Gradient estimates and Liouville-type theorems for the semilinear elliptic equation involving the nonlinear gradient source

We study local and global properties of positive solutions to the equation $-Δu=u^p+M|\nabla u|^q$ in a domain $Ω$ of $\mathbb R^N$, where $p,q$ are parameters and $M>0$. By constructing a linear operator, we establish the differential inequality containing an auxiliary function. By selecting approp