Existence of multiple positive solutions of semilinear elliptic boundary value problems in the half space

In this paper, we prove that if QQ and ff satisfy some suitable conditions, then −Δu+u=Q(x)f(u)−Δu+u=Q(x)f(u) in R+N with the boundary condition u(y,0)=λg(y)u(y,0)=λg(y) has at least two positive solutions if 0<λ<λ∗0<λ<λ∗, a unique positive solution if λ=λ∗λ=λ∗ and no positive solution if λ>λ∗λ>λ∗.