I'm reading the paper Remarks concerning the conformal deformation of riemannian structures on compact manifolds by NEIL S. TRUDINGER.
I'm stuck with the Theorem 3, which says that let $u$ be a $W_{2}^{1}(M)$ solution of an equation of the form $$\frac{4(n-1)}{n-2} \Delta u-R u=-\bar{R} u^{\frac{n+2}{n-2}}.$$ Then $u \in C^{\infty}(M)$.
The proof is as follows:
The function $u$ satisfies \begin{equation} \int_{M}\left(\frac{4(n-1)}{n-2} g^{i j} u_{i} \xi_{j}+R u \xi\right) d V=\bar{R} \int_{M}|u|^{N-1} \xi d V \label{1}\tag{1} \end{equation} for all $\xi \in W_{2}^{1}(M)$, then construct a test function.
Define $\bar{u}=\sup (u, 0)$ and for a fixed $\beta>1$define the functions $$ \begin{gathered} G(\bar{u})= \begin{cases}\bar{u}^{\beta} & \text { if } \bar{u} \leq l \\ l^{q-1}\left(q l^{q-1} \bar{u}-(q-1) l^{q}\right) & \text { if } \bar{u}>l\end{cases} \\ F(\bar{u})= \begin{cases}\bar{u}^{q} & \text { if } \bar{u} \leq l \\ q l^{q-1} \bar{u}-(q-1) l^{q} & \text { if } \bar{u}>l \end{cases} \end{gathered} $$ where $2 q=\beta+1$.
The function $G(\bar{u})$ is a uniformly Lipshitz continuous function of $u$ and hence belongs to $W_{2}^{1}(M)$. Likeuise $F(\bar{u})$. Observe also that $G$ and $F$ vanish for negative $u$ and that $$ \left(F^{\prime}(\bar{u})\right)^{2} \leq q G^{\prime}(\bar{u}), \quad(F(\bar{u}))^{2} \geq \bar{u} G(\bar{u}). $$ Let us now substitute in \eqref{1} test functions $$ \xi=\eta^{2} G(\bar{u}) $$ we have (this is my calculation, I don't know how to get the result in paper) $$\int_{M}\left(\frac{4(n-1)}{n-2} g^{i j} u_{i} (2\eta \eta_{j}G(\bar{u})+\eta^{2}G^{\prime}(\bar{u})u_{j})+R u \eta^{2} G(\bar{u})\right) d V=\bar{R} \int_{M}|u|^{N-1} \eta^{2} G(\bar{u}) d V.$$
Then $$\frac{4(n-1) \mu}{n-2} \int_{M} \eta^{2} G^{\prime}(\bar{u}) u_{i}^{2} d V = \int_{M}\bar{R} |u|^{N-1} \eta^{2} G(\bar{u})-\frac{4(n-1)}{n-2} g^{i j} u_{i}\left(2 \eta \eta_{j} G(\bar{u})\right)-Ru\eta ^{2} G(\bar{u}) dV.$$
Then he gets $$\begin{aligned} \frac{4(n-1) \mu}{n-2} \int_{\dot{M}} \eta^{2} G^{\prime}(\bar{u}) u_{i}^{2} d V \leq \int_{M}\left(\frac{4(n-1)}{n-2} \sup \left|g^{i j}\right| \eta\left|\eta_{i}\right|+\sup |R| u \eta^{2}+\bar{R} u^{N-2} \eta^{2}\right) G d V. \end{aligned}$$
Did this term miss something? Should it be: $$\begin{aligned} \frac{4(n-1) \mu}{n-2} \int_{\dot{M}} \eta^{2} G^{\prime}(\bar{u}) u_{i}^{2} d V \leq \int_{M}\left(\frac{4(n-1)}{n-2} \sup \left|g^{i j}\right| \eta\left|\eta_{j}\right|u_{i} +\sup |R| u \eta^{2}+\bar{R} u^{N-1} \eta^{2}\right) G d V? \end{aligned}$$ Then he gets $$\int_{M} \eta^{2} G^{\prime}(\bar{u}) u_{i}^{2} d V \leq C \int_{M}\left\{\left(\left|\eta_{i}\right|^{2}+\eta^{2}\right)(\bar{u}) G(\bar{u})+\eta^{2} \bar{u}^{ N-2} \bar{u} G(\bar{u})\right\} d V.$$ I'm stuck with these steps.
I tried like this:
Since we have $$\frac{4(n-1) \mu}{n-2} \int_{M} \eta^{2} G^{\prime}(\bar{u}) u_{i}^{2} d V = \int_{M}\bar{R} |u|^{N-1} \eta^{2} G(\bar{u})-\frac{4(n-1)}{n-2} g^{i j} u_{i}\left(2 \eta \eta_{j} G(\bar{u})\right)-Ru\eta ^{2} G(\bar{u}) dV.$$ And since M is compact we have
$$-\int_{M} \eta u_{i} G=\int_{M} u\left(\eta_{i} G+\eta G_{i}\right)=\int_{M} u \eta_{i} G+\int_{M} u \eta G_{i}$$
then I compute the last term
$$\int_{M} u \eta G_{i}=\int_{M} u \eta G^{\prime}(\bar{u}) u_{i}\leqslant \int_{M}\left(\frac{u^{2}+u_{i}^{2}}{2}\right) \eta G^{\prime}(\bar{u})$$
then put the $\int_{M} \frac{u_{i}^{2}}{2} \eta G^{\prime}(\bar{u})$ to the LHS(the problem is that if I put this to the LHS to combine with $\frac{4(n-1) \mu}{n-2} \int_{M} \eta^{2} G^{\prime}(\bar{u}) u_{i}^{2} d V$, is the coefficient still positive? It seems to be depending on the elliptic constant $\mu$.)
And notice that $$G^{\prime}(\bar{u})= \begin{cases}\beta\bar{u}^{\beta-1} & \text { if } \bar{u} \leq l \\ l^{q-1}\left(q l^{q-1}\right) & \text { if } \bar{u}>l\end{cases}$$
So we have
$$\left\{\begin{array}{l}\frac{\bar{u} G^{\prime}(\bar{u})}{\beta}=G(\bar{u}) \quad \bar{u} \leqslant l \\ \bar{u} G^{\prime}(\bar{u})-C_{0}=G(\bar{u}) \quad \bar{u}>l\end{array}\right.$$
so we can turn $u^{2}$$\eta G^{\prime}(\bar{u})$ to be something about $\bar{u} \eta G(\bar{u})$.
