Nonexistence of entire positive solutions for conformal Hessian quotient inequalities

Abstract

<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">$ \mathbb{R}^n $</tex-math></inline-formula>. We prove the nonexistence results of entire positive <inline-formula><tex-math id="M2">$ k $</tex-math></inline-formula>-admissible solution to a conformal Hessian quotient inequality, and entire <inline-formula><tex-math id="M3">$ (k, k') $</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>