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The aim of this paper is to study negative classical solutions to a$k$-Hessian equation involving a nonlinearity with a general weight\begin{equation} \label{Eq:Ma:0} \tag{$P$} \begin{cases} S_k(D^2u)= \lambda\rho(|x|) (1-u)^q &\mbox{in }\;\; B,\\ u=0 &\mbox{on }\partial B. \end{cases}\end{equation} Here, $B$ denotes the unit ball in $\mathbb R^n\!$, $n>2k$,$\lambda$ is a positive parameter and $q>k$ with $k\in \mathbb N$. The function$r\rho'(r)/\rho(r)$ satisfies very general conditions in the radial direction$r=|x|$. We show the existence, nonexistence, and multiplicity of solutions toProblem \eqref{Eq:Ma:0}. The main technique used for the proofs is aphase-plane analysis related to a non-autonomous dynamical system associated tothe equation in \eqref{Eq:Ma:0}. Further, using the aforementionednon-autonomous system, we give a comprehensive characterization of $P_2$-,$P_3^+$-, $P_4^+$-solutions to the related problem \begin{equation*}\begin{cases} S_k(D^2 w)= \rho(|x|) (-w)^q, \\ w<0, \end{cases} \end{equation*}given on the entire space $\mathbb R^n\!$. In particular, we describe newclasses of solutions: fast decay $P^+_3$-solutions and $P_4^+$-solutions.

Existence, multiplicity and classification results for solutions to $k$-Hessian equations with general weights