$(M,g)$ is a compact Riemannian manifold. Let $u$ be a solution of $\Delta u = F(u)$ on $M$. $$ f:=\frac{|\nabla u|^2}{(\mu\sup u-u)^2} $$ Assume $x_0\in M$ st $f$ achieves its supremum. Then, why I can assume $\partial_\alpha u =0$ for $\alpha >1$ by choosing suitable orthonormal frame at $x_0$ ?
This question is from 209 page of Li and Yau's Estimates of eigenvalues of a compact Riemannian manifold.
Unless I'm misunderstanding, none of your assumptions on $u$ are required. Just choose any orthonormal frame $E_\alpha$ with $E_1 = C \mathrm{grad}(u),$ and you naturally have $$\partial_{E_\alpha} u= \langle\mathrm{grad}(u), E_\alpha\rangle=\frac 1 C\langle E_1, E_\alpha \rangle = \frac 1 C\delta_{1 \alpha}.$$