Skip to main content
SpringerLink
Log in

Bubbles over Bubbles: A C 0-theory for the Blow-up of Second Order Elliptic Equations of Critical Sobolev Growth

  • Conference paper
Variational Problems in Riemannian Geometry

Part of the book series: Progress in Nonlinear Differential Equations and Their Applications ((PNLDE,volume 59))

  • 529 Accesses

Abstract

Let (M, g,) be a smooth compact Riemannian manifold of dimension n ≥ 3, and △g=-divg▽ be the Laplace-Beltrami operator. Let also 2* be the critical Sobolev exponent for the embedding of the Sobolev space H 21 (M) into Lebesgue’s spaces, and h be a smooth function on M. Elliptic equations of critical Sobolev growth like

$$\Delta _g u + hu = u^{2^* - 1} $$

have been the target of investigation for decades. A very nice H 21 -theory for the asymptotic behaviour of solutions of such an equation is available since the 1980’s. We discuss here the C 0-theory recently developed by Druet, Hebey and Robert. Such a theory provides sharp pointwise estimates for the asymptotic behaviour of solutions of the above equation.

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

Access this chapter

Log in via an institution
Chapter
USD 29.95
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
eBook
USD 84.99
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
Softcover Book
USD 109.99
Price excludes VAT (USA)
  • Compact, lightweight edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info
Hardcover Book
USD 109.99
Price excludes VAT (USA)
  • Durable hardcover edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info

Tax calculation will be finalised at checkout

Purchases are for personal use only

Institutional subscriptions

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. Atkinson, F.V., and Peletier, L.A., Elliptic equations with nearly critical growth, J. Diff. Eq., 70, 349–365, 1987.

    Article  MathSciNet  MATH  Google Scholar 

  2. Bliss, G. A., An integral inequality, J. London Math. Soc., 5, 40–46, 1930.

    Article  MathSciNet  MATH  Google Scholar 

  3. Brézis, H., and Coron, J.M., Convergence de solutions de H-systèmes et applications aux surfaces à courbure moyenne constante, C.R. Acad. Sci. Paris, 298, 389–392, 1984.

    MATH  Google Scholar 

  4. Brézis, H., and Coron, J.M., Convergence of solutions of H-systems or how to blow bubbles, Arch. Rational Mech. Anal., 89, 21–56, 1985.

    Article  MathSciNet  MATH  Google Scholar 

  5. Brézis, H., and Peletier, L.A., Asymptotics for elliptic equations involving critical Sobolev exponents, in Partial Differential Equations and the Calculus of Variations, eds. F. Colombini, A. Marino, L. Modica and S. Spagnalo, Basel: Birkhäuser, 1989.

    Google Scholar 

  6. Caffarelli, L. A., Gidas, B., and Spruck, J., Asymptotic symmetry and local behavior of semilinear elliptic equations with critical Sobolev growth, Comm. Pure Appl. Math., 42, 271–297, 1989.

    Article  MathSciNet  MATH  Google Scholar 

  7. Druet, O., The best constants problem in Sobolev inequalities, Math. Ann., 314, 327–346, 1999.

    Article  MathSciNet  MATH  Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

  9. Druet, O., From one bubble to several bubbles. The low-dimensional case, J. Differential Geom., To appear.

    Google Scholar 

  10. Druet, O., and Hebey, E., The AB program in geometric analysis. Sharp Sobolev inequalities and related problems, Memoirs of the American Mathematical Society, MEMO/160/761, 2002.

    Google Scholar 

  11. Druet, O., and Hebey, E., Blow-up examples for second order elliptic PDEs of critical Sobolev growth, Trans. Amer. Math. Soc., to appear.

    Google Scholar 

  12. Druet, O., Hebey, E., and Robert, F., Blow-up theory for elliptic PDEs in Riemannian geometry, Mathematical Notes, Princeton University Press, to appear.

    Google Scholar 

  13. Druet, O., Hebey, E., and Robert, F., A C 0 -theory for the blow-up of second order elliptic equations of critical Sobolev growth, Electronic Research Announcements of the AMS, 9, 19–25, 2003.

    Article  MathSciNet  MATH  Google Scholar 

  14. Druet, O., and Robert, F., Asymptotic profile for the sub-extremals of the sharp Sobolev inequality on the sphere, Comm. P.D.E., 25, 743–778, 2001.

    Article  MathSciNet  Google Scholar 

  15. Gagliardo, E., Proprietà di alcune classi di funzioni in più variabili, Ricerche Mat., 7, 102–137, 1958.

    MathSciNet  MATH  Google Scholar 

  16. Han, Z.C., Asymptotic approach to singular solutions for nonlinearelliptic equations involving critical Sobolev exponent, Ann. Inst. H. Poincaré. Anal. Non Linéaire, 8, 159–174, 1991.

    MATH  Google Scholar 

  17. Hebey, E., Nonlinear analysis on manifolds: Sobolev spaces and inequalities, CIMS Lecture Notes, Courant Institute of Mathematical Sciences, Vol. 5, 1999. Second edition published by the American Mathematical Society, 2000.

