# A fractional p-Laplacian problem with multiple critical Hardy-Sobolev nonlinearities

## TL;DR

This paper proves the existence of solutions for a fractional p-Laplacian problem with multiple critical Hardy-Sobolev nonlinearities involving singularities, using a refined concentration-compactness principle and extremal functions.

## Contribution

It introduces a refined concentration-compactness principle and demonstrates the attainment of extremals for the Sobolev inequality in this context.

## Key findings

- Existence of weak solutions for the problem.
- Development of a refined concentration-compactness principle.
- Proof that extremals for the Sobolev inequality are attained.

## Abstract

In this work, we study the existence of weak solution to the following quasi linear elliptic problem involving the fractional $p$-Laplacian operator, a Hardy potential and multiple critical Sobolev nonlinearities with singularities, \begin{align*} (-\Delta_p)^su - \mu \dfrac{\vert u \vert^{p-2} u}{\vert x \vert^{ps}} = \dfrac{\vert u\vert^{p^*_s(\beta)-2}u}{\vert x \vert^{\beta}} + \dfrac{\vert u\vert^{p^*_s(\alpha )-2} u}{\vert x \vert^\alpha }, \end{align*} where $ x \in \mathbb{R}^N$, $u\in D^{s,p}(\mathbb{R}^N)$, $0<s<1$, $1<p<+\infty$, $N>sp$, $0<\alpha<sp$, $0<\beta<sp$, $\beta\neq\alpha$, $\mu < \mu_H := \inf_{u \in D^{s,p}(\mathbb{R}^N) \backslash \{ 0 \}} [u]_{s,p}^p / \vert\vert u \vert\vert_{s,p}^p > 0$. To prove the existence of solution to the problem we have to formulate a refined version of the concentration-compactness principle and, as an independent result, we have to show that the extremals for the Sobolev inequality are attained.