Skip to main content
SpringerLink
Log in

Existence and uniqueness results for large solutions of general nonlinear elliptic equations

  • Chapter
Nonlinear Evolution Equations and Related Topics

Abstract

We study under what condition there exists a solution of —Δu +f (u) =0 in a domain Ω which blows-up on the boundary, independently of the regularity of the boundary, and we provide criteria for uniqueness. We apply our results to the case f(u) = e au.

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

Access this chapter

Log in via an institution
Chapter
USD 29.95
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
eBook
USD 109.00
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
Softcover Book
USD 139.99
Price excludes VAT (USA)
  • Compact, lightweight edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info

Tax calculation will be finalised at checkout

Purchases are for personal use only

Institutional subscriptions

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. Bandle, C. and Marcus, M.Large solutions of semilinear elliptic equations:existence, uniqueness and asymptotic behavior, J. Anal. Math.58(1992), 9–24.

    Article  MathSciNet  MATH  Google Scholar 

  2. Bandle, C. and Marcus, M.Asymptotic behavior of solutions and their derivative for semilinear elliptic problems with blow-up on the boundaryAnn. I.H.P., Analyse Non Linéaire12(1995), 155–171.

    Google Scholar 

  3. BIEBERBACH, L.Du=e“ und die automorphen funktionenMath. Annalen 77 (1916), 173–212.

    Article  MathSciNet  Google Scholar 

  4. Brezis, H.Some variational problems of the Thomas-Fermi type in variational Inequalitieseds. R. W. Cottle, F. Giannessi and J. L. Lions, Wiley, Chichester 1980,53–73.

    Google Scholar 

  5. Brezis, H. and VERGN, L.Removable singularities for some nonlinear elliptic equationsArch. Rat. Mech. Anal. 75 (1980), 1–6.

    MathSciNet  MATH  Google Scholar 

  6. Chasseigne, E. and Vazquez, J. L.Theory of extended solutions for fast diffusion equations. Radiation from singularitiesArch. Rat. Mech. Anal.164(2002), 133–187.

    MathSciNet  MATH  Google Scholar 

  7. Dhersin, J. L. and LE GALLJ.F.Wiener’s test for super-Brownian motion and the Brownian snakeProbab. Theory Rel Fields108(1997), 103–129

    Article  MATH  Google Scholar 

  8. FAbbri, J. Personal communication 2002.

    Google Scholar 

  9. Abbri,J.andLicgis,J.R.BoundarybehaviourofsolutionsofsomeweaklysuperlinearellipticequationsAdvanced Nonlinear Studies 2 (2002), 147–176.

    MathSciNet  Google Scholar 

  10. Frank, P. and VON Mises, RDie Differential undlntegralgleichungen derMechanik und Physik 1Second Edit., Rosenberg, New-York 1943.

    Google Scholar 

  11. Gilbarg, D. and Trudinger, N. S.Partial Differential Equations of Second Order2nd Ed. Springer-Verlag, Berlin/New-York 1983.

    Book  MATH  Google Scholar 

  12. Grillot, M. and Veron, L.Boundary trace of solutions of the Prescribed Gaussian curvature equationProc. Roy. Soc. Edinburgh130A (2000), 1–34.

    Article  MathSciNet  Google Scholar 

  13. ISCOE, I.On the support of measure-valued critical branching Brownian motionAnn. Prob.16(1988), 200–221.

    Article  MathSciNet  MATH  Google Scholar 

  14. Keller, J. B.On solutions of Au = f (u)Comm. Pure Appl. Math.10(1957), 503–510.

    Article  MathSciNet  MATH  Google Scholar 

  15. LABUTIN, D.Wiener regularity for large solutions of Nonlinear equationspreprint.

    Google Scholar 

  16. Loewner, C. and Nirenberg, L.Partial differential equations invariant under conformal or projective transformationsinContibutions to AnalysisL. Ahlfors and al., eds. 1972, 245–272.

