Exactly two positive solutions of nonhomogeneous semilinear elliptic equations in unbounded cylinder domains

In this paper, we consider the nonhomogeneous semilinear elliptic equation -Δu + u = λK(x)u^p + h(x) in Ω, u > 0 in Ω, u ∈ H₀¹(Ω), (*) _λ; where λ ≥ 0, N ≥ 3, 1 < p < N + 2/N - 2, and Ω is an unbounded cylinder domain. Under some suitable conditions on K and h, we show that there exists a positive constant λ* such that (*)_λ has exactly two solutions if λ ∈ (0, λ*) and no solution if λ > λ*. Furthermore, (*)_λ has at least one solution for λ = λ* provided that h(x) ∈ L^2N/N+2 (Ω) ∩ L^∞ (Ω).