  In this paper we give conditions on $f$ so that problem $$ \Delta u
+f(u)=0,\quad x\in \mathbb{R}^N, N\ge 2, $$ 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 $(0,\infty)$. 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.
