I'm reading the paper Remarks concerning the conformal deformation of riemannian structures on compact manifolds by NEIL S. TRUDINGER.enter link description here

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 \quad(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 (1) test functions $$ \xi=\eta^{2} G(\bar{u}) $$ we have(This is my caculation, 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 $$\begin{aligned} \frac{4(n-1) \mu}{n-2} \int_{\dot{M}} \eta^{2} G^{\prime}(\bar{u}) u_{i}^{2} d V \leq \int_{M}\left(\frac{4(n-1)}{n-2} \sup \left|g^{i j}\right| \eta\left|\eta_{i}\right|\right. &\left.+\sup |R| u \eta^{2}+\bar{R} u^{N-2} \eta^{2}\right) G d V \end{aligned}$$

Did this term miss something? Should it be: $$\begin{aligned} \frac{4(n-1) \mu}{n-2} \int_{\dot{M}} \eta^{2} G^{\prime}(\bar{u}) u_{i}^{2} d V \leq \int_{M}\left(\frac{4(n-1)}{n-2} \sup \left|g^{i j}\right| \eta\left|\eta_{i}\right|\right.u_{i} &\left.+\sup |R| u \eta^{2}+\bar{R} u^{N-1} \eta^{2}\right) G d V \end{aligned}$$ 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 have tried to integrate by parts but all fail.