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

Consider the following mean curvature equation in the static spacetime: div (f(x)∇ u1−f2(x)|∇ u|2)+∇ u∇ f(x)1−f2(x)|∇ u|2=λNH 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.