On a power-type coupled system with mean curvature operator in Minkowski space

Abstract

Abstract We study the Dirichlet problem for the prescribed mean curvature equation in Minkowski space { M ( u ) + v α = 0 in B , M ( v ) + u β = 0 in B , u | ∂ B = v | ∂ B = 0 , $$ \textstyle\begin{cases} \mathcal{M}(u)+ v^{\alpha }=0\quad \text{in } B, \\ \mathcal{M}(v)+ u^{\beta }=0\quad \text{in } B, \\ u|_{\partial B}=v|_{\partial B}=0, \end{cases} $$ where M ( w ) = div ( ∇ w 1 − | ∇ w | 2 ) $\mathcal{M}(w)=\operatorname{div} ( \frac{\nabla w}{\sqrt{1-|\nabla w|^{2}}} )$ and B is a unit ball in R N ( N ≥ 2 ) $\mathbb{R}^{N} (N\geq 2)$ . We use the index theory of fixed points for completely continuous operators to obtain the existence, nonexistence and uniqueness results of positive radial solutions under some corresponding assumptions on α, β.

Keywords

• Minkowski curvature operator
• System
• Positive radial solution
• Uniqueness