Optimal Liouville theorems for the Lane-Emden equation on Riemannian manifolds

Abstract: We study degenerate quasilinear elliptic equations on Riemannian manifolds and obtain several Liouville theorems. Notably, we provide rigorous proof asserting the nonexistence of positive solutions to the subcritical Lane-Emden-Fowler equations over complete Riemannian manifolds with nonnegative Ricci curvature. These findings serve as a significant generalization of Gidas and Spruck's pivotal work (Comm. Pure Appl. Math. 34, 525-598, 1981) which focused on the semilinear case, as well as Serrin and Zou's contributions (Acta Math. 189, 79-142, 2002) within the context of Euclidean geometries.