Critical quasilinear equations on Riemannian manifolds

TL;DR

The article addresses the classification of positive solutions to the critical quasilinear equations on complete Riemannian manifolds with nonnegative Ricci curvature, focusing on the critical $p$-Laplace equation and the Liouville equation. It develops a sharp nonlinear Kato inequality and Cheng-Yau gradient estimates, coupled with integral inequalities and a Moser-iteration framework, to obtain Liouville-type rigidity results. Under natural energy, decay, or polynomial-growth hypotheses, it proves that the ambient manifold must be isometric to $\, olinebreak \mathbb{R}^n$ and the solutions reduce to Aubin-Talenti bubbles, with explicit forms and sharp constants. The Liouville case yields analogous rigidity and explicit bubbles, leading to nonexistence of Sobolev minimizers on such manifolds. Collectively, these results extend Euclidean Liouville theory to noncompact manifolds with nonnegative Ricci curvature and have implications for conformal geometry and Sobolev inequalities on curved spaces.

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.

Theorems & Definitions (39)

• Definition 1.1
• Theorem I
• Corollary 1.1
• Corollary 1.2
• Corollary 1.3
• Definition 1.2
• Theorem II
• Lemma 2.1
• Remark 2.2
• proof : Proof of \ref{['lem:preliminary-1']}