The interior gradient estimate of prescribed Hessian quotient curvature equations

In this paper, we establish the interior gradient estimate of k-admissible solutions of prescribed Hessian quotient curvature equations σk(aij)σl(aij)=f(x)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\frac{\sigma _k (a_{ij})}{\sigma _l (a_{ij})} = f(x)$$\end{document} with 0≤l<k≤n\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$0 \le l < k \le n$$\end{document}. As an application, we get a Liouville type theorem.