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

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 the unit ball in ${\mathbb{R}}^n$, $n>2k$ ($k\in \mathbb{N}$), $\lambda >0$, $q>k$ and $\sigma \ge 0$. We study the existence, uniqueness and multiplicity of negative bounded radially symmetric solutions of (1). The methodology to obtain our results is based on a dynamical system approach. For this, we introduce a new transformation which reduces problem (1) to an autonomous two dimensional generalized Lotka-Volterra system.