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

In this article, we consider the following eigenvalue problems − u+ u = λ f (u) + h(x) in Ω, u>0 in Ω, u ∈ H1 0 (Ω), (∗)λ where λ >0, N 2 and Ω ⊂ RN is the upper semi-strip domain with a hole in RN. 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. © 2006 Elsevier Inc. All rights reserved