Existence and bifurcation for some elliptic problems on exteriorstrip domains

We consider the semilinear elliptic problem - Δu + u = λK(x)u^p + 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 ℝ^N. 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*.