Existence of multiple positive solutions for inhomogeneous Neumann problem ✩

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 Dγ u = ϕ(x), (∗) under some assumptions on the boundary ∂Ω and the function ϕ(x). For ϕ(x) 0, ϕ(x) ≡ 0, ϕ(x) ∈ Cα( ¯ Ω), 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 ∗ ).  2002 Elsevier Science (USA). All rights reserved