A stronger constant rank theorem

Motivated from one-dimensional rigidity results of entire solutions to Liouville equation, we consider the semilinear equation \begin{align} \label{liouvilleequationab} \Delta u=G(u) \quad \mbox{in $\mathbb{R}^n$}, \end{align}where $G>0, G'<0$ and $GG^{''}\le A(G')^2$, with $A>0$. Let $u$ be a smooth convex solution and $\sigma_k(D^2 u)$ be the $k$-th elementary symmetric polynomial with respect to $D^2u$. We prove stronger constant rank theorems in the following sense. (1) When $A\le 2$, if $\sigma_2(D^2u)$ takes a local minimum, then $D^2 u$ has constant rank $1$. (2) When $A\le \frac{n}{n-1}$, if $\sigma_n(D^2 u)$ takes a local minimum, then $\sigma_n(D^2 u)$ is always zero in the domain.