Gradient estimates for $\Delta_pu-|\nabla u|^q+b(x)|u|^{r-1}u=0$ on a complete Riemannian manifold and Liouville type theorems

In this paper the Nash-Moser iteration method is used to study the gradient estimates of solutions to the quasilinear elliptic equation $\Delta_p u-|\nabla u|^q+b(x)|u|^{r-1}u=0$ defined on a complete Riemannian manifold $(M,g)$. When $b(x)\equiv0$, a unified Cheng-Yau type estimate of the solutions to this equation is derived. Regardless of whether this equation is defined on a manifold or a region of Euclidean space, certain technical and geometric conditions posed in \cite[Theorem E, F]{MR3261111} are weakened and hence some of the estimates due to Bidaut-V\'eron, Garcia-Huidobro and V\'eron (see \cite[Theorem E, F]{MR3261111}) are improved. In addition, we extend their results to the case $p>n=\dim(M)$. When $b(x)$ does not vanish, we can also extend some estimates for positive solutions to the above equation defined on a region of the Euclidean space due to Filippucci-Sun-Zheng \cite{filippucci2022priori} to arbitrary solutions to this equation on a complete Riemannian manifold. Even in the case of Euclidean space, the estimates for positive solutions in \cite{filippucci2022priori} and our results can not cover each other.