Existence of solutions for gradient coupled Dirichlet systems

Abstract

In this paper, we prove existence of weak solutions in W1,20 (Ω) ∩L∞(Ω) for the gradient coupled Dirichlet system {u ∈ W1,20 (Ω): –div(M(x)∇u) + u + a(x)∇u · ∇ψ = f(x), {ψ ∈ W1,20 (Ω): –div(M(x)∇ψ) + ψ + a(x)∇u · ∇ψ = g(x). We also prove that if f (x), g(x) ≥ 0 (of course ≢ 0 a.e.), then u(x), ψ(x) ≥ 0 and the sets {u = 0} and {ψ = 0} have zero Lebesgue measure.