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

  • Nicolas SaintierEmail author
Original Article


We prove the SH1 p —theory for critical equations involving the p-Laplace operator on compact manifolds. We also prove pointwise estimates for these equations.


System Theory Compact Manifold Asymptotic Estimate Pointwise Estimate Critical Equation 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. 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)CrossRefzbMATHMathSciNetGoogle Scholar
  2. 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)CrossRefMathSciNetGoogle Scholar
  3. 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)MathSciNetGoogle Scholar
  4. 4.
    Brezis, H., Nirenberg, L.: Positive solutions of nonlinear elliptic equations involving critical Sobolev exponents. Comm. Pure Appl. Math. 36(4), 437–477 (1983)MathSciNetGoogle Scholar
  5. 5.
    Damascelli, L., Pacella, F.: Monotonicity and symmetry results for p-Laplace equations and applications. Adv. Differential Equations 5, 1179–1200 (2000)MathSciNetGoogle Scholar
  6. 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)CrossRefMathSciNetGoogle Scholar
  7. 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)MathSciNetGoogle Scholar
  8. 8.
    Druet, O.: Generalized scalar curvature type equations on compact Riemannian manifolds. Proc. Roy. Soc. Edinburgh Sect. A 130(4), 767–788 (2000)zbMATHMathSciNetGoogle Scholar
  9. 9.
    Druet, O.: Isoperimetric inequalities on compact manifolds. Geom. Dedicata 90, 217–236 (2002)CrossRefzbMATHMathSciNetGoogle Scholar
  10. 10.
    Druet, O.: Sharp local isoperimetric inequalities involving the scalar curvature. Proc. Amer. Math. Soc. 130(8), 2351–2361 (2002)CrossRefzbMATHMathSciNetGoogle Scholar
  11. 11.
    Druet, O.: The best constants problem in Sobolev inequalities. Math. Ann. 314(2), 327–346 (1999)CrossRefzbMATHMathSciNetGoogle Scholar
  12. 12.
    Druet, O., Hebey, E.: The AB program in geometric analysis: sharp Sobolev inequalities and related problems. Mem. Amer. Math. Soc. 160(761), (2002)Google Scholar
  13. 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)CrossRefMathSciNetGoogle Scholar
  14. 14.
    Evans, L.C.: Weak convergence methods for nonlinear partial differential equations. Conference Board of the Mathematical Sciences 74 (1990)Google Scholar
  15. 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)CrossRefMathSciNetGoogle Scholar
  16. 16.
    Guedda, M., Veron, L.: Quasilinear elliptic equations involving critical Sobolev exponents. Nonlinear Anal. 13(8), 879–902 (1989)CrossRefMathSciNetGoogle Scholar
  17. 17.
    Hebey, E.: Nonlinear analysis on manifolds: Sobolev spaces and inequalities. Courant Lecture Notes in Mathematics 5 (1999)Google Scholar
  18. 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)CrossRefMathSciNetGoogle Scholar
  19. 19.
    Ladyzhenskaya, O., Ural'tseva, N.: Linear and Quasilinear Elliptic Equations. Academic Press (1968)Google Scholar
  20. 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. 21.
    Sacks, J., Uhlenbeck, K.: The existence of minimal immersions of two-spheres. Ann. of Math. (2), 113(1), 1–24 (1981)MathSciNetGoogle Scholar
  22. 22.
    Schoen, R., Zhang, D.: Prescribed scalar curvature on the n-sphere. Calc. Var. Partial Differential Equations 4(1), 1–25 (1996)MathSciNetGoogle Scholar
  23. 23.
    Struwe, M.: Variational methods. Applications to Nonlinear Partial Differential Equations and Hamiltonian Systems. Third edition. Springer-Verlag (2000)Google Scholar
  24. 24.
    Tolksdorf, P.: Regularity for a more general class of quasilinear elliptic equations. J. Differential Equations 51(1), 126–150 (1984)CrossRefzbMATHMathSciNetGoogle Scholar

Copyright information

© Springer-Verlag 2005

Authors and Affiliations

  1. 1.UFR de Mathématiques, Equipe Géométrie et DynamiqueUniversité Paris 7Paris Cedex 05France

Personalised recommendations