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

TL;DR

This paper investigates the existence, multiplicity, and classification of negative solutions to a $k$-Hessian equation with general weights, using phase-plane analysis to understand solution behaviors and characterize new classes of solutions.

Contribution

It introduces a comprehensive phase-plane analysis for $k$-Hessian equations with general weights, providing new classifications and detailed solution existence results.

Findings

Existence and nonexistence results for solutions depending on parameters.

Identification of multiple solution branches including fast decay solutions.

Characterization of new classes of solutions such as $P_3^+$ and $P_4^+$.

Abstract

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 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)= \rho(|x|) (-w)^q, \\ w<0, \end{cases} \end{equation*} given on the entire space $\mathbb R^n\!$. In particular, we describe new classes of solutions: fast decay $P^+_3$-solutions and $P_4^+$-solutions.