Abstract and Applied Analysis
Volume 2007 (2007), Article ID 43018, 19 pages
doi:10.1155/2007/43018

Multiple positive solutions of nonhomogeneous elliptic equations in unbounded domains

Abstract

We will show that under suitable conditions on f and h, there exists a positive number λ∗ such that the nonhomogeneous elliptic equation −Δu+u=λ(f(x,u)+h(x)) in Ω, u∈H01(Ω), N≥2, has at least two positive solutions if λ∈(0,λ∗), a unique positive solution if λ=λ∗, and no positive solution if λ>λ∗, where Ω is the entire space or an exterior domain or an unbounded cylinder domain or the complement in a strip domain of a bounded domain. We also obtain some properties of the set of solutions.