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

TL;DR

This paper investigates the existence, uniqueness, and multiplicity of radially symmetric solutions to a weighted k-Hessian PDE in a unit ball, using a dynamical system approach via a novel transformation to a Lotka-Volterra system.

Contribution

It introduces a new transformation that reduces the weighted k-Hessian PDE to an autonomous Lotka-Volterra system, enabling analysis of solutions.

Findings

Existence and multiplicity results for solutions depending on parameters.

Uniqueness of solutions under certain conditions.

Development of a dynamical system framework for the PDE.

Abstract

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 $\sigma\geq 0$. 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.