Multiplicity results for sign changing bound state solutions of a semilinear equation

In this paper we give conditions on $f$ so that problem $$\mathrm{\Delta }u+f(u)=0,x\in {\mathbb{R}}^N,N\ge 2,$$ has at least two radial bound state solutions with any prescribed number of zeros, and such that $u(0)$ belongs to a specific subinterval of $(0,\mathrm{\infty })$. This property will allow us to give conditions on $f$ so that this problem has at least any given number of radial solutions having a prescribed number of zeros.