Hardy-Sobolev inequality with higher dimensional singularity

TL;DR

This paper investigates the existence of positive solutions to a Hardy-Sobolev type equation with higher-dimensional singularities, considering the influence of geometry and potential functions in a bounded domain.

Contribution

It extends the analysis of Hardy-Sobolev inequalities to higher-dimensional singularities and establishes existence results based on geometric and potential conditions.

Findings

Existence of solutions depends on local geometry of the singularity.

Solutions are influenced by the potential function h.

The study covers the critical Hardy-Sobolev exponent case.

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 $h$.