Necessary and sufficient conditions to Bernstein theorem of a Hessian equation

Abstract:

The Hessian quotient equations \begin{equation} S_{k,l}(D^2u)\equiv \frac {S_k(D^2u)}{S_l(D^2u)}=1, \ \ \forall x\in {\mathbb {R}}^n \end{equation} were studied for $k-$th symmetric elementary function $S_k(D^2u)$ of eigenvalues $\lambda (D^2u)$ of the Hessian matrix $D^2u$, where $0\leq l<k\leq n$. For $l=0$, (0.1) is reduced to a $k-$Hessian equation \begin{equation} S_k(D^2u)=1, \ \ \forall x\in {\mathbb {R}}^n. \end{equation} Two quadratic growth conditions were found by Bao-Cheng-Guan-Ji [American J. Math. 125 (2013), pp. 301–316] ensuring the Bernstein properties of (0.1) and (0.2) respectively. In this paper, we will drop the point wise quadratic growth condition of Bao-Cheng-Guan-Ji and prove three necessary and sufficient conditions to Bernstein property of (0.1) and (0.2), using a reverse isoperimetric type inequality, volume growth or $L^p$-integrability respectively. Our new volume growth or $L^p-$integrable conditions improve largely various previously known point wise conditions provided Bao et al.; Chen and Xiang [J. Differential Equations 267 (2019), pp. 52027–5219]; Cheng and Yau [Comm. Pure Appl. Math. 39 (1986), pp. 8397–866]; Li, Ren, and Wang [J. Funct. Anal. 270 (2016), pp. 26917–2714]; Yuan [Invent. Math. 150 (2002), pp. 1177–125], etc.