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(P)Δu+f(u)=0,x∈Rn. We assume that the nonlinearity f∈C(−∞,∞) is an odd function satisfying some convexity and growth conditions, and has one zero at b>0, is nonpositive and not-identically 0 in (0,b), positive in [b,∞), and is differentiable in (0,∞).