Advertisement

Quasilinear and Hessian Lane-Emden Type Systems with Measure Data

  • Marie-Françoise Bidaut-Véron
  • Quoc-Hung Nguyen
  • Laurent VéronEmail author
Article
  • 9 Downloads

Abstract

We study nonlinear systems of the form \(-{\Delta }_{p}u=v^{q_{1}}+\mu , -{\Delta }_{p}v=u^{q_{2}}+\eta \) and \(F_{k}[-u]=v^{s_{1}}+\mu , F_{k}[-v]=u^{s_{2}}+\eta \) in a bounded domain Ω or in \(\mathbb {R}^{N}\) where μ and η are nonnegative Radon measures, Δp and Fk are respectively the p-Laplacian and the k-Hessian operators and q1, q2, s1 and s2 positive numbers. We give necessary and sufficient conditions for existence expressed in terms of Riesz or Bessel capacities.

Keywords

p-Laplacian k-Hessian Bessel and Riesz capacities Measures Maximal functions 

Mathematics Subject Classification 2010

35J70 35J60 45G15 31C15 

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

Notes

References

  1. 1.
    Adams, D.R., Heberg, L.I.: Function Spaces and Potential Theory, Grundlehren der Mathematischen Wisenschaften, vol. 31. Springer, Berlin (1999)Google Scholar
  2. 2.
    Adams, D.R., Pierre, M.: Capacitary strong type estimates in semilinear problems. Ann. Inst. Fourier (Grenoble) 41, 117–135 (1991)MathSciNetCrossRefGoogle Scholar
  3. 3.
    Baras, P., Pierre, M.: Critère d’existence des solutions positives pour des équations semi-linéaires non monotones. Ann. Inst. H. Poincaré, Anal. Non Lin. 3, 185–212 (1985)MathSciNetCrossRefGoogle Scholar
  4. 4.
    Bidaut-Véron, M. F.: Local and global behavior of solutions of quasilinear equations of Emden-Fowler type. Arch. Ration. Mech. Anal. 107, 293–324 (1989)MathSciNetCrossRefGoogle Scholar
  5. 5.
    Bidaut-Véron, M. F.: Necessary conditions of existence for an elliptic equation with source term and measure data involving p-Laplacian. In: Proceedings of 2001 Luminy Conf. on Quasilinear Elliptic and Parabolic Equations and Systems, Electron. Journal of Difference Equations in Conference, Vol. 8, pp. 23–34 (2002)Google Scholar
  6. 6.
    Bidaut-Véron, M. F.: Removable singularities and existence for a quasilinear equation with absorption or source term and measure data. Adv. Nonlinear Stud. 3, 25–63 (2003)MathSciNetCrossRefGoogle Scholar
  7. 7.
    Bidaut-Véron, M. F., Nguyen, Q.-H., Véron, L.: Quasilinear Lane-Emden equations with absorption and measure data. J. Math. Pures Appl. 102, 315337 (2014)MathSciNetCrossRefGoogle Scholar
  8. 8.
    Bidaut-Véron, M.F., Nguyen, Q.-H., Véron, L.: Quasilinear elliptic equations with source mixed term and measure data, preprintGoogle Scholar
  9. 9.
    Bidaut-Véron, M. F., Pohozaev, S.: Nonexistence results and estimates for some nonlin- ear elliptic problems. J. Anal. Math. 84, 1–49 (2001)MathSciNetCrossRefGoogle Scholar
  10. 10.
    Bidaut-Véron, M. F., Yarur, C.: Semilinear elliptic equations and systems with measure data: existence and a priori estimates. Adv. Diff. Equ. 7, 257–296 (2002)MathSciNetzbMATHGoogle Scholar
  11. 11.
    Birindelli, I., Demengel, F.: Some Liouville theorems for the p-Laplacian. In: Proceedings of 2001 Luminy Conference on Quasilinear Elliptic and Parabolic Equations and Systems, Electronic Journal of Difference Equations in Conference, vol. 8, pp. 35–46 (2002)Google Scholar
