Critical quasilinear equations on Riemannian manifolds

TL;DR

This paper studies critical quasilinear elliptic equations on Riemannian manifolds, providing classification and rigidity results by developing new inequalities and gradient estimates, extending previous work in the field.

Contribution

It introduces a sharp nonlinear Kato inequality and Cheng-Yau type gradient estimates to classify solutions and establish rigidity on Riemannian manifolds with nonnegative Ricci curvature.

Findings

Classification of positive solutions to the critical p-Laplace equation.

Rigidity results for the ambient manifold.

Extension of results to quasilinear Liouville equations.

Abstract

In this paper, we investigate critical quasilinear elliptic partial differential equations on a complete Riemannian manifold with nonnegative Ricci curvature. By exploiting a new and sharp nonlinear Kato inequality and establishing some Cheng-Yau type gradient estimates for positive solutions, we classify positive solutions to the critical $p$-Laplace equation and show rigidity concerning the ambient manifold. Our results extend and improve some previous conclusions in the literature. Similar results are obtained for solutions to the quasilinear Liouville equation involving the $n$-Laplace operator, where $n$ corresponds to the dimension of the ambient manifold.