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.