Revista Matemática Iberoamericana


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Volume 35, Issue 5, 2019, pp. 1559–1582
DOI: 10.4171/rmi/1092

Published online: 2019-07-22

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

Yasuhito Miyamoto[1], Justino Sánchez[2] and Vicente Vergara[3]

(1) University of Tokyo, Japan
(2) Universidad de La Serena, Chile
(3) Universidad de Concepción, Chile

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)\quad\begin{cases} S_k(D^2u)= \lambda \,\frac{|x|^{\mu-2}}{(1+|x|^2)^{{\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.

Keywords: $k$-Hessian operator, radial solutions, non-autonomous Lotka–Volterra system, phase analysis, critical exponents, singular solution, intersection number

Miyamoto Yasuhito, Sánchez Justino, Vergara Vicente: Multiplicity of bounded solutions to the $k$-Hessian equation with a Matukuma-type source. Rev. Mat. Iberoam. 35 (2019), 1559-1582. doi: 10.4171/rmi/1092