Hardy and Hardy-Sobolev inequalities on Riemannian manifolds

Abstract

Let $(M,g)$ be a smooth compact Riemannian manifold of dimension $N\ge 3$. 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} μ_{λ,σ}=\inf_{u \in H^1(M)\setminus \lbrace0\rbrace} \frac{\displaystyle\int_M |\nabla u|^2 dv_g -λ\int_M u^2 dv_g }{\biggl(\displaystyle\int_M ρ^{-σ} |u|^{2^*(σ)} dv_g\biggl)^{2/2^*(σ)}}, \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.

References in corpus (1)

• On the Hardy-Poincaré inequality with boundary singularities

Cited by in corpus (2)

• Influence of an $L^p$-perturbation on Hardy-Sobolev inequality with singularity a curve
• Optimal Hardy-Sobolev Inequalities on Compact Riemannain Manifolds