Harnack type inequality and Liouville theorem for subcritical fully nonlinear equations

Abstract

We consider this equation ${\sigma }_k\left(A^u\right)=u^{\left(p-\frac{n+2}{n-2}\right)k},$where $n\ge 3$ and $p\in \left(\frac{n}{n-2},\frac{n+2}{n-2}\right)$. Here ${\sigma }_k$ denotes the $k$th elementary symmetric function of the eigenvalues of $A^u$, and $A^u=-\frac{2}{n-2}{u}^{-\frac{n+2}{n-2}}{D}^{2}u+\frac{2n}{{(n-2)}^2}{u}^{-\frac{2n}{n-2}}\nabla u\otimes \nabla u-\frac{2}{{(n-2)}^2}{u}^{-\frac{2n}{n-2}}{|\nabla u|}^{2}I,$where $\nabla u$ denotes the gradient of $u$ and $D^{2}u$ denotes the Hessian of $u$. This equation has fruitful backgrounds in geometry and physics. We then obtain Schoen’s Harnack type inequality in Euclidean balls, and asymptotic behavior of an entire solution. Based on the asymptotic behavior, we give another proof of the Liouville theorem obtained by A. Li and Y.Y. Li (2005).