International Journal of Mathematics and Mathematical Sciences
Volume 2006 (2006), Issue 15, Article ID 73278, 26 pages
doi:10.1155/IJMMS/2006/73278

Existence and bifurcation for some elliptic problems on exterior strip domains

Abstract

We consider the semilinear elliptic problem −Δu+u=λK(x)up+f(x) in Ω, u>0 in Ω, u∈H01(Ω), where λ≥0, N≥3, 1<p<(N+2)/(N−2), and Ω is an exterior strip domain in ℝN. Under some suitable conditions on K(x) and f(x), we show that there exists a positive constant λ∗ such that the above semilinear elliptic problem has 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∗.