A remark on the uniqueness of positive solutions for some Dirichlet problems

In this paper we prove the uniqueness of positive weak solutions to the problem-Δpu=g(x,u),x∈Ω,u=0,x∈∂Ω,by a direct argument. Here ΩΩ is a bounded domain in RN(N⩾3)RN(N⩾3) with smooth boundary ∂Ω∂Ω, ΔpΔp denotes the p-Laplacian operator defined by Δpz=div(|∇z|p-2∇z)Δpz=div(|∇z|p-2∇z); p>1p>1, g(x,0)=0g(x,0)=0, for a.e. x∈Ω,x∈Ω, and g(x,u)g(x,u) is a Caratheodory function. We provide a direct uniqueness proof for this problem, under certain conditions.