Two new constant rank theorems

Abstract

Motivated from one-dimensional rigidity results of entire solutions to Liouville equation, we consider the semilinear equation(0.1)$\Delta u=G\left(u\right)\text{in }{R}^n,$where $G>0,G^{\prime }<0$ and $G{G}^{″}\le A{\left(G^{\prime }\right)}^2$, with $A>0$. Let u be a smooth convex solution to (0.1) and ${\sigma }_k\left(D^{2}u\right)$ be the k-th elementary symmetric polynomial with respect to $D^{2}u$. Under the above conditions, we prove the following two new constant rank theorems:

• (1)
  If ${\sigma }_2\left(D^{2}u\right)$ has a local minimum, then $D^{2}u$ has constant rank 1 for $A\le 2$.
• (2)
  If ${\sigma }_n\left(D^{2}u\right)$ has a local minimum, then ${\sigma }_n\left(D^{2}u\right)$ is always zero and $D^{2}u$ must have constant rank $r\le n-1$ in the domain for $A\le \frac{n}{n-1}$.