# A minimization problem involving a fractional Hardy-Sobolev type inequality

## TL;DR

This paper proves the existence of solutions to a minimization problem involving a fractional Hardy-Sobolev inequality with an inner singularity, extending understanding of optimal constants in fractional Sobolev spaces.

## Contribution

It establishes the attainability of the optimal constant in a fractional Hardy-Sobolev inequality with inner singularity for bounded domains.

## Key findings

- Existence of nontrivial solutions for the minimization problem.
- Attainability of the optimal constant in the fractional Hardy-Sobolev inequality.
- Analysis of the problem in the case of inner singularity.

## Abstract

In this work, we obtain an existence of nontrivial solutions to a minimization problem involving a fractional Hardy-Sobolev type inequality in the case of inner singularity. Precisely, for $\lambda>0$ we analyze the attainability of the optimal constant $$ \mu_{\alpha, \lambda}(\Omega):=\inf\left\{ [u]^2_{s,\Omega}+\lambda\int_{\Omega}|u|^2 \, dx \colon u\in H^s(\Omega), \, \int_{\Omega} \frac{|u(x)|^{2_{s,\alpha}}}{|x|^{\alpha}} \, dx=1 \right\}, $$ where $0<s<1, n>4s, 0<\alpha<2s$, $2_{s,\alpha}=\frac{2(n-\alpha)}{n-2s}$, and $\Omega \subset \mathbb{R}^n$ be a bounded domain such that $0\in \Omega$.