The Dirichlet problem for mixed Hessian quotient equations

Abstract

This paper is concerned with the solvability of the Dirichlet problem of mixed Hessian quotient equations, which precisely are${\sigma }_m\left(D^{2}u\right)={\sigma }_m\left(D^{2}u\right)\left[\sum _{k=1}^m\sum _{j=0}^{k-1}{l}_{kj}^{k-j+1}\right(x\left)\frac{{\sigma }_j}{{\sigma }_k}\right(D^{2}u\left)\right],\forall 1\le m\le d,$ which include k-Hessian equation, Hessian quotient equation and mixed k-Hessian equation. We establish the existence of global $C^{1,1}$ solution under degenerate case, and furthermore the existence of global $C^{2,\alpha }$ solution under nondegenerate case. The uniqueness is also obtained.