# Fractional Kirchhoff equation with a general critical nonlinearity

## TL;DR

This paper investigates the existence and asymptotic behavior of solutions to a fractional Kirchhoff equation with critical nonlinearity in  small, using a perturbation approach without the Ambrosetti-Rabinowitz condition.

## Contribution

It introduces a novel perturbation method to prove solution existence for fractional Kirchhoff equations without the Ambrosetti-Rabinowitz condition when parameter b is small.

## Key findings

- Existence of solutions for small b
- Asymptotic behavior of solutions as b approaches zero
- Solution properties without the Ambrosetti-Rabinowitz condition

## Abstract

In this paper, we study the fractional Kirchhoff equation with critical nonlinearity \begin{align*} \left(a+b\int_{\mathbb R^N}|(-\Delta)^{\frac{s}{2}}u|^2dx\right)(-\Delta)^su+u=f(u)\ \ \mbox{in}\ \ \mathbb R^N, \end{align*} where $N>2s$ and $(-\Delta)^s$ is the fractional Laplacian with $0<s<1$. By using a perturbation approach, we prove the existence of solutions to the above problem without the Ambrosetti-Rabinowitz condition when the parameter $b$ small. What's more, we obtain the asymptotic behavior of solutions as $b\to 0$.