Let $ (M,g) $ be a smooth compact Riemannian manifold of dimension $ N \geq 3 $. Given $p_0 \in M$, $\lambda \in \mathcal{ R}$ and $\sigma \in (0,2]$, we study existence and non existence of minimizers of the following quotient: \begin{ equation}\label{ Paper Equation} \mu_{ \lambda,\sigma}=\inf_{ u \in H^1(M)\setminus \lbrace0\rbrace} \frac{ \displaystyle\int_M |\nabla u|^2 dv_g -\lambda \int_M u^2 dv_g }{ \biggl(\displaystyle\int_M \rho^{ -\sigma} |u|^{ 2^*(\sigma)} dv_g\biggl)^{ 2/2^*(\sigma)}}, \end{ equation} where $\rho(.):=dist(p_0,.)$ denoted the geodesic distance from $p \in M$ to $p_0$. In particular for $\sigma=2$, we provide sufficient and necessary conditions of existence of minimizers in terms of $\lambda$. For $\sigma\in (0,2)$ we prove existence of minimizers under scalar curvature pinching.
Categories: math.AP