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 ∈ H 0 (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∗ .