In this paper, we study the global $C^{1, 1}$ C 1 , 1 regularity for viscosity solution of the degenerate Monge–Ampère type equation $\det [D^{2}u-A(x, Du)]=B(x, u, Du)$ det [ D 2 u − A ( x , D u ) ] = B ( x , u , D u ) with the Neumann boundary value condition $D_{\nu }u=\varphi (x)$ D ν u = φ ( x ) , where the matrix A is under the regular condition and some structure conditions, and the right-hand term B is nonnegative.

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<p style='text-indent:20px;'>In this paper, we consider the nonexistence problem for conformal Hessian quotient inequalities in <inline-formula><tex-math id="M1">\begin{document}$ \mathbb{R}^n $\end{document}</tex-math></inline-formula>. We prove the nonexistence results of entire positive <inline-formula><tex-math id="M2">\begin{document}$ k $\end{document}</tex-math></inline-formula>-admissible solution to a conformal Hessian quotient inequality, and entire <inline-formula><tex-math id="M3">\begin{document}$ (k, k') $\end{document}</tex-math></inline-formula>-admissible solution pair to a system of Hessian quotient inequalities, respectively. We use the contradiction method combining with the integration by parts, suitable choices of test functions, Taylor's expansion and Maclaurin's inequality for Hessian quotient operators.</p>