On the uniqueness of the second bound state solution of a semilinear equation

We establish the uniqueness of the second radial bound state solution ofequation(P)Δu+f(u)=0,x∈Rn. We assume that the nonlinearity f∈C(−∞,∞)f∈C(−∞,∞) is an odd function satisfying some convexity and growth conditions of superlinear type, and either has one zero at b>0b>0, is nonpositive and not identically 0 in (0,b)(0,b), and is differentiable and positive [b,∞)[b,∞), or is positive and differentiable in [0,∞)[0,∞).