Applied Mathematics and Optimization

, Volume 12, Issue 1, pp 191–202

A Strong Maximum Principle for some quasilinear elliptic equations

  • J. L. Vázquez

DOI: 10.1007/BF01449041

Cite this article as:
Vázquez, J.L. Appl Math Optim (1984) 12: 191. doi:10.1007/BF01449041


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.

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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