Multiplicity of bounded solutions to the $k$-Hessian equation with a Matukuma-type source

TL;DR

This paper investigates the existence and multiplicity of bounded solutions to a $k$-Hessian equation with a Matukuma-type source, using dynamical systems and sub/supersolution methods to analyze the problem.

Contribution

It introduces a transformation to reduce the $k$-Hessian problem to a generalized Lotka-Volterra system, enabling new results on solution multiplicity.

Findings

Multiple solutions exist depending on parameters.

Transformation simplifies the analysis of the nonlinear PDE.

Dynamical systems methods reveal solution structure.

Abstract

The aim of this paper is to deal with the $k$-Hessian counterpart of the Laplace equation involving a nonlinearity studied by Matukuma. Namely, our model is the problem \begin{equation*} (1)\;\;\;\begin{cases} S_k(D^2u)= \lambda \frac{|x|^{\mu-2}}{(1+|x|^2)^{\frac{\mu}{2}}} (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 $\mathbb{R}^n,n>2k$ ($k\in\mathbb{N}$), $\lambda>0$ is an additional parameter, $q>k$ and $\mu\geq 2$. In this setting, through a transformation recently introduced by two of the authors that reduces problem (1) to a non-autonomous two-dimensional generalized Lotka-Volterra system, we prove the existence and multiplicity of solutions for the above problem combining dynamical-systems tools, the intersection number between a regular and a singular solution and the super and subsolution method.