Global bifurcation of solutions of the mean curvature spacelike equation in certain standard static spacetimes

Guowei Dai; Alfonso Romero; Pedro J. Torres

doi:10.3934/dcdss.2020118

Article Contents

2020, Volume 13, Issue 11: 3047-3071. Doi: 10.3934/dcdss.2020118

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Global bifurcation of solutions of the mean curvature spacelike equation in certain standard static spacetimes

1.
School of Mathematical Sciences, Dalian University of Technology, Dalian 116024, China
2.
Departamento de Geometría y Topología, Universidad de Granada, 18071 Granada, Spain
3.
Departamento de Matemática Aplicada, & Research Unit Modeling Nature (MNat), Universidad de Granada, 18071 Granada, Spain

^* Corresponding author
^* Corresponding author

Received: March 2019

Early access: October 31, 2019

Published online: October 31, 2019

The first author is supported by NNSF of China (No. 11871129) and Xinghai Youqing funds from Dalian University of Technology, the second one by Spanish MINECO Grant with FEDER funds MTM2016-78807-C2-1-P and the third author by Spanish MINECO Grant with FEDER funds MTM2017-82348-C2-1-P.

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Abstract

We study the existence/nonexistence and multiplicity of spacelike graphs for the following mean curvature equation in a standard static spacetime

$ \begin{eqnarray} \text{div} \left(\frac{a\nabla u}{\sqrt{1-a^2\vert \nabla u\vert^2}}\right)+\frac{g(\nabla u, \nabla a)}{\sqrt{1-a^2\vert \nabla u\vert^2}} = \lambda NH \end{eqnarray} $

with $ 0 $-Dirichlet boundary condition on the unit ball. According to the behavior of $ H $ near $ 0 $, we obtain the global structure of one-sign radial spacelike graphs for this problem. Moreover, we also obtain the existence and multiplicity of entire spacelike graphs.

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Mathematics Subject Classification: 35B32, 53A10.

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Figure 1. Bifurcation diagrams of Theorem 1.1

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