Skip to main content
Log in

Linking solutions for quasilinear equations at critical growth involving the “1-Laplace” operator

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

Abstract

We show that the problem at critical growth, involving the 1-Laplace operator and obtained by relaxation of \({-\Delta_1 u=\lambda |u|^{-1}u+|u|^{1^*-2} u}\) , admits a nontrivial solution \({u \in BV(\Omega)}\) for any λ ≥ λ1. Nonstandard linking structures, for the associated functional, are recognized.

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

Access this article

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. Ambrosetti A., Rabinowitz P.H.: Dual variational methods in critical point theory and applications. J. Funct. Anal. 14, 349–381 (1973)

    Article  MATH  MathSciNet  Google Scholar 

  2. Ambrosio L., Fusco N., Pallara D.: Functions of Bounded Variation and Free Discontinuity Problems. Oxford University Press, New York (2000)

    MATH  Google Scholar 

  3. Arioli G., Gazzola F.: Some results on p-Laplace equations with a critical growth term. Differ. Int. Equ. 11, 311–326 (1998)

    MATH  MathSciNet  Google Scholar 

  4. Aubin T.: Problèmes isopérimétriques et espaces de Sobolev. J. Differ. Geom. 11, 573–598 (1976)

    MATH  MathSciNet  Google Scholar 

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

    Article  MATH  MathSciNet  Google Scholar 

  6. Campa I., Degiovanni M.: Subdifferential calculus and nonsmooth critical point theory. SIAM J. Optim. 10, 1020–1048 (2000)

    Article  MATH  MathSciNet  Google Scholar 

  7. Capozzi A., Fortunato D., Palmieri G.: An existence result for nonlinear elliptic problems involving critical Sobolev exponent. Ann. Inst. H. Poincaré Anal. Non Linéaire 2, 463–470 (1985)

    MATH  MathSciNet  Google Scholar 

  8. Chang K.-C.: Variational methods for nondifferentiable functionals and their applications to partial differential equations. J. Math. Anal. Appl. 80, 102–129 (1981)

    Article  MATH  MathSciNet  Google Scholar 

  9. Chang K.-C.: Infinite-dimensional Morse Theory and Multiple Solution Problems. Birkhäuser, Boston (1993)

    MATH  Google Scholar 

  10. Clarke F.H.: Optimization and Nonsmooth Analysis. Wiley, New York (1983)

    MATH  Google Scholar 

  11. Corvellec J.-N., Degiovanni M., Marzocchi M.: Deformation properties for continuous functionals and critical point theory. Topol. Methods Nonlinear Anal. 1, 151–171 (1993)

    MATH  MathSciNet  Google Scholar 

  12. Degiovanni M., Lancelotti S.: Linking over cones and nontrivial solutions for p-Laplace equations with p-superlinear nonlinearity. Ann. Inst. H. Poincaré Anal. Non Linéaire 24, 907–919 (2007)

    Article  MATH  MathSciNet  Google Scholar 

  13. Degiovanni M., Lancelotti S.: Linking solutions for p-Laplace equations with nonlinearity at critical growth. J. Funct. Anal. 256, 3643–3659 (2009)

    Article  MATH  MathSciNet  Google Scholar 

  14. Degiovanni M., Marzocchi M.: A critical point theory for nonsmooth functionals. Ann. Mat. Pura Appl. 167, 73–100 (1994)

    Article  MATH  MathSciNet  Google Scholar 

  15. Degiovanni M., Marzocchi M., Rădulescu V.D.: Multiple solutions of hemivariational inequalities with area-type term. Calc. Var. Partial Differ. Equ. 10, 355–387 (2000)

    Article  MATH  Google Scholar 

  16. Degiovanni M., Schuricht F.: Buckling of nonlinearly elastic rods in the presence of obstacles treated by nonsmooth critical point theory. Math. Ann. 311, 675–728 (1998)

    Article  MATH  MathSciNet  Google Scholar 

  17. Demengel F.: On some nonlinear partial differential equations involving the “1”-Laplacian and critical Sobolev exponent. ESAIM Control Optim. Calc. Var. 4, 667–686 (1999)

