关键词: 梯度估计, 拟线性椭圆方程, 黎曼流形, Liouville型定理, Nash-Moser迭代

Abstract: This paper investigates gradient estimates for smooth solutions to a class of quasilinear elliptic equations of the form $\operatorname{div} \left((1 + |\nabla u|^2)^\theta \nabla u \right) + a|\nabla u|^r + h(u) = 0$ on a complete Riemannian manifold, where $a$ is a constant and $h(u)$ is a nonlinear term. By employing the Saloff-Coste type Sobolev inequality and the Nash-Moser iteration method, we derive, under the assumption of a lower bound on the Ricci curvature, a unified and concise local gradient estimate for solutions satisfying the structural conditions $-\frac{1}{2} < \theta \le 0$ and $r > 1 + 2\theta$. Moreover, an explicit upper bound for the gradient of entire solutions is obtained. As a direct consequence of these gradient estimates, we prove a Liouville-type theorem: if the manifold is noncompact and has nonnegative Ricci curvature, then any globally defined solution must be constant.

Key words: Gradient estimate, Quasilinear elliptic equation, Riemannian manifold, Liouville-type theorem, Nash-Moser iteration