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


In its simplest form the Strong Maximum Principle says that a nonnegative superharmonic continuous function in a domain Ω ⊂ ℝ n ,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 − Δ p u + β(u) =f where Δ p (u) = div(|Du|p-2Du), 1 <p < ∞. Our main result characterizes the nonexistence of a dead core in some reaction-diffusion systems.


Continuous Function Weak Solution System Theory Mathematical Method Simple Form 


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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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