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

In this paper, we prove that if QQ is a positive continuous function in RNRN and satisfies the suitable conditions, then for sufficiently small λ>0λ>0,−Δu+u=Q(z)up−1−Δu+u=Q(z)up−1 with the boundary condition u(x,zN)=λg(x)u(x,zN)=λg(x) admits at least three positive solutions in R+N.