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

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}.