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We consider the problem \begin{equation}(1)\;\;\; \begin{cases} S_k(D^2u)=\lambda |x|^{\sigma} (1-u)^q &\mbox{in }\;\; B,\\ u <0 & \mbox{in }\;\; B,\\u=0 &\mbox{on }\partial B, \end{cases} \end{equation} where $B$ denotes theunit ball in $\mathbb{R}^n$, $n>2k$ ($k\in \mathbb{N}$), $\lambda>0$, $q > k$and $\sigma\geq 0$. We study the existence, uniqueness and multiplicity ofnegative bounded radially symmetric solutions of (1). The methodology to obtainour results is based on a dynamical system approach. For this, we introduce anew transformation which reduces problem (1) to an autonomous two dimensionalgeneralized Lotka-Volterra system.

Radially bounded solutions of a $k$-Hessian equation involving a weighted nonlinear source