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 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.