# Problems arising from the Trudinger's paper in 1968 "Remarks concerning the conformal deformation of riemannian structures on compact manifolds"

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 $$\bar{u} \eta G(\bar{u})$$.