Hardy-Sobolev inequality with higher dimensional singularity

Abstract

For $N\geq 4$, we let $\Omega$ to be a smooth bounded domain of $\mathbb{R}^N$, $\Gamma$ a smooth closed submanifold of $\Omega$ of dimension $k$ with $1\leq k \leq N-2$ and $h$ a continuous function defined on $\Omega$. We denote by $\rho_\Gamma\left(\cdot\right):=\dist_g\left(\cdot, \Gamma\right)$ the distance function to $\Gamma$. For $\sigma\in (0,2)$, we study existence of positive solutions $u \in H^1_0\left(\Omega\right)$ to the nonlinear equation $$ -\Delta u+h u=\rho_\Gamma^{-\sigma} u^{2^*(\sigma)-1} \qquad \textrm{in } \Omega, $$ where $2^*(\sigma):=\frac{2(N-\sigma)}{N-2}$ 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 $h$.