Uniqueness of positive radial solutions for Δu−u+up=0 on an annulus

We prove uniqueness of positive radial solutions to the semilinear elliptic equation Δu−u+up=0, p>1, subject to the Dirichlet boundary condition on an annulus in . As a by-product, our argument also provides a much simpler, if not the simplest, new proof for the uniqueness of positive solutions to the same problem in a finite ball or in the whole space .