local isoperimetric inequalities involving the scalar curvature. Proc. Amer. Math. Soc. 130(8), 2351–2361 (2002)

    Article  MATH  MathSciNet  Google Scholar 

  11. Druet, O.: The best constants problem in Sobolev inequalities. Math. Ann. 314(2), 327–346 (1999)

    Article  MATH  MathSciNet  Google Scholar 

  12. Druet, O., Hebey, E.: The AB program in geometric analysis: sharp Sobolev inequalities and related problems. Mem. Amer. Math. Soc. 160(761), (2002)

  13. Druet, O., Hebey, E., Robert, F.: A C0-theory for the blow-up of second order elliptic equations of critical Sobolev growth. Electron. Res. Announc. Amer. Math. Soc. 9, 19–25 (2003)

    Article  MathSciNet  Google Scholar 

  14. Evans, L.C.: Weak convergence methods for nonlinear partial differential equations. Conference Board of the Mathematical Sciences 74 (1990)

  15. Ghoussoub, N., Yuan, C.: Multiple solutions for quasi-linear PDEs involving the critical Sobolev and Hardy exponents. Trans. Amer. Math. Soc. 352(12), 5703–5743 (2000)

    Article  MathSciNet  Google Scholar 

  16. Guedda, M., Veron, L.: Quasilinear elliptic equations involving critical Sobolev exponents. Nonlinear Anal. 13(8), 879–902 (1989)

    Article  MathSciNet  Google Scholar 

  17. Hebey, E.: Nonlinear analysis on manifolds: Sobolev spaces and inequalities. Courant Lecture Notes in Mathematics 5 (1999)

  18. Hebey, E., Robert, F.: Coercivity and Struwe's compactness for Paneitz type operators with constant coefficients. Calc. Var. Partial Differential Equations 13(4), 491–517 (2001)

    Article  MathSciNet  Google Scholar 

  19. Ladyzhenskaya, O., Ural'tseva, N.: Linear and Quasilinear Elliptic Equations. Academic Press (1968)

  20. Lions, P.L: The concentration-compactness principle in the calculus of variations, the limit case, parts 1 and 2. Rev. Mat. Iberoamericana 1(1/2), 145–201, 45–121 (1985)

    Google Scholar 

  21. Sacks, J., Uhlenbeck, K.: The existence of minimal immersions of two-spheres. Ann. of Math. (2), 113(1), 1–24 (1981)

    MathSciNet  Google Scholar 

  22. Schoen, R., Zhang, D.: Prescribed scalar curvature on the n-sphere. Calc. Var. Partial Differential Equations 4(1), 1–25 (1996)

    MathSciNet  Google Scholar 

  23. Struwe, M.: Variational methods. Applications to Nonlinear Partial Differential Equations and Hamiltonian Systems. Third edition. Springer-Verlag (2000)

  24. Tolksdorf, P.: Regularity for a more general class of quasilinear elliptic equations. J. Differential Equations 51(1), 126–150 (1984)

    Article  MATH  MathSciNet  Google Scholar 

Download references

Author information

Authors and Affiliations

Authors

Corresponding author

Correspondence to Nicolas Saintier.

Rights and permissions

Reprints and permissions

About this article

Cite this article

Saintier, N. Asymptotic estimates and blow-up theory for critical equations involving the p-Laplacian. Calc. Var. 25, 299–331 (2006). https://doi.org/10.1007/s00526-005-0344-7

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s00526-005-0344-7

Keywords

Navigation