The Neumann problem for a class of Hessian quotient type equations

Abstract

In this paper, we consider the Neumann problem for a class of Hessian quotient equations involving a gradient term on the right-hand side in Euclidean space. More precisely, we derive the interior gradient estimates for the $(\Lambda ,k)$-convex solution of Hessian quotient equation $\frac{{\sigma }_k\left(\Lambda \right(D^{2}u\left)\right)}{{\sigma }_l\left(\Lambda \right(D^{2}u\left)\right)}=\psi (x,u,Du)$ with $0\le l<k\le C_{n-1}^{p-1}$ under the assumption of the growth condition. As an application, we obtain the global a priori estimates and the existence theorem for the Neumann problem of this Hessian quotient type equation.