Elliptic equations with absorption in a half-space

Article
  • 75 Downloads

Abstract

We give a necessary and sufficient condition, in the spirit of the classical works by Keller and Osserman, for the elliptic equation Δu = f (u) to have a solution in a half-space of RN. The function f is supposed to be nondecreasing and nonnegative, and we are interested in solutions whose range is where f > 0. The possibility of obtaining such a necessary and sufficient condition has been an open question for a long time.

Keywords

Elliptic equations non-existence half-space coercive problems subsolutions 

Mathematical subject classification

35J60 35J61 

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. [1]
    S.N. Armstrong and B. Sirakov. Nonexistence of positive supersolutions of elliptic equations via the maximum principle. Comm. Partial Differential Equations, 36 (2011), 2011–2047.MathSciNetCrossRefMATHGoogle Scholar
  2. [2]
    C. Bandle and M. Essen. On positive solutions of Emden equations in cone-like domains. Arch. Rational Mech. Anal., 112(4) (1990), 319–338.MathSciNetCrossRefMATHGoogle Scholar
  3. [3]
    H. Berestycki, I. Capuzzo-Dolcetta and L. Nirenberg. Superlinear indefinite elliptic problems and nonlinear Liouville theorems. Topol. Methods Nonlinear Anal., 4(1) (1994), 59–78.MathSciNetCrossRefMATHGoogle Scholar
  4. [4]
    H. Berestycki, L. Caffarelli and L. Nirenberg. Further qualitative properties for elliptic equations in unbounded domains. Ann. ScuolaNorm. Sup. Pisa, 25 (1997), 69–94.MathSciNetMATHGoogle Scholar
  5. [5]
    Z. Chen, C.-S. Lin and W. Zou. Monotonicity and nonexistence results to cooperative systems in the half space. J. Funct. Anal., 266 (2014), 1088–1105.MathSciNetCrossRefMATHGoogle Scholar
  6. [6]
    M.G. Crandall, H. Ishii and P.-L. Lions. User’s guide to viscosity solutions of second-order partial differential equations. Bull. Amer. Math. Soc., 27(1) (1992), 1–67.MathSciNetCrossRefMATHGoogle Scholar
  7. [7]
    N. Dancer. Some notes on the method of moving planes. Bull. Austral.Math. Soc., 46 (1992), 425–434.MathSciNetCrossRefMATHGoogle Scholar
  8. [8]
    Y. Du. Order structure and topological methods in nonlinear partial differential equations. Series in PDE and Applications, World Scientific (2006).CrossRefMATHGoogle Scholar
  9. [9]
    A. Farina. Liouville-type theorems for elliptic problems, in “Handbook of Differential Equations: Stationary Partial Differential Equations”, Vol. 4 (Michel Chipot, Editor) (2007), 483–591.MATHGoogle Scholar
  10. [10]
    P. Felmer, M. Montenegro and A. Quaas. A note on the strong maximum principle and the compact support principle. J. Differential Equations, 246 (2009), 39–49.MathSciNetCrossRefMATHGoogle Scholar
  11. [11]
    D. Gilbarg and N. Trudinger. Elliptic partial differential equations of second order. Springer-Verlag, 2nd edition, Berlin-Heidelberg (2001).MATHGoogle Scholar
  12. [12]
    J.B. Keller. On solutions of Δu = f (u). Comm. Pure Appl. Math., 10 (1957), 503–510.MathSciNetCrossRefMATHGoogle Scholar
  13. [13]
    V. Kondratiev, V. Liskevich and V. Moroz. Positive solutions to superlinear second-order divergence type elliptic equations in cone-like domains. Ann. Inst. H. Poincare Anal. Non Lineaire, 22(1) (2005), 25–43.MathSciNetCrossRefMATHGoogle Scholar
  14. [14]
    M. Marcus and L. Véron. Existence and uniqueness results for large solutions of general nonlinear elliptic equations. J. Evol. Equ., 3 (2004), 637–652.MathSciNetCrossRefMATHGoogle Scholar
  15. [15]
    R. Osserman. On the inequality Δu ≥ f (u). Pac. J. Math., 7 (1957), 1641–1647.CrossRefMATHGoogle Scholar
  16. [16]
    P. Pucci and J. Serrin. The maximum principle. Progress in Nonlinear Differential Equations and their Applications, 73. Birkhauser Verlag, Basel (2007).Google Scholar
  17. [17]
    V. Rădulescu. Singular phenomena in nonlinear elliptic problems: from boundary blow-up solutions to equations with singular nonlinearities, in “Handbook of Differential Equations: Stationary Partial Differential Equations”, Vol. 4 (Michel Chipot, Editor) (2007), 483–591.Google Scholar
  18. [18]
    J.L. Vázquez. A strongmaximum principle for some quasilinear elliptic equations. Appl.Math. Optim., 12(3) (1984), 191–202.MathSciNetCrossRefMATHGoogle Scholar

Copyright information

© Sociedade Brasileira de Matemática 2016

Authors and Affiliations

  1. 1.Departamento de AnálisisMatemáticoUniversidad de La LagunaLa LagunaSpain
  2. 2.Instituto Universitario de Estudios Avanzados (IUdEA) en Física Atómica, Molecular y FotónicaUniversidad de La LagunaLa LagunaSpain
  3. 3.Departamento de MatemáticaUniversidad Técnica Federico Santa María Casilla V-110, Avda. Espa na, 1680ValparaísoChile
  4. 4.Departamento de Matemática PUC-RioRio de JaneiroBrazil

Personalised recommendations