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

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