EXISTENCE AND REGULARITY OF SPACELIKE HYPERSURFACES FOR MEAN CURVATURE EQUATION IN THE STATIC SPACETIME

Abstract

Consider the following mean curvature equation in the static spacetime:

$$\text{div }\left(\frac{f(x)\text{∇ }u}{\sqrt{1-f^2(x)|\text{∇ }u{|}^2}}\right)+\frac{\text{∇ }u\text{∇ }f(x)}{\sqrt{1-f^2(x)|\text{∇ }u{|}^2}}=\lambda NH$$

with Dirichlet boundary condition on a bounded domain. We investigate the existence and uniqueness of classical solution. By the variational method, we also establish the multiplicity of strong solutions. Moreover, according to the behavior of $H$ near $0$, we obtain the global structure of positive solutions for this problem. The symmetry of positive solutions is also investigated.