We will show that under suitable conditions on f and h, there exists a positive number λ∗ such that the nonhomogeneous elliptic equation −Δu + u = λ( f (x,u) + h(x)) in Ω, u ∈ H 0 (Ω), N ≥ 2, has at least two positive solutions if λ ∈ (0,λ∗), a unique positive solution if λ = λ∗ , and no positive solution if λ > λ∗ , where Ω is the entire space or an exterior domain or an unbounded cylinder dom...