Uniqueness of sign-changing radial solutions for Δu − u + |u|p − 1u = 0 in some ball and annulus

The following Dirichlet problem{Δu−u+|u|p−1u=0inΩ,u=0on∂Ω, is considered, where Ω is either an annulus or a ball in RNRN and p>1p>1. The uniqueness of radial solutions having exactly k−1k−1 nodes is shown for the following cases: Ω is a sufficiently thin annulus; Ω is a certain small ball, N≥4N≥4 and 11p>1 is sufficiently close to 1 and N=3N=3, 5 or 7.