Classification of positive solutions to a class of Laplace equations with a gradient term
Jingbo Dou, Benfeng Shi, Tian Wu, Hua Zhu
TL;DR
This work studies positive solutions to the gradient-augmented Laplace equation $- abla^{2}u = f(u)| abla u|^{q}$ on complete noncompact manifolds with nonnegative Ricci curvature. The authors develop and apply the invariant tensor technique to derive differential identities and a pseudo-invariant function to classify solutions across three regimes: the first subcritical, the second subcritical, and the second critical case. They prove a Serrin-type Liouville theorem in the first subcritical range, establish a subcritical rigidity result using the invariant tensor in the second subcritical case, and achieve a rigidity/classification in the second critical case for dimensions $n\,\in\{3,4,5\}$ and $0<q<1/(n-1)$, showing that either $u$ is constant or $(M^n,g)$ is isometric to ${\mathbb R}^n$ with $u$ matching an explicit radial profile. This constitutes the first rigidity result for equations with gradient terms in the second critical case and extends known classifications to gradient-augmented Laplace equations on manifolds.
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 \[-Δu = f(u)|\nabla u|^q\quad\text{in }~M^n,\] 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}.