A note on the critical Laplace Equation and Ricci curvature

We study strictly positive solutions to the critical Laplace equation $$- \Delta u = n(n-2) u^{\frac{n+2}{n-2}},$$ decaying at most like $d(o, x)^{-(n-2)/2}$, on complete noncompact manifolds $(M, g)$ with nonnegative Ricci curvature, of dimension $n \geq 3$. We prove that, under an additional mild assumption on the volume growth, such a solution does not exist, unless $(M, g)$ is isometric to $\mathbb{R}^n$ and $u$ is a Talenti function. The method employs an elementary analysis of a suitable function defined along the level sets of $u$.