Classification of positive solutions to a class of Laplace equations with a gradient term

TL;DR

This paper establishes Liouville theorems and classification results for positive solutions of a class of Laplace equations with gradient terms on noncompact Riemannian manifolds, revealing rigidity and uniqueness in critical cases.

Contribution

It provides the first rigidity results for equations with gradient terms in the second critical case on manifolds, expanding understanding of solution behavior.

Findings

Proves Liouville theorems using a differential identity.

Classifies positive solutions in second critical case without additional conditions.

Shows all solutions are of a specific known form.

Abstract

In this paper, we investigate positive solutions to a class of Laplace equations with a gradient term on a complete, connected, and noncompact Riemannian manifold \((M^n,g)\) with nonnegative Ricci curvature, namely \[-\Delta u = f(u)|\nabla u|^q\quad\text{in }~M^n,\] where \(n\geqslant 3\), \(q>0,\) and \(f\) is a positive continuous function. We prove some Liouville theorems employing a key differential identity derived via the invariant tensor technique. In particular, for \(f(u)=u^{\frac{2-q}{n-2}(n+\frac{q}{1-q})-1}\) is the second critical case in dimension \(n=3,4,5\), without any additional conditions, such as integrable conditions on \(u\), we show the rigidity for the ambient manifold and classification result of positive solutions. To our knowledge, this is the first rigidity result for equations with gradient terms in the second critical case. Moreover, this result confirms that all solutions must be of the form found in \cite{BV-GH-V2019}.