Applied Mathematics and Optimization

, Volume 12, Issue 1, pp 191–202 | Cite as

A Strong Maximum Principle for some quasilinear elliptic equations

  • J. L. Vázquez
Article

Abstract

In its simplest form the Strong Maximum Principle says that a nonnegative superharmonic continuous function in a domain Ω ⊂ ℝn,n ⩾ 1, is in fact positive everywhere. Here we prove that the same conclusion is true for the weak solutions of − Δu + β(u) = f withβ a nondecreasing function ℝ → ℝ,β(0)=0, andf⩾0 a.e. in Ω if and only if the integral∫(β(s)s)−1/2ds diverges ats=0+. We extend the result to more general equations, in particular to − Δpu + β(u) =f where Δp(u) = div(|Du|p-2Du), 1 <p < ∞. Our main result characterizes the nonexistence of a dead core in some reaction-diffusion systems.

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References

  1. 1.
    Aris R (1975) The Mathematical Theory of Diffusion and Reaction in Permeable Catalysts. Clarendon Press, OxfordGoogle Scholar
  2. 2.
    Bandle C, Sperb RP, Stakgold I (in press) Diffusion-reaction with monotone kynetics. J Nonlinear AnalysisGoogle Scholar
  3. 3.
    Bénilan P (1978) Opérateurs accrétifs et semigroupes dansL p (1⩽p⩽∞). In: Fujita H (ed) Japan-France Seminar 1976. Japan Society for the Promotion of Science: TokyoGoogle Scholar
  4. 4.
    Bénilan P, Brézis H, Crandall MG (1975) A semilinear equation inL 1(ℝn). Ann Scuola Norm Sup Pisa 4:523–555Google Scholar
  5. 5.
    Bertsch M, Kersner R, Peletier LA (in press) Positivity versus localization in degenerate diffusion equationsGoogle Scholar
  6. 6.
    Brézis H, Véron L (1980) Removable singularities of some nonlinear elliptic equations. Arch Rat Mech Anal75 1–6Google Scholar
  7. 7.
    Díaz JI, Hernández J (to appear) On the existence of a free boundary for a class of reactiondiffusion systems. Madison Res Center TS Report 2330. SIAM J Math AnalGoogle Scholar
  8. 8.
    Díaz JI, Herrero MA (1981) Estimates on the support of the solutions of some nonlinear elliptic and parabolic problems. Proc Royal Soc Ed 89A:249–258Google Scholar
  9. 9.
    di Benedetto E (1983)C 1+α local regularity of weak solutions of degenerate elliptic equations, Nonlinear Analysis 7:827–850Google Scholar
  10. 10.
    Friedman A, Phillips D (in press) The free boundary of a semilinear elliptic equation. Tr Amer Math SocGoogle Scholar
  11. 11.
    Gilbarg D, Trudinger NS (1977) Elliptic Differential Equations of Second Order. Springer Verlag, BerlinGoogle Scholar
  12. 12.
    Hopf E (1927) Elementare Bemerkungen über die Lösungen partieller Differentialgleichungen zweiter Ordnung vom elliptischen Typus. Sitz Ber Preuss Akad Wissensch, Berlin. Math Phys kl 19Google Scholar
  13. 13.
    Kato T (1972) Schrödinger operators with singular potentials. Israel J Math 13:135–148Google Scholar
  14. 14.
    Tolksdorf P (1984) Regularity for a more general class of quasilinear elliptic equations. J Diff Equations 51:126–150Google Scholar
  15. 15.
    Vázquez JL, Véron L (in press) Isolated singularities of some semilinear elliptic equations. J Diff EqGoogle Scholar
  16. 16.
    Vázquez JL, Véron L (in press) Singularities of elliptic equations with an exponential nonlinearity. Mathematische AnnalenGoogle Scholar

Copyright information

© Springer-Verlag New York Inc. 1984

Authors and Affiliations

  • J. L. Vázquez
    • 1
  1. 1.División de MatemáticasUniversidad AutónomaMadrid-34Spain

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