doi:10.1155/2007/14731 Research Article 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 ∈H10 (Ω), where λ> 0, N =m+ n ≥ 2, n ≥ 1, 0 ∈ ω ⊆ Rm is a smooth bounded domain, S = ω×Rn, D is a smooth bounded domain in RN 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 ∗. Copyright © 2007 Tsing-San Hsu. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. 1