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

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

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 $\mu \ge 2$. 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.