Boundary Value Problems
Volume 2007 (2007), Article ID 14731, 25 pages
doi:10.1155/2007/14731

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

Abstract

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⊆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∗.