Uniqueness for the Brezis-Nirenberg problem on compact Einstein manifolds

Abstract

We consider the positive solution of the following semi-linear elliptic equation on the compact Einstein manifolds $M^{n}$ with positive scalar curvature $R_{0}$ \begin{equation*} \Delta_{0}u-\lambda u+f(u)u^{(n+2)/(n-2)}=0, \end{equation*} where $\Delta_{0}$ is the Laplace-Beltrami operator on $M^{n}$. We prove that for $0<\lambda\leq (n-2)R_{0}/(4(n-1))$ and $f'(u)\leq 0$, and at least one of two inequalities is strict, the only positive solution to the above equation is constant. The method here is intrinsic.