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.