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

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)= λρ(|x|) (1-u)^q &\mbox{in };; B,\ u=0 &\mbox{on }\partial B. \end{cases} \end{equation} Here, $B$ denotes the unit ball in ${\mathbb{R}}^n\text{\hspace{0.17em}⁣}$, $n>2k$, $\lambda$ is a positive parameter and $q>k$ with $k\in \mathbb{N}$. The function $r{\rho }^{\mathrm{\prime }}(r)\mathrm{/}\rho (r)$ satisfies very general conditions in the radial direction $r=\mathrm{\mid }x\mathrm{\mid }$. We show the existence, nonexistence, and multiplicity of solutions to Problem \eqref{Eq:Ma:0}. The main technique used for the proofs is a phase-plane analysis related to a non-autonomous dynamical system associated to the equation in \eqref{Eq:Ma:0}. Further, using the aforementioned non-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)= ρ(|x|) (-w)^q, \ w<0, \end{cases} \end{equation*} given on the entire space ${\mathbb{R}}^n\text{\hspace{0.17em}⁣}$. In particular, we describe new classes of solutions: fast decay $P_3^{+}$-solutions and $P_4^{+}$-solutions.