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

TL;DR

This paper develops gradient estimates for solutions to a class of quasilinear elliptic equations on complete Riemannian manifolds, improving existing results and extending them to broader geometric and functional settings.

Contribution

It introduces new gradient estimates for solutions to the equation, generalizes previous results to cases where $p>n$, and extends estimates to solutions on complete Riemannian manifolds.

Findings

Unified Cheng-Yau type estimate derived for $b(x) ot eq 0$

Improved estimates under weakened geometric conditions

Extended estimates to the case $p>n$ and on complete manifolds

Abstract

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.