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A Strong Maximum Principle for some quasilinear elliptic equations

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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) −1/2 ds 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-2 Du), 1 <p < ∞. Our main result characterizes the nonexistence of a dead core in some reaction-diffusion systems.

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This work was partly done while the author was visiting the University of Minnesota as a Fulbright Scholar.

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Vázquez, J.L. A Strong Maximum Principle for some quasilinear elliptic equations. Appl Math Optim 12, 191–202 (1984). https://doi.org/10.1007/BF01449041

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