A simple proof of Aubin’s theorem in Yamabe problem

Theorem(Aubin) Let $(M, g)$ be a compact Riemannian manifold with $Y(M, g) < Y (S_n)$, where $Y(M, g)$ is called the Yamabe invariant and defined by

$\displaystyle Y(M,g)=\inf_{u\in C^{\infty}(M)}Q_g(u)=\inf_{ u\in C^{\infty}(M)}\frac{\int_M(|\nabla u|^2+\frac{n-2}{4(n-1)}R_gu^2)d\mu_g}{\|u\|^2_{L^{2^*}}}.$

Then the infimum of the functional $Q_g(u)$ is attained. Namely, the Yamabe problem can be solved.

The original proof of Theorem used the subcritical equations. There is another simple proof by Druet and Hebey using the  Brezis and Lieb’s lemma.

Proof: After passing to a subsequence, we may also assume that there exists $u\in H^1(M)$ such that $u_i\rightharpoonup u$ weakly in $H^1(M)$, applying the imbedding theorem, we obtain $u_i\to u$ strongly in $L^2(M)$, furthermore $u_i\to u$ almost everywhere as $i\to\infty$. In particular, $u$ is nonnegative. It easy form the weakly convergence that

$\displaystyle \|\nabla u_i\|^2_{L^2}=\|\nabla(u_i-u)\|_{L^2}^2+\|\nabla u\|^2_{L^2}+o(1)$

for all $i$, where $o(1)\to0$ as $i\to\infty$. We also have that($p=2^*=\frac{2n}{n-2}$) by  Brezis and Lieb’s lemma

$\displaystyle \|u_i\|^p_{L^p}=\|u_i-u\|_{L^p}^p+\|u\|_{L^p}^p+o(1).$

Thanks to the sharp Sobolev inequality of Hebey and Vaugon, there exists $B>0$ such that for any $i$,

$\displaystyle \|u_i-u\|^2_{L^p}\le K_n^2\|\nabla(u_i-u)\|^2_{L^2}+B\|u_i-u\|^2_{L^2}.$

Since $u_i\in \mathcal{H}=\{u\in H^1(M):\int_M|u|^pd\mu_g=1\}$, it follows that

$\displaystyle (1-\|u\|^p_{L^p})^{\frac 2p}\le K_n^2\left(\|\nabla u_i\|^2_{L^2}-\|\nabla u\|^2_{L^2}\right)+o(1).$

Since $Q_g(u_i)\to Y_g(M)$, and since $u_i\to u$ in $L^2(M)$, we also have that

$\displaystyle\begin{gathered} K_n^2\left(\|\nabla u_i\|^2_{L^2}-\|\nabla u\|^2_{L^2}\right)\hfill\\ \qquad=K_n^2Y(M)-K_n^2\left(\int_M|\nabla u|^2d\mu+\int_MaRu^2d\mu\right)+o(1)\hfill\\ \qquad\le K_n^2Y(M)-K_n^2Y(M)\|u\|^2_{L^p}+o(1)\hfill\\ \end{gathered}$

Hence,

$\displaystyle (1-\|u\|^p_{L^p})^{\frac 2p}\le K_n^2Y_g(M)(1-\|u\|^2_{L^p})+o(1).$

We assume that $Y_g(M) and note that

$\displaystyle 1-\|u\|^2_{L^p}\le (1-\|u\|^p_{L^p})^{\frac 2p}$

this implies that

$\displaystyle \|u\|^2_{L^p}=1.$

Then,

$\displaystyle \|\nabla u_i\|_{L^2}\to \|\nabla u\|_{L^2}$

as $i\to\infty$. and since

$\displaystyle \|\nabla u_i\|^2_{L^2}=\|\nabla(u_i-u)\|_{L^2}^2+\|\nabla u\|^2_{L^2}+o(1)$

We obtain that $u_i\to u$ strongly in $H^1$ as $i\to\infty$. In particular, $u$ is a minimizer for $Y_g(M)$ and $u$ is a weak nonnegative solution of the Yamabe equation

$\displaystyle \Delta_gu-\frac{n-2}{4(n-1)}R_gu=Y_g(M)u^{2^*-1}.$

Regularity argument and the maximum principle then give that $u$ is smooth and positive. This prove the Theorem.