Eigenvalue Problems and Bifurcation of Nonhomogeneous Semilinear Elliptic Equations in Exterior Strip Domains

• Main Author: Tsing-San Hsu
• Format: Article
• Language: English
• Published: SpringerOpen 2006-12-01
• Series: Boundary Value Problems
• Online Access: http://dx.doi.org/10.1155/2007/14731

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∈É⊆â„Âm is a smooth bounded domain, ð•Š=É×â„Ân, D is a smooth bounded domain in â„ÂN such that D⊂⊂ð•Š,Ω=ð•ŠD¯. 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∗.