A Note on the Critical Laplace Equation and Ricci Curvature

Abstract

AbstractWe study strictly positive solutions to the critical Laplace equation $$\begin{aligned} - \Delta u = n(n-2) u^{\frac{n+2}{n-2}}, \end{aligned}$$ - Δ u = n ( n - 2 ) u n + 2 n - 2 , decaying at most like $$d(o, x)^{-(n-2)/2}$$ d ( o , x ) - ( n - 2 ) / 2 , on complete noncompact manifolds (M, g) with nonnegative Ricci curvature, of dimension $$n \ge 3$$ n ≥ 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 $$\mathbb R^n$$ 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.