Skip to main content
SpringerLink
Log in

Quasilinear and Hessian Type Equations with Exponential Reaction and Measure Data

  • Published:
Archive for Rational Mechanics and Analysis Aims and scope Submit manuscript

Abstract

We prove existence results concerning equations of the type \({-\Delta_pu=P(u)+\mu}\) for p > 1 and F k [−u] = P(u) + μ with \({1 \leqq k < \frac{N}{2}}\) in a bounded domain Ω or the whole \({\mathbb{R}^N}\), where μ is a positive Radon measure and \({P(u)\sim e^{au^\beta}}\) with a > 0 and \({\beta \geqq 1}\). Sufficient conditions for existence are expressed in terms of the fractional maximal potential of μ. Two-sided estimates on the solutions are obtained in terms of some precise Wolff potentials of μ. Necessary conditions are obtained in terms of Orlicz capacities. We also establish existence results for a general Wolff potential equation under the form \({u={\bf W}_{\alpha, p}^R[P(u)]+f}\) in \({\mathbb{R}^N}\), where \({0 < R \leqq \infty}\) and f is a positive integrable function.

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. Adams, D.R., Hedberg, L.I.: Function Spaces and Potential Theory. Grundlehren der Mathematischen Wisenschaften, vol. 31. Springer, New York, (1999)

  2. Adams D.R., Pierre M.: Capacitary strong type estimates in semilinear problems. Ann. Inst. Fourier (Grenoble), 41, 117–135 (1991)

    Article  MATH  MathSciNet  Google Scholar 

  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)

    MathSciNet  Google Scholar 

  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)

    Article  MATH  Google Scholar 

  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 Conference on Quasilinear Elliptic and Parabolic Equations and Systems. Electron. J. Differ. Equ. Conf. 8, 23–34 (2002)

  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)

    MATH  MathSciNet  Google Scholar 

  7. Bidaut-Véron M.F., Pohozaev S.: Nonexistence results and estimates for some nonlinear elliptic problems. J. Anal. Math. 84, 1–49 (2001)

    Article  MATH  MathSciNet  Google Scholar 

  8. Bidaut-Véron, M.F., Nguyen Quoc, H., Véron, L.: Quasilinear Lane–Emden equations with absorption and measure data.J. des Math. Pures et Appl. (2014). doi:10.1016/j.matpur.2013.11.011

  9. 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. Electron. J. Differ. Equ. Conf. 8, 35–46 (2002)

  10. 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)

    MATH  MathSciNet  Google Scholar 

  11. Fefferman R.: Strong differentiation with respect to measure. Am. J. Math. 103, 33–40 (1981)

    Article  MATH  MathSciNet  Google Scholar 

  12. Ferrari F., Franchi B., Verbitsky I.: Hessian inequality and the fractional Laplacian. J. Reine Angew. Math. 667, 133–148 (2012)

    MATH  MathSciNet  Google Scholar 

  13. Grenon N.: Existence results for semilinear elliptic equations with small measure data. Ann. Inst. H. Poincaré, Anal. Non Linear 19, 1–11 (2002)

    Article  ADS  MATH  MathSciNet  Google Scholar 

  14. Heinonen, J., Kilpelinen, T., Martio, O.: Nonlinear Potential Theory of Degenerate Elliptic Equations. Unabridged republication of the 1993 original. Dover Publications, Inc., Mineola, 2006 (xii+404 pp)

  15. Honzik P., Jaye B.: On the good-γ inequality for nonlinear potentials. Proc. Am. Math. Soc. 140, 4167–4180 (2012)

    Article  MATH  MathSciNet  Google Scholar 

  16. Kilpelainen T., Malý J.: Degenerate elliptic equation with measure data and nonlinear potentials. Ann. Sc. Norm. Super. Pisa, Cl. Sci. 19, 591–613 (1992)

    Google Scholar 

  17. Kilpelainen T., Malý J.: The Wiener test and potential estimates for quasilinear elliptic equations. Acta Math. 172, 137–161 (1994)

    Article  MathSciNet  Google Scholar 

  18. Labutin D.: Potential estimates for a class of fully nonlinear elliptic equations. Duke Math. J. 111, 1–49 (2002)

    Article  MATH  MathSciNet  Google Scholar 

  19. Phuc N.C., Verbitsky I.E.: Quasilinear and Hessian equations of Lane–Emden type. Ann. Math. 168, 859–914 (2008)

    Article  MATH  MathSciNet  Google Scholar 

  20. Phuc N.C., Verbitsky I.E.: Singular quasilinear and Hessian equation and inequalities. J. Funct. Anal. 256, 1875–1906 (2009)

    Article  MATH  MathSciNet  Google Scholar 

  21. Rao, M.M., Ren, Z.D.: Theory of Orlicz Spaces. Textb. Pure Appl. Math. (1991)

  22. Richard Y., Véron L.: Isotropic singularities of solutions of nonlinear elliptic inequalities. Ann. I.H.P. Anal. Non Linéaire 6, 37–72 (1989)

    MATH  Google Scholar 

  23. Serrin J., Zou H.: Cauchy–Liouville and universal boundedness theorems for quasilinear elliptic equations and inequalities. Acta Math. 189, 79–142 (2002)

    Article  MATH  MathSciNet  Google Scholar 

  24. Trudinger N.S., Wang X.J.: Hessian measures. Topol. Methods Nonlinear Anal. 10, 225–239 (1997)

    MATH  MathSciNet  Google Scholar 

  25. Trudinger N.S., Wang X.J.: Hessian measures II. Ann. Math. 150, 579–604 (1999)

    Article  MATH  MathSciNet  Google Scholar 

  26. Trudinger N.S., Wang X.J.: Hessian measures III. J. Funct. Anal. 193, 1–23 (2002)

    Article  MATH  MathSciNet  Google Scholar 

  27. Trudinger N.S., Wang X.J.: On the weak continuity of elliptic operators and applications to potential theory. Am. J. Math. 124, 369–410 (2002)

    Article  MATH  MathSciNet  Google Scholar 

  28. Véron, L.: Elliptic equations involving measures. In: Stationary Partial Differential Equations, Handbook of Equations, vol. I, pp. 593–712. Elsevier B.V., London (2004)

Download references

Author information

Authors and Affiliations

  1. Laboratoire de Mathématiques et Physique Théorique, Université François Rabelais, Tours, France

    Quoc-Hung Nguyen & Laurent Véron

Authors
  1. Quoc-Hung Nguyen
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Laurent Véron
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Laurent Véron.

Additional information

Communicated by G. Dal Maso

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Nguyen, QH., Véron, L. Quasilinear and Hessian Type Equations with Exponential Reaction and Measure Data. Arch Rational Mech Anal 214, 235–267 (2014). https://doi.org/10.1007/s00205-014-0756-7

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s00205-014-0756-7

Keywords

Search

Navigation