  We study sign-changing radial solutions for the following semi-linear
elliptic equation \begin{align*} \Delta u-u+|u|^{p-1}u=0\quad{\rm{in}}\
\mathbb{R}^N,\quad u\in H^1(\mathbb{R}^N), \end{align*} where
$1<p<\frac{N+2}{N-2}$, $N\geq3$. It is well-known that this equation has a
unique positive radial solution and sign-changing radial solutions with exactly
$k$ nodes. In this paper, we show that such sign-changing radial solution is
also unique when $p$ is close to $\frac{N+2}{N-2}$. Moreover, those solutions
are non-degenerate, i.e., the kernel of the linearized operator is exactly
$N$-dimensional.
