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 $\Gamma$, one seeks a complete hypersurface $\Sigma$ in $\mathbb{H}^{n+1}$ satisfying $\sigma_{n-1}(\kappa)+\alpha\sigma_{n}(\kappa)=\sigma\in (0,n),\,\,\partial \Sigma=\Gamma$ where $\alpha$ is a non-negative number.