Eigenvalue problems of nonhomogeneous semilinear elliptic equations in Esteban-Lions domains with holes

In this article, we consider the following eigenvalue problems(*)_λ- Δ u + u = λ (f (u) + h (x)) in Ω, u > 0 in Ω, u ∈ H₀¹ (Ω), where λ > 0, N ≥ 2 and Ω ⊂ R^N is the upper semi-strip domain with a hole in R^N. Under some suitable conditions on f and h, we show that there exists a positive constant λ^* such that Eq. (*)_λ has at least two solutions if λ ∈ (0, λ^*), a unique positive solution if λ = λ^*, and no positive solution if λ > λ^*. We also obtain some further properties of the positive solutions of (*)_λ.