Gradient estimates for Δ_pu-|∇ u|^q+b(x)|u|^r-1u=0 on a complete Riemannian manifold and Liouville type theorems

2023/09/07 by Han, Dong, He, Jie, Wang, Youde
#Analysis of PDEs (math.AP) #FOS: Mathematics

paper · doi:10.48550/arxiv.2309.03510

Abstract

In this paper the Nash-Moser iteration method is used to study the gradient estimates of solutions to the quasilinear elliptic equation Δ_p u-|∇ u|^q+b(x)|u|^r-1u=0 defined on a complete Riemannian manifold (M,g). When b(x)≡0, 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-Véron, Garcia-Huidobro and Véron (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 \citefilippucci2022priori to arbitrary solutions to this equation on a complete Riemannian manifold. Even in the case of Euclidean space, the estimates for positive solutions in \citefilippucci2022priori and our results can not cover each other.

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