    Article  MATH  MathSciNet  Google Scholar 

  18. Demengel F.: Some existence’s results for noncoercive “1-Laplacian” operator. Asymptot. Anal. 43, 287–322 (2005)

    MATH  MathSciNet  Google Scholar 

  19. Egnell H.: Existence and nonexistence results for m-Laplace equations involving critical Sobolev exponents. Arch. Ration. Mech. Anal. 104, 57–77 (1988)

    Article  MATH  MathSciNet  Google Scholar 

  20. Fadell E.R., Rabinowitz P.H.: Bifurcation for odd potential operators and an alternative topological index. J. Funct. Anal. 26, 48–67 (1977)

    Article  MATH  MathSciNet  Google Scholar 

  21. 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, 139–174 (1978)

    Article  MATH  MathSciNet  Google Scholar 

  22. García Azorero J., Peral Alonso I.: Existence and nonuniqueness for the p-Laplacian: nonlinear eigenvalues. Comm. Partial Differ. Equ. 12, 1389–1430 (1987)

    MATH  Google Scholar 

  23. Gazzola F., Ruf B.: Lower-order perturbations of critical growth nonlinearities in semilinear elliptic equations. Adv. Differ. Equ. 2, 555–572 (1997)

    MATH  MathSciNet  Google Scholar 

  24. Giusti E.: Minimal Surfaces and Functions of Bounded Variation. Birkhäuser Verlag, Basel (1984)

    MATH  Google Scholar 

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

    Article  MATH  MathSciNet  Google Scholar 

  26. Ioffe A., Schwartzman E.: Metric critical point theory. I. Morse regularity and homotopic stability of a minimum. J. Math. Pures Appl. 75, 125–153 (1996)

    MATH  MathSciNet  Google Scholar 

  27. Katriel G.: Mountain pass theorems and global homeomorphism theorems. Ann. Inst. H. Poincaré Anal. Non Linéaire 11, 189–209 (1994)

    MATH  MathSciNet  Google Scholar 

  28. Kawohl B., Schuricht F.: Dirichlet problems for the 1-Laplace operator, including the eigenvalue problem. Commun. Contemp. Math. 9, 515–543 (2007)

    Article  MATH  MathSciNet  Google Scholar 

  29. Marzocchi M.: Multiple solutions of quasilinear equations involving an area-type term. J. Math. Anal. Appl. 196, 1093–1104 (1995)

    Article  MATH  MathSciNet  Google Scholar 

  30. Milbers, Z., Schuricht, F.: Existence of a sequence of eigensolutions for the 1-Laplace operator. Technische Universität Dresden. MATH-AN-04-2008 (2008. preprint)

  31. Rabinowitz, P.H.: Minimax Methods in Critical Point Theory with Applications to Differential Equations. American Mathematical Society, Providence (1986)

  32. Rockafellar R.T.: Generalized directional derivatives and subgradients of nonconvex functions. Can. J. Math. 32, 257–280 (1980)

    MATH  MathSciNet  Google Scholar 

Download references

Author information

Authors and Affiliations

Authors

Corresponding author

Correspondence to Marco Degiovanni.

Additional information

Communicated by L. Ambrosio.

The research of M. Degiovanni was partially supported by the PRIN project “Variational and topological methods in the study of nonlinear phenomena” and by Gruppo Nazionale per l’Analisi Matematica, la Probabilità e le loro Applicazioni (INdAM).

The research of P. Magrone was partially supported by the PRIN project “Variational methods and nonlinear differential equations”.

Rights and permissions

Reprints and permissions

About this article

Cite this article

Degiovanni, M., Magrone, P. Linking solutions for quasilinear equations at critical growth involving the “1-Laplace” operator. Calc. Var. 36, 591 (2009). https://doi.org/10.1007/s00526-009-0246-1

Download citation

  • Received:

  • Accepted:

  • Published:

  • DOI: https://doi.org/10.1007/s00526-009-0246-1

Mathematics Subject Classification (2000)

Navigation