Critical Sobolev Exponent ¹

Brezis' first open problem - An elliptic equation involving the critical exponent in 3D

Yamabe problem # Yamabe problem: Suppose $(\mathcal{M}, g_0)$ is a compact closed Riemannian manifold with dimension $N \geq 3$, does there exist a conformal metric $g = u^{\frac{4}{N-2}}g_0$ which has constant scalar curvature $R_g \equiv C$? Find $u > 0$ on $\mathcal{M}$ such that $$ -\frac{4(N-1)}{N-2}\Delta_{g_0}u + R_{g_0}u = Cu^{\frac{N+2}{N-2}}\qquad\text{on }\mathcal{M}. $$ Some results: Trudinger [1968]: if $g$ has non-positive scalar curvature. Aubin [1976]: $N \geq 6$ and $(\mathcal{M}, g)$ not locally conformally flat. Schoen [1984]: any dimension, the remaining cases, assuming the Positive Mass Theorem by Schoen-Yau [1979]. A special case # Consider the special case where $\mathcal{M}$ is a bounded domain $\Omega$ in $\mathbb{R}^{N}$: $$ \begin{cases} -\Delta u = u^{\frac{N+2}{N-2}}\qquad\text{in }\Omega, \\ u > 0\qquad\text{in }\Omega, \\ u = 0\qquad\text{on }\partial\Omega. \end{cases} $$