The Neumann problem for a class of degenerate Hessian quotient type equations

Abstract

In this paper, we obtain some important inequalities for a class of Hessian quotient type operators $\frac{{\sigma }_k(\Lambda (D^{2}u))}{{\sigma }_l(\Lambda (D^{2}u))}$, which can be regarded as a generalization of the classical Hessian quotient operators. As an application, we establish global a priori estimates and prove an existence theorem for the Neumann problem of the corresponding degenerate Hessian quotient type equation, in which the admissible range of $k$ is extended to $0<k\le C_n^{\mathbf{p}}$ with $1\le \mathbf{p}\le n-1$.