Hardy-Sobolev inequality with higher dimensional singularity

Abstract

For $N\ge 4$, we let $\Omega$ to be a smooth bounded domain of ${\mathbb{R}}^N$, $\Gamma$ a smooth closed submanifold of $\Omega$ of dimension $k$ with $1\le k\le N-2$ and $h$ a continuous function defined on $\Omega$. We denote by $ρ_Γ\left(\cdot\right):=\dist_g\left(\cdot, Γ\right)$ the distance function to $\Gamma$. For $\sigma \in (0,2)$, we study existence of positive solutions $u\in H_0^1\left(\Omega \right)$ to the nonlinear equation $$-\Delta u+hu={\rho }_{\Gamma }^{-\sigma }{u}^{2^{\ast }(\sigma )-1}\text{in }\Omega ,$$ where $2^{\ast }(\sigma ):=\frac{2(N-\sigma )}{N-2}$ is the critical Hardy-Sobolev exponent. In particular, we provide existence of solution under the influence of the local geometry of $\G$ and the potential $h$.

arXiv admin note: text overlap with arXiv:1702.02202