On the uniqueness of sign changing bound state solutions of a semilinear equation

We establish the uniqueness of the higher radial bound state solutions of $$\mathrm{\Delta }u+f(u)=0,x\in {\text{RR}}^n\mathrm{.}\text{leqno}(P)$$ We assume that the nonlinearity $f\in C(-\mathrm{\infty },\mathrm{\infty })$ 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 $[b,\mathrm{\infty })$, or is positive and differentiable in $[0,\mathrm{\infty })$.