    Google Scholar 

  18. Hebey, E., Nonlinear elliptic equations of critical Sobolev growth from a dynamical view-, point, Preprint, Conference in honor of H. Brézis and F. Browder, Rutgers University, 2001.

    Google Scholar 

  19. Hebey, E., and Vaugon, M., The best constant problem in the Sobolev embedding theorem for complete Riemannian manifolds, Duke Math. J., 79, 235–279, 1995.

    Article  MathSciNet  MATH  Google Scholar 

  20. Li, Y.Y., Prescribing scalar curvature on S n and related problems, Part I, J. Diff. Equations, 120, 319–410, 1995.

    Article  MATH  Google Scholar 

  21. Li, Y.Y., Prescribing scalar curvature on S n and related problems, Part II: existence and compactness, Comm. Pure Appl. Math., 49, 541–597, 1996.

    Article  MathSciNet  MATH  Google Scholar 

  22. Lions, P.L., The concentration-compactness principle in the calculus of variations. The limit case. I, II, Rev. Mat. Iberoamericana, 1: no. 1, 145–201; no. 2, 45–121, 1985.

    Google Scholar 

  23. Nirenberg, L., On elliptic partial differential equations, Ann. Scuola Norm. Sup. Pisa, 13, 116–162, 1959.

    MathSciNet  Google Scholar 

  24. Obata, M., The conjectures on conformal transformations of Riemannian manifolds, J. Differential Geom., 6, 247–258, 1971/72.

    MathSciNet  Google Scholar 

  25. Robert, F.Asymptotic behaviour of a nonlinear elliptic equation with critical Sobolev exponent - The radial case I, Adv. Differential Equations, 6, 821–846, 2001.

    MathSciNet  MATH  Google Scholar 

  26. Robert, F., Asymptotic behaviour of a nonlinear elliptic equation with critical Sobolev exponent - The radial case II, Nonlinear Differ. Eq. Appl., 9, 361–384, 2002.

    Article  MATH  Google Scholar 

  27. Sacks, P., and Uhlenbeck, K., On the existence of minimal immersions of 2-spheres, Ann. of Math., 113, 1–24, 1981.

    Article  MathSciNet  MATH  Google Scholar 

  28. Schoen, R., Variational theory for the total scalar curvature functional for Riemannian metrics and related topics, Topics in calculus of variations (Montecatini Terme, 1987), 120–154. Lecture Notes in Mathematics, 1365, Springer, Berlin-New York, 1989.

    Google Scholar 

  29. Schoen, R., On the number of constant scalar curvature metrics in a conformal class, Differential Geometry: A symposium in honor of Manfredo Do Carmo, H.B.Lawson and K.Tenenblat eds., Pitman Monogr. Surveys Pure Appl. Math., 52, 311–320, 1991.

    MathSciNet  Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

  31. Sobolev, S.L., Sur un théorème d’analyse fonctionnelle, Mat. Sb., 46, 471–496, 1938.

    Google Scholar 

  32. Struwe, M., A global compactness result for elliptic boundary problems involving limiting nonlinearities, Math. Z., 187, 511–517, 1984.

    Article  MathSciNet  MATH  Google Scholar 

  33. Wente, H.C., Large solutions to the volume constrained Plateau problem, Arch. Rational Mech. Anal., 75, 59–77, 1980.

    Article  MathSciNet  MATH  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Département de Mathématiques Site de Saint-Martin, Université de Cergy-Pontoise, 2 avenue Adolphe Chauvin, Cergy-Pontoise cedex, F-95302, France

    Emmanuel Hebey

Authors
  1. Emmanuel Hebey
    View author publications

    You can also search for this author in PubMed Google Scholar

Editor information

Editors and Affiliations

  1. Département de Mathématiques, Université de Bretagne Occidentale, UFR Science et Technique 6, avenue Victor Le Gorgeu, B.P. 809, Brest Cedex, 29285, France

    Paul Baird , Ali Fardoun  & Rachid Regbaoui ,  & 

  2. Laboratoire de Mathématiques et Physique Théorique, UMR 6083 du CNRS Université de Tours, Parc de Grandmont, Tours Cedex, 37200, France

    Ahmad El Soufi

Rights and permissions

Reprints and permissions

Copyright information

© 2004 Springer Basel AG

About this paper

Cite this paper

Hebey, E. (2004). Bubbles over Bubbles: A C 0-theory for the Blow-up of Second Order Elliptic Equations of Critical Sobolev Growth. In: Baird, P., Fardoun, A., Regbaoui, R., El Soufi, A. (eds) Variational Problems in Riemannian Geometry. Progress in Nonlinear Differential Equations and Their Applications, vol 59. Birkhäuser, Basel. https://doi.org/10.1007/978-3-0348-7968-2_1

Download citation

  • DOI: https://doi.org/10.1007/978-3-0348-7968-2_1

  • Publisher Name: Birkhäuser, Basel

  • Print ISBN: 978-3-0348-9640-5

  • Online ISBN: 978-3-0348-7968-2

  • eBook Packages: Springer Book Archive

Publish with us

Policies and ethics

Search

Navigation