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 $-\mathrm{\Delta }u=u^p+M\mathrm{\mid }\mathrm{\nabla }u{\mathrm{\mid }}^q$ in a domain $\mathrm{\Omega }$ 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 appropriate auxiliary functions over various regions and employing the maximum principle, we derive the local gradient estimates for all $(p,q)\in {\mathbb{R}}^2$, and further establish Liouville-type theorems. As an application, we acquire universal estimates for local solutions of elliptic equations with general nonlinearities. Our results extend partial conclusions established in Bidaut-Véron, Garcia-Huidobro and Véron [Math. Ann. 378 (1-2) (2020) 13-56].