A Strong Maximum Principle for some quasilinear elliptic equations

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 with β a nondecreasing function ℝ → ℝ, β(0)=0, and f≥0 a.e. in Ω if and only if the integral ∫(β(s)s)-1/2ds diverges at s=0+. We extend the result to more general equations, in particular to - Δpu + β(u) =f where Δp(u) = div(|Du|p-2Du), 1 < ∞. Our main result characterizes the nonexistence of a dead core in some reaction-diffusion systems. © 1984 Springer-Verlag New York Inc.