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 − Δ u + u = λ ( f ( x , u ) + h ( x ) ) in Ω , u ∈ H 0 1 ( 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.