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 , $$ \textstyle\begin{cases} \mathcal{M}(u)+ v^{\alpha }=0\quad \text{in } B, \\ \mathcal{M}(v)+ u^{\beta }=0\quad \text{i...
| Main Authors: | , , |
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| Format: | Article |
| Language: | English |
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SpringerOpen
2021-11-01
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| Series: | Boundary Value Problems |
| Subjects: | |
| Online Access: | https://doi.org/10.1186/s13661-021-01572-z |
| _version_ | 1818928391603093504 |
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| author | Zhiqian He Yanzhong Zhao Liangying Miao |
| author_facet | Zhiqian He Yanzhong Zhao Liangying Miao |
| author_sort | Zhiqian He |
| collection | DOAJ |
| description | Abstract We study the Dirichlet problem for the prescribed mean curvature equation in Minkowski space { M ( u ) + v α = 0 in B , M ( v ) + u β = 0 in B , u | ∂ B = v | ∂ B = 0 , $$ \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} $$ where M ( w ) = div ( ∇ w 1 − | ∇ w | 2 ) $\mathcal{M}(w)=\operatorname{div} ( \frac{\nabla w}{\sqrt{1-|\nabla w|^{2}}} )$ and B is a unit ball in R N ( N ≥ 2 ) $\mathbb{R}^{N} (N\geq 2)$ . We use the index theory of fixed points for completely continuous operators to obtain the existence, nonexistence and uniqueness results of positive radial solutions under some corresponding assumptions on α, β. |
| first_indexed | 2024-12-20T03:28:10Z |
| format | Article |
| id | doaj.art-98a5f53631bd4b9e8eaa9e8004cb510e |
| institution | Directory Open Access Journal |
| issn | 1687-2770 |
| language | English |
| last_indexed | 2024-12-20T03:28:10Z |
| publishDate | 2021-11-01 |
| publisher | SpringerOpen |
| record_format | Article |
| series | Boundary Value Problems |
| spelling | doaj.art-98a5f53631bd4b9e8eaa9e8004cb510e2022-12-21T19:55:03ZengSpringerOpenBoundary Value Problems1687-27702021-11-01202111910.1186/s13661-021-01572-zOn a power-type coupled system with mean curvature operator in Minkowski spaceZhiqian He0Yanzhong Zhao1Liangying Miao2Department of Basic Teaching and Research, Qinghai UniversityDepartment of Basic Teaching and Research, Qinghai UniversitySchool of Mathematics and Statistics, Qinghai Nationalities UniversityAbstract We study the Dirichlet problem for the prescribed mean curvature equation in Minkowski space { M ( u ) + v α = 0 in B , M ( v ) + u β = 0 in B , u | ∂ B = v | ∂ B = 0 , $$ \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.https://doi.org/10.1186/s13661-021-01572-zMinkowski curvature operatorSystemPositive radial solutionUniqueness |
| spellingShingle | Zhiqian He Yanzhong Zhao Liangying Miao On a power-type coupled system with mean curvature operator in Minkowski space Boundary Value Problems Minkowski curvature operator System Positive radial solution Uniqueness |
| title | On a power-type coupled system with mean curvature operator in Minkowski space |
| title_full | On a power-type coupled system with mean curvature operator in Minkowski space |
| title_fullStr | On a power-type coupled system with mean curvature operator in Minkowski space |
| title_full_unstemmed | On a power-type coupled system with mean curvature operator in Minkowski space |
| title_short | On a power-type coupled system with mean curvature operator in Minkowski space |
| title_sort | on a power type coupled system with mean curvature operator in minkowski space |
| topic | Minkowski curvature operator System Positive radial solution Uniqueness |
| url | https://doi.org/10.1186/s13661-021-01572-z |
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