Existence and bifurcation for some elliptic problems on exterior strip domains

We consider the semilinear elliptic problem − Δ u + u = λ K ( x ) u p + f ( x ) in Ω , u > 0 in Ω , u ∈ H 0 1 ( 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 ∗ .