Gradient bounds and Liouville theorems for quasi-linear equations on compact manifolds with nonnegative Ricci curvature

In this work we establish a gradient bound and Liouville-type theorems for solutions to Quasi-linear elliptic equations on compact Riemannian manifolds with nonnegative Ricci curvature. Also, we provide a local splitting theorem when the inequality in the gradient bound becomes equality at some point. Moreover, we prove a Harnack-type inequality and an ABP estimate for the gradient of solutions in domains contained in the manifold.