Exactly two positive solutions of nonhomogeneous semilinear elliptic equations in the half-space

In this article, we shall show that under suitable conditions on ff and hh, there exists a positive number λ∗λ∗ such that the nonhomogeneous semilinear elliptic equation −Δu+σ2u=λ(f(u)+h(x))in R+N,u∈H01(R+N),N≥3, has exactly two positive solutions if λ∈(0,λ∗)λ∈(0,λ∗), a unique positive solution if λ=λ∗λ=λ∗ and no positive solution if λ>λ∗λ>λ∗. We also obtain some properties of the set of solutions.