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

Abstract

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.

Keywords

• infinitely many radial solutions
• mean curvature operator
• minkowski space
• topological method
• bifurcation