Asymptotic in Sobolev spaces for symmetric Paneitz-type equations on Riemannian manifolds

  • Nicolas SaintierEmail author


We describe the asymptotic behaviour in Sobolev spaces of sequences of solutions of Paneitz-type equations [Eq. (E α ) below] on a compact Riemannian manifold (M, g) which are invariant by a subgroup of the group of isometries of (M, g). We also prove pointwise estimates.

Mathematics Subject Classification (2000)

35B40 35J20 35B33 


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. 1.
    Bredon G.E.: Introduction to compact transformation groups. Pure and Applied Mathematics, vol. 46. Academic Press, London (1972)Google Scholar
  2. 2.
    Chang, S.Y.A.: On Paneitz Operator—a fourth order differential operator in conformal geometry. In: Christ, M., Kenig, C., Sadorsky, C. (Eds.) Harmonic Analysis and Partial Differential Equations, Essays in honor of Alberto P. Calderon. Chicago Lectures in Mathematics, pp. 127–150 (1999)Google Scholar
  3. 3.
    Chang, S.Y.A, Yang, P.C.: On a fourth order curvature invariant. In: Branson, T. (Ed.) Spectral Problems in Geometry and Arithmetic, Comp. Math. vol. 237, pp. 9–28. AMS (1999)Google Scholar
  4. 4.
    Clapp, M.: A global compactness result for elliptic problems with critical nonlinearity on symmetric domains. Nonlinear Equations: Methods, Models and Applications, Progr. Nonlinear Differential Equations Applications, vol. 54, pp. 117–126. Birkhäuser, Boston (2003)Google Scholar
  5. 5.
    Djadli Z., Hebey E., Ledoux M.: Paneitz-type operators and applications. Duke Math. J. 104(1), 129–169 (2000)zbMATHCrossRefMathSciNetGoogle Scholar
  6. 6.
    Edmunds D.E., Fortunato F., Janelli E.: Critical exponents, critical dimensions, and the biharmoni operator. Arch. Rational. Mech. Anal. 112, 269–289 (1990)zbMATHCrossRefMathSciNetGoogle Scholar
  7. 7.
    Faget Z.: Best constant in Sobolev inequalities on Riemannian manifolds in the presence of symmetries. Potential Anal. 17(2), 105–124 (2002)zbMATHCrossRefMathSciNetGoogle Scholar
  8. 8.
    Faget Z.: Optimal constants in critical Sobolev inequalities on Riemannian manifolds in the presence of symmetries. Ann. Global Anal. Geom. 24, 161–200 (2003)zbMATHCrossRefMathSciNetGoogle Scholar
  9. 9.
    Hebey, E.: Nonlinear analysis on manifolds: Sobolev spaces and inequalities. Courant Lecture Notes in Mathematics, vol. 5 (1999)Google Scholar
  10. 10.
    Hebey E.: Sharp Sobolev inequalities of second order. J. Geom. Anal. 13(1), 145–162 (2003)zbMATHMathSciNetGoogle Scholar
  11. 11.
    Hebey E., Robert F.: Coercivity and Struwe’s compactness for Paneitz type operators with constant coefficients. Calc. Var. Partial Differ. Equ. 13(4), 491–517 (2001)zbMATHCrossRefMathSciNetGoogle Scholar
  12. 12.
    Hebey E., Vaugon M.: Sobolev spaces in the presence of symmetries. J. Math. Pures Appl. 76(10), 859–881 (1997)MathSciNetGoogle Scholar
  13. 13.
    Lieb E.H.: Sharp constants in the Hardy–Littlewood–Sobolev and related inequalities. Ann. Math. 118(2), 349–374 (1983)CrossRefMathSciNetGoogle Scholar
  14. 14.
    Lin C.S.: A classification of solutions of a conformally invariant fourth order equation in \({\mathbb{R}^n}\) . Comment Math. Helv. 73, 206–231 (1998)zbMATHCrossRefMathSciNetGoogle Scholar
  15. 15.
    Lions P.L.: The concentration-compactness principle in the calculus of variations, the limit case, parts 1 and 2. Rev. Mat. Iberoamericana 1, 145–201 (1985) 45–121zbMATHMathSciNetGoogle Scholar
  16. 16.
    Robert F.: Positive solutions for a fourth order equation invariant under isometries. Proc. Am. Math. Soc. 131(5), 1423–1431 (2003)zbMATHCrossRefGoogle Scholar
  17. 17.
    Saintier N.: Asymptotic estimates and blow-up theory for critical equations involving the p-laplacian. Calc. Var. Partial Differ. Equ. 25(3), 299–331 (2006)zbMATHCrossRefMathSciNetGoogle Scholar
  18. 18.
    Saintier N.: Changing sign solutions of a conformally invariant fourth order equation in the Euclidean space. Commun. Anal. Geometry 14(4), 613–624 (2006)zbMATHMathSciNetGoogle Scholar
  19. 19.
    Saintier, N.: Best constant in critical Sobolev inequalities of second order in the presence of symmetries on Riemannian manifolds (in preparation)Google Scholar
  20. 20.
    Struwe M.: A global compactness result for elliptic boundary value problem involving limiting nonlinearities. Math. Z. 187(4), 511–517 (1984)zbMATHCrossRefMathSciNetGoogle Scholar
  21. 21.
    Struwe M.: Variational methods. Applications to nonlinear partial differential equations and Hamiltonian systems. Springer, Berlin (2000)zbMATHGoogle Scholar

Copyright information

© Springer-Verlag 2008

Authors and Affiliations

  1. 1.Departamento de Matemática, FCEyNUniversidad de Buenos-AiresBuenos AiresArgentina
  2. 2.Universidad Nacional de General SarmientoLos Polvorines, Pcia de Buenos AiresArgentina

Personalised recommendations