Skip to main content
SpringerLink
Log in

Maximal solutions of semilinear elliptic equations with locally integrable forcing term

  • Published:
Israel Journal of Mathematics Aims and scope Submit manuscript

Abstract

We study the existence of a maximal solution of −Δu+g(u)=f(x) in a domain Ω ∈ ℝN with compact boundary, assuming thatf ∈ (L 1 loc (Ω))+ and thatg is nondecreasing,g(0)≥0 andg satisfies the Keller-Osserman condition. We show that if the boundary satisfies the classicalC 1,2 Wiener criterion, then the maximal solution is a large solution, i.e., it blows up everywhere on the boundary. In addition, we discuss the question of uniqueness of large solutions.

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. C. Bandle and M. Marcus,Asymptotic behavior of solutions and their derivative for semilinear elliptic problems with blow-up on the boundary, Annales de l’Institut Henri Poincaré. Analyse Non Linéaire12 (1995), 155–171.

    MATH  MathSciNet  Google Scholar 

  2. Ph. Benilan and H. Brezis,Nonlinear preoblems related to the Thomas-Fermi equation, Journal of Evolution Equations3 (2003), 673–770.

    Article  MATH  MathSciNet  Google Scholar 

  3. H. Brezis,Semilinear equations in ℝN without condition at infinity, Applied Mathematics and Optimization12 (1985), 271–282.

    MathSciNet  Google Scholar 

  4. H. Brezis and W. Strauss,Semi-linear second-order elliptic equations in L 1, Journal of the Mathematical Society of Japan25 (1973), 565–590.

    MATH  MathSciNet  Google Scholar 

  5. J. B. Keller,On solutions of Δu=f(u), Communications on Pure and Applied Mathematics10 (1957), 503–510.

    Article  MATH  MathSciNet  Google Scholar 

  6. D. Labutin,Wiener regularity for large solutions of nonlinear equations, Arkiv för Matematik41 (2003), 307–339.

    Article  MATH  MathSciNet  Google Scholar 

  7. C. Loewner and L. Nirenberg,Partial differential equations invariant under conformal or projective transformations, inContributions to Analysis (L. V. Ahlfors, I. Kra, B. Maskit and L. Nirenberg, eds.), Academic Press, New York and London, 1974, pp. 245–272.

    Google Scholar 

  8. M. Marcus and L. Véron,Uniqueness and asymptotic behaviour of solutions with boundary blow-up for a class of nonlinear elliptic equations, Annales de l’Institut Henri Poincaré14 (1997), 237–274.

    Article  MATH  Google Scholar 

  9. M. Marcus and L. Véron,The boundary trace of positive solutions of semilinear elliptic equations: the subcritical case, Archive for Rational Mechanics and Analysis144 (1998), 201–231.

    Article  MATH  MathSciNet  Google Scholar 

  10. M. Marcus and L. Véron,Existence and uniqueness results for large solutions of general nonlinear elliptic equations, Journal of Evolution Equations3 (2003), 637–652.

    Article  MATH  MathSciNet  Google Scholar 

  11. M. Marcus and L. Véron,The boundary trace and generalized B.V.P. for semilinear elliptic equations with coercive absorption, Communications on Pure and Applied Mathematics56 (2003), 689–731.

    Article  MATH  MathSciNet  Google Scholar 

  12. R. Osserman,On the inequality Δu ≥f(u), Journal of Mathematics7 (1957), 1641–1647.

    MATH  MathSciNet  Google Scholar 

  13. L. Véron,Generalized boundary value problems for nonlinear elliptic equations, Electronic Journal of Differential Equations Conference6 (2000), 313–342.

    Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Department of Mathematics, Technion - Israel Institute of Technology, 32000, Haifa, Israel

    Moshe Marcus

  2. Laboratoire de Mathématiques, Faculté des Sciences, Parc de Grandmont, 37200, Tours, France

    Laurent Véron

Authors
  1. Moshe Marcus
    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 Moshe Marcus.

Additional information

This research was partially supported by an EC Grant through the RTN Program “Front-Singularities”, HPRN-CT-2002-00274.

Rights and permissions

Reprints and permissions

About this article

Cite this article

Marcus, M., Véron, L. Maximal solutions of semilinear elliptic equations with locally integrable forcing term. Isr. J. Math. 152, 333–348 (2006). https://doi.org/10.1007/BF02771990

Download citation

  • Received:

  • Issue Date:

  • DOI: https://doi.org/10.1007/BF02771990

Keywords

Search

Navigation