Gradient estimates and Liouville theorems for the \(\Phi\)-Laplacian equations on Riemannian manifolds

Abstract

This paper establishes gradient estimates and Liouville-type theorems for the \(\Phi\)-Laplacian equation \(\Delta_{\Phi}(u) = G(|\nabla u|^2)\) on complete Riemannian manifolds and its parabolic counterpart \(\partial_t u = \Lambda_{\Phi}(u)\) on compact Riemannian manifolds. Using a nonlinear \(\Phi\)-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 \(\Phi\) 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.