Multiplicity results for ground state solutions of a semilinear equation via abrupt changes in magnitude of the nonlinearity

Given $k\in \mathbb{N}$, we define a class of continuous piecewise functions $f$ having abrupt but controlled magnitude changes so that the problem $$\mathrm{\Delta }u+f(u)=0,x\in {\mathbb{R}}^N,N>2,$$ has at least $k$ radially symmetric ground state solutions.