Multiplicity results for bound state solutions of a semilinear equation

We consider the problem of multiplicity and uniqueness of radial solutions of a nonlinear elliptic equation of the form \begin{eqnarray*} \begin{gathered} \Delta u +f(u)=0,\quad x\in \mathbb{R}^N, N\geq 2, \lim\limits_{|x|\to\infty}u(x)=0. \end{gathered} \end{eqnarray*} where $f$ is a prescribed function, satisfying appropriate conditions. This paper is a review of recent developments on the multiplicity of solutions, with a special emphasis on the effect of the mid-range behavior of the nonlinearity $f$ over the existence and multiplicity of solutions.