JournalsrmiVol. 35, No. 5pp. 1559–1582

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

  • Yasuhito Miyamoto

    University of Tokyo, Japan
  • Justino Sánchez

    Universidad de La Serena, Chile
  • Vicente Vergara

    Universidad de Concepción, Chile
Multiplicity of bounded solutions to the $k$-Hessian equation with a Matukuma-type source cover
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Abstract

The aim of this paper is to deal with the kk-Hessian counterpart of the Laplace equation involving a nonlinearity studied by Matukuma. Namely, our model is the problem

(1){Sk(D2u)=λxμ2(1+x2)μ/2(1u)q\mboxinB,u<0\mboxinB,u=0\mboxonB,(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}

where BB denotes the unit ball in Rn\mathbb{R}^n, n>2kn > 2k (kNk\in\mathbb{N}), λ>0\lambda > 0 is an additional parameter, q>kq > k and μ2\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.

Cite this article

Yasuhito Miyamoto, Justino Sánchez, Vicente Vergara, Multiplicity of bounded solutions to the kk-Hessian equation with a Matukuma-type source. Rev. Mat. Iberoam. 35 (2019), no. 5, pp. 1559–1582

DOI 10.4171/RMI/1092