    Google Scholar 

  17. Marcus, M.and Veron, L.Uniqueness and asymptotic behaviour of solutions with boundary blow-up for a class of nonlinear elliptic equationsAnn. Inst. H. Poincaré14(1997), 237–274.

    MathSciNet  MATH  Google Scholar 

  18. Marcus, M. and Veron, L.The boundary trace of positive solutions of semilinear elliptlic equations: the subcritical caseArch. Rat. Mech. Anal.144(1998), 201–231.

    MathSciNet  MATH  Google Scholar 

  19. Nt, W. M.On the elliptic equation Au + K (x)u (n + 2) i ( “ -2)Indiana Univ. Math.J 31(1982), 493–539.

    Google Scholar 

  20. Osserman, R.On the inequality Au > f (u)Pacific J. Math. 7 (1957), 1641–1647.

    MathSciNet  MATH  Google Scholar 

  21. Rademacher, H.Einige besondere Probleme partieller Differentialgleichungensee [10, pp. 838–845].

    Google Scholar 

  22. Ratto, A., Rigoli, M. and Veron, L.Scalar curvature and conformal deformation of hyperbolic spaceJ. Funct. Anal.121(1994), 543–572.

    Article  MathSciNet  Google Scholar 

  23. Richard, Y. and Veron, L.Isotropic singularities of solutions of some nonlinear elliptic inequalitiesAnn. lnst. H. Poincaré6(1989), 37–72.

    MathSciNet  MATH  Google Scholar 

  24. Vazquez, I. L.On a semilinear equation in R 2 involving bounded measuresProc. Roy. Soc. Edinburgh95A(1983), 181–202.

    Article  Google Scholar 

  25. Veron, L.Singularities of Solutions of Second Order Quasilinear EquationsPitman Research Notes in Math.353(1996), Addison Wesley Longman Inc.

    Google Scholar 

  26. Veron, L.Semilinear elliptic equations with uniform blow-up on the boundary J.Analyse Math.59(1992), 231–250.

    Article  MathSciNet  MATH  Google Scholar 

  27. Veron, L.Weak and strong singularities of nonlinear elliptic equationsProc. Symp. Pure Math.45–2(1986), 477–495.

    Google Scholar 

  28. Veron, L.Generalized boundary value problems for nonlinear elliptic equationsElectr. J. Diff. Equ. Conf. 6 (2000), 313–342.

    MathSciNet  Google Scholar 

Download references

Author information

Authors and Affiliations

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

    Moshe Marcus

  2. Laboratoire de Mathématiques et Physique Théorique CNRS UMR 6083, Université François Rabelais, Tours, 37200, 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

Editor information

Editors and Affiliations

  1. Department of Applied Analysis, University of Ulm, 89069, Ulm, Germany

    Wolfgang Arendt

  2. Analyse Numérique, Université Pierre et Marie Curie, B.C. 187, 4, place Jussieu, 75252, Paris Cedex 05, France

    Haïm Brézis

  3. Department of Mathematics Hill Center, Busch Campus, Rutgers University, 110 Frelinghuysen RD, Piscataway, NJ, 08854, USA

    Haïm Brézis

  4. Campus de Ker Lann, Antenne de Bretagne de l’ENS Cachan, 35170, Bruz, France

    Michel Pierre

Additional information

Dedicated to the memory of Philippe Bénilan

Rights and permissions

Reprints and permissions

Copyright information

© 2004 Springer Basel AG

About this chapter

Cite this chapter

Marcus, M., Véron, L. (2004). Existence and uniqueness results for large solutions of general nonlinear elliptic equations. In: Arendt, W., Brézis, H., Pierre, M. (eds) Nonlinear Evolution Equations and Related Topics. Birkhäuser, Basel. https://doi.org/10.1007/978-3-0348-7924-8_33

Download citation

  • DOI: https://doi.org/10.1007/978-3-0348-7924-8_33

  • Publisher Name: Birkhäuser, Basel

  • Print ISBN: 978-3-7643-7107-4

  • Online ISBN: 978-3-0348-7924-8

  • eBook Packages: Springer Book Archive

Publish with us

Policies and ethics

Search

Navigation