Electronic Journal of Qualitative Theory of Differential Equations (2020-04-01)

Existence of infinitely many radial nodal solutions for a Dirichlet problem involving mean curvature operator in Minkowski space

  • Man Xu,
  • Ruyun Ma

DOI
https://doi.org/10.14232/ejqtde.2020.1.27
Journal volume & issue
Vol. 2020, no. 27
pp. 1 – 14

Abstract

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In this paper, we show the existence of infinitely many radial nodal solutions for the following Dirichlet problem involving mean curvature operator in Minkowski space \begin{equation*} \begin{cases} -\text{div}\left(\frac{\nabla y}{\sqrt{1-|\nabla y|^2}}\right)=\lambda h(y)+g(|x|,y)\quad\text{in}~B,\\ y=0\quad\text{on}~\partial B, \end{cases} \end{equation*} where $B=\{x\in \mathbb{R}^N: |x|<1\}$ is the unit ball in $\mathbb{R}^N$, $N\geq1$, $\lambda\geq0$ is a parameter, $h\in C(\mathbb{R})$ and $g\in C(\mathbb{R}^+\times\mathbb{R})$. By bifurcation and topological methods, we prove the problem possesses infinitely many component of radial solutions branching off at $\lambda=0$ from the trivial solution, each component being characterized by nodal properties.

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