Gradient estimates and Liouville theorems for the $\mathrm{\Phi }$-Laplacian equations on Riemannian manifolds

This paper establishes gradient estimates and Liouville-type theorems for the $\mathrm{\Phi }$-Laplacian equation ${\mathrm{\Delta }}_{\mathrm{\Phi }}(u)=G(\mathrm{\mid }\mathrm{\nabla }u{\mathrm{\mid }}^2)$ on complete Riemannian manifolds and its parabolic counterpart ${\mathrm{\partial }}_{t}u={\mathrm{\Lambda }}_{\mathrm{\Phi }}(u)$ on compact Riemannian manifolds. Using a nonlinear $\mathrm{\Phi }$-Bochner formula and the Nash-Moser iteration technique, we prove local gradient bounds under the lower bound assumption of Ricci curvature and suitable conditions on $\mathrm{\Phi }$ and $G$, which leads to Liouville theorems for global solutions. For the parabolic case, we employ the maximum principle to derive gradient estimates on compact Riemannian manifolds, and subsequently obtain Liouville-type results. Our work provides a unified framework that generalizes prior results for $p$-harmonic functions and other quasilinear equations.