  We establish the uniqueness of the higher radial bound state solutions of $$
\Delta u +f(u)=0,\quad x\in \RR^n. \leqno(P) $$ We assume that the nonlinearity
$f\in C(-\infty,\infty)$ is an odd function satisfying some convexity and
growth conditions, and either has one zero at $b>0$, is non positive and not
identically 0 in $(0,b)$, and is differentiable and positive $[b,\infty)$, or
is positive and differentiable in $[0,\infty)$.
