Liouville-type theorems and universal estimates for semilinear elliptic equations involving the product of the function and its gradient

In this paper, we study local and global properties of positive solutions to the equation $-Δu=u^p|\nabla u|^q$ in a domain $Ω$ of $\mathbb R^N$, where $p$ and $q$ are parameters. We introduce a linear operator to construct the differential inequality to obtain gradient estimates, and further establ