EXISTENCE AND BIFURCATION FOR SOME ELLIPTIC PROBLEMS ON EXTERIOR STRIP DOMAINS

We consider the semilinear elliptic problem −Δu + u = λK(x)up + f (x) in Ω, u> 0 in Ω, u ∈ H10 (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 ∗. Copyright © 2006 Hindawi Publishing Corporation. All rights reserved. 1