Multiple Positive Solutions of Nonhomogeneous Elliptic Equations in Unbounded Domains

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∈H01(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 domain or the complement in a strip domain of a bounded domain. We also obtain some properties of the set of solutions.