Eigenvalue problems and bifurcation of nonhomogeneous semilinear elliptic equations in exterior strip domains

We consider the following eigenvalue problems: - Δu + u = λ(f(u) + h(x)) in Ω, u > 0 in Ω, u ∈ H₀¹ (Ω), where λ > 0, N = m + n ≥ 2, n ≥ 1, 0 ∈ ω ⊆ ℝ^m is a smooth bounded domain, S = ω × ℝⁿ, D is a smooth bounded domain in ℝ^N such that D ⊂ ⊂ S, Ω = S\D. Under some suitable conditions on f and h, we show that there exists a positive constant λ* such that the above-mentioned problems have at least two solutions if λ ∈ (0,λ*), a unique positive solution if λ = λ*, and no solution if λ > λ*. We also obtain some bifurcation results of the solutions at λ = λ*.