In this paper, we study the existence and nonexistence of multiple positive solutions for the inhomogeneous Neumann boundary value problem Δu+up−λu=0, with Dγu=ϕ(x), (*) under some assumptions on the boundary ∂Ω and the function ϕ(x). For ϕ(x)≥0, ϕ(x) 0, ϕ(x) ∈Cα( ), it is shown that there exists a constant λ*＞0 such that problem (*) possesses at least two positive solutions if λ∈ (λ*,∞) and at least one positive solution if λ=λ*. Furthermore, there are no positive solutions for problem (*) if λ∈ (-∞, λ*).