Skip to main content
SpringerLink
Log in

On the Dependence on p of the Variational Eigenvalues of the p-Laplace Operator

  • Published:
Potential Analysis Aims and scope Submit manuscript

Abstract

We study the behavior of the variational eigenvalues of the p-Laplace operator, with homogeneous Dirichlet boundary condition, when p is varying. After introducing an auxiliary problem, we characterize the continuity answering, in particular, a question raised in Lindqvist (Potential Anal. 2(3), 199–218 1993).

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

Access this article

Log in via an institution

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

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

  1. Attouch, H.: Variational convergence for functions and operators Applicable mathematics series. Pitman, Boston (1984)

    Google Scholar 

  2. Bartsch, T.: Topological methods for variational problems with symmetries Lecture Notes in Mathematics, vol. 1560. Springer, Berlin (1993)

    Google Scholar 

  3. Belloni, M., Kawohl, B.: A direct uniqueness proof for equations involving the p-Laplace operator. Manuscripta Math. 109(2), 229–231 (2002)

    Article  MathSciNet  MATH  Google Scholar 

  4. Brasco, L., Franzina, G.: A note on positive eigenfunctions and hidden convexity. Arch. Math. (Basel) 99(4), 367–374 (2012)

    Article  MathSciNet  MATH  Google Scholar 

  5. Brezis, H.: Functional analysis, Sobolev spaces and partial differential equations Universitext. Springer, New York (2011)

    Google Scholar 

  6. Champion, T., De Pascale, L.: Asymptotic behaviour of nonlinear eigenvalue problems involving p-Laplacian-type operators. Proc. Roy. Soc. Edinburgh Sect. A 137 (6), 1179–1195 (2007)

    Article  MathSciNet  MATH  Google Scholar 

  7. Dal Maso, G.: An introduction to Γ-convergence Progress in nonlinear differential equations and their applications, vol. 8. Birkhäuser, Boston (1993)

    Google Scholar 

  8. Degiovanni, M., Marzocchi, M.: Limit of minimax values under Γ-convergence. Electron. J. Differential Equations 2014(266), 19 (2014)

    MathSciNet  Google Scholar 

  9. DiBenedetto, E.: C 1+α local regularity of weak solutions of degenerate elliptic equations. Nonlinear Anal. 7 (8), 827–850 (1983)

    Article  MathSciNet  MATH  Google Scholar 

  10. El Khalil, A., El Manouni, S., Ouanan, M.: On the principal eigencurve of the p-Laplacian: stability phenomena. Canad. Math. Bull. 49(3), 358–370 (2006)

    Article  MathSciNet  MATH  Google Scholar 

  11. El Khalil, A., Lindqvist, P., Touzani, A.: On the stability of the first eigenvalue of A p u+λ g(x)|u|p−2 u=0 with varying p. Rend. Mat. Appl. (7) 24(2), 321–336 (2004)

    MathSciNet  MATH  Google Scholar 

  12. Fadell, E. R., Rabinowitz, P. H.: Bifurcation for odd potential operators and an alternative topological index. J. Functional Analysis 26(1), 48–67 (1977)

    Article  MathSciNet  MATH  Google Scholar 

  13. Fadell, E. R., Rabinowitz, P. H.: Generalized cohomological index theories for Lie group actions with an application to bifurcation questions for Hamiltonian systems. Invent. Math. 45(2), 139–174 (1978)

    Article  MathSciNet  MATH  Google Scholar 

  14. Kawohl, B., Lindqvist, P.: Positive eigenfunctions for the p-Laplace operator revisited. Analysis (Munich) 26(4), 545–550 (2006)

    MathSciNet  MATH  Google Scholar 

  15. Kawohl, B., Lucia, M., Prashanth, S.: Simplicity of the principal eigenvalue for indefinite quasilinear problems. Adv. Differential Equations 12(4), 407–434 (2007)

    MathSciNet  MATH  Google Scholar 

  16. Krasnosel’skiı̆, M.A.: Topological methods in the theory of nonlinear integral equations. A Pergamon press book. The Macmillan Co., New York (1964)

    Google Scholar 

  17. Lindqvist, P.: On the equation div (|∇u|p −2∇u)+λ|u|p−2u=0. Proc. Amer. Math. Soc. 109(1), 157–164 (1990). and 116 (1992), no 2, 583–584

  18. Lindqvist, P.: On nonlinear Rayleigh quotients. Potential Anal. 2(3), 199–218 (1993)

    Article  MathSciNet  MATH  Google Scholar 

  19. Lindqvist, P.: A nonlinear eigenvalue problem. In: Ciatti, P., Gonzalez, E., Lanza de Cristoforis, M., Leonardi, G.P. (eds.) Topics in mathematical analysis, Ser. Anal. Appl. Comput., 3, 175–203. World Sci. Publ., Hackensack (2008)

    Google Scholar 

  20. Lucia, M., Prashanth, S.: Simplicity of principal eigenvalue for p-Laplace operator with singular indefinite weight. Arch. Math. (Basel) 86(1), 79–89 (2006)

    Article  MathSciNet  MATH  Google Scholar 

  21. Lucia, M., Schuricht, F.: A class of degenerate elliptic eigenvalue problems. Adv. Nonlinear Anal. 2(1), 91–125 (2013)

    MathSciNet  MATH  Google Scholar 

  22. Parini, E.: Continuity of the variational eigenvalues of the p-Laplacian with respect to p. Bull. Aust. Math. Soc. 83(3), 376–381 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  23. Rabinowitz, P. H.: Minimax methods in critical point theory with applications to differential equations, CBMS Regional Conference Series in Mathematics, vol. 65. Published for the Conference Board of the Mathematical Sciences, Washington (1986)

    Book  Google Scholar 

  24. Szulkin, A., Willem, M.: Eigenvalue problems with indefinite weight. Studia Math. 135(2), 191–201 (1999)

    MathSciNet  MATH  Google Scholar 

  25. Talenti, G.: Best constant in Sobolev inequality. Ann. Mat. Pura Appl. (4) 110, 353–372 (1976)

    Article  MathSciNet  MATH  Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

  27. Vázquez, J.L.: A strong maximum principle for some quasilinear elliptic equations. Appl. Math. Optim. 12(3), 191–202 (1984)

    Article  MathSciNet  MATH  Google Scholar 

  28. Willem, M.: Minimax theorems, Progress in nonlinear differential equations and their applications, vol. 24. Birkhäuser, Boston (1996)

Download references

Author information

Authors and Affiliations

  1. Dipartimento di Matematica e Fisica, Università Cattolica del Sacro Cuore, Via dei Musei 41, 25121, Brescia, Italy

    Marco Degiovanni & Marco Marzocchi

Authors
  1. Marco Degiovanni
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Marco Marzocchi
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Marco Degiovanni.

Additional information

The research of the authors was partially supported by Gruppo Nazionale per l’Analisi Matematica, la Probabilità e le loro Applicazioni (INdAM)

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Degiovanni, M., Marzocchi, M. On the Dependence on p of the Variational Eigenvalues of the p-Laplace Operator. Potential Anal 43, 593–609 (2015). https://doi.org/10.1007/s11118-015-9487-0

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s11118-015-9487-0

Keywords

Mathematics Subject Classification (2010)

Search

Navigation