A note on the critical Laplace Equation and Ricci curvature

We study strictly positive solutions to the critical Laplace equation -Δu=n(n-2)un+2n-2,decaying at most like d(o, x) -(n-2)/2, on complete noncompact manifolds (M, g) with nonnegative Ricci curvature, of dimension 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 Rn and u is a Talenti function. The method employs an elementary analysis of a suitable function defined along the level sets of u.

We study strictly positive solutions to the critical Laplace equation −∆u = n(n − 2)u 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 ≥ 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 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.