On a power-type coupled system with mean curvature operator in Minkowski space
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 , $$ \textstyle\begin{cases} \mathcal{M}(u)+ v^{\alpha }=0\quad \text{in } B, \\ \mathcal{M}(v)+ u^{\beta }=0\quad \text{i...
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| Autores principales: | , , |
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| Formato: | article |
| Lenguaje: | EN |
| Publicado: |
SpringerOpen
2021
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| Materias: | |
| Acceso en línea: | https://doaj.org/article/98a5f53631bd4b9e8eaa9e8004cb510e |
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| Sumario: | Abstract We study the Dirichlet problem for the prescribed mean curvature equation in Minkowski space { M ( u ) + v α = 0 in B , M ( v ) + u β = 0 in B , u | ∂ B = v | ∂ B = 0 , $$ \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. |
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