  12. 12.
    Dal Maso, G., Murat, F., Orsina, L., Prignet, A.: Renormalized solutions of elliptic equations with general measure data. Ann. Sc. Norm. Sup. Pisa 28, 741–808 (1999)MathSciNetzbMATHGoogle Scholar
  13. 13.
    Heinonen, J., Kilpelainen, T., Martio, O.: Nonlinear potential theory of degenerate elliptic equations. Unabridged republication of the 1993 original. Dover Publications, Inc., Mineola (2006)Google Scholar
  14. 14.
    Kilpelainen, T., Malý, J.: Degenerate elliptic equation with measure data and nonlinear potentials. Ann. Sc. Norm. Super. Pisa, Cl. Sci. 19, 591–613 (1992)MathSciNetzbMATHGoogle Scholar
  15. 15.
    Kilpelainen, T., Malý, J.: The Wiener test and potential estimates for quasilinear elliptic equations. Acta Math. 172, 137–161 (1994)MathSciNetCrossRefGoogle Scholar
  16. 16.
    Labutin, D.: Potential estimates for a class of fully nonlinear elliptic equations. Duke Math. J. 111, 1–49 (2002)MathSciNetCrossRefGoogle Scholar
  17. 17.
    Marcus, M., Véron, L.: Nonlinear Second Order Elliptic Equations Involving Measures. De Gruyter Series in Nonlinear Analysis and Applications, vol. 21. De Gruyter, Berlin (2014)Google Scholar
  18. 18.
    Phuc, N.C., Verbitsky, I.E.: Quasilinear and Hessian equations of Lane-Emden type. Ann. Math. 168, 859–914 (2008)MathSciNetCrossRefGoogle Scholar
  19. 19.
    Phuc, N.C., Verbitsky, I.E.: Singular quasilinear and Hessian equation and inequalities. J. Funct. Anal. 256, 1875–1906 (2009)MathSciNetCrossRefGoogle Scholar
  20. 20.
    Serrin, J., Zou, H.: Cauchy-liouville and universal boundedness theorems for quasilinear elliptic equations and inequalities. Acta Math. 189, 79–142 (2002)MathSciNetCrossRefGoogle Scholar
  21. 21.
    Trudinger, N.S., Wang, X.J.: Hessian measures. Topol. Methods Nonlinear Anal. 10, 225–239 (1997)MathSciNetCrossRefGoogle Scholar
  22. 22.
    Trudinger, N.S., Wang, X.J.: Hessian measures II. Ann. Math. 150, 579–604 (1999)MathSciNetCrossRefGoogle Scholar
  23. 23.
    Trudinger, N.S., Wang, X.J.: Hessian measures III. J. Funct. Anal. 193, 1–23 (2002)MathSciNetCrossRefGoogle Scholar
  24. 24.
    Trudinger, N.S., Wang, X.J.: On the weak continuity of elliptic operators and applications to potential theory. Amer. J. Math. 124, 369–410 (2002)MathSciNetCrossRefGoogle Scholar
  25. 25.
    Véron, L.: Elliptic equations involving measures. In: Stationary Partial Differential Equations, Handbook of Equations, vol. I, pp. 593–712. Elsevier B.V. (2004)Google Scholar
  26. 26.
    Véron, L.: Local and global aspects of quasilinear degenerate elliptic equations. Quasilinear elliptic singular problems, p xv+ 457. World Scientific Publishing Co. Pte. Ltd., Hackensack (2017)CrossRefGoogle Scholar

Copyright information

© Springer Nature B.V. 2018

Authors and Affiliations

  • Marie-Françoise Bidaut-Véron
    • 1
  • Quoc-Hung Nguyen
    • 2
  • Laurent Véron
    • 2
    Email author
  1. 1.Scuola Normale SuperioreCentro Ennio de GiorgiPisaItaly
  2. 2.Laboratoire de Mathématiques et Physique ThéoriqueUniversité François RabelaisToursFrance

Personalised recommendations