Hypersurfaces of constant sum Hessian curvature in Hyperbolic space

In this paper, we study the asymptotic Plateau problem in hyperbolic space for constant sum Hessian curvature. More precisely, given a asymptotic boundary $\mathrm{\Gamma }$, one seeks a complete hypersurface $\mathrm{\Sigma }$ in ${\mathbb{H}}^{n+1}$ satisfying ${\sigma }_{n-1}(\kappa )+\alpha {\sigma }_n(\kappa )=\sigma \in (0,n),\mathrm{\partial }\mathrm{\Sigma }=\mathrm{\Gamma }$ where $\alpha$ is a non-negative number.