# A fractional Kirchhoff problem involving a singular term and a critical nonlinearity

## TL;DR

This paper studies a complex fractional Kirchhoff problem with a singular term and critical nonlinearity, establishing the existence of two solutions using variational methods and truncation techniques.

## Contribution

It introduces a novel approach to handle the degenerate Kirchhoff case with critical nonlinearity and singular terms, proving existence of solutions.

## Key findings

- Existence of two solutions for the problem.
- Effective handling of the degenerate Kirchhoff case.
- Application of variational methods with truncation.

## Abstract

In this paper we consider the following critical nonlocal problem $$ \left\{\begin{array}{ll} M\left(\displaystyle\iint_{\mathbb{R}^{2N}}\frac{|u(x)-u(y)|^2}{|x-y|^{N+2s}}dxdy\right)(-\Delta)^s u = \displaystyle\frac{\lambda}{u^\gamma}+u^{2^*_s-1}&\quad\mbox{in } \Omega,\\ u>0&\quad\mbox{in } \Omega,\\ u=0&\quad\mbox{in } \mathbb{R}^N\setminus\Omega, \end{array}\right. $$ where $\Omega$ is an open bounded subset of $\mathbb R^N$ with continuous boundary, dimension $N>2s$ with parameter $s\in (0,1)$, $2^*_s=2N/(N-2s)$ is the fractional critical Sobolev exponent, $\lambda>0$ is a real parameter, exponent $\gamma\in(0,1)$, $M$ models a Kirchhoff type coefficient, while $(-\Delta)^s$ is the fractional Laplace operator. In particular, we cover the delicate degenerate case, that is when the Kirchhoff function $M$ is zero at zero. By combining variational methods with an appropriate truncation argument, we provide the existence of two solutions.