Normalized ground states for fractional Kirchhoff equations with critical or supercritical nonlinearity

The aim of this paper is to study the existence of ground states for a class of fractional Kirchhoff type equations with critical or supercritical nonlinearity

$$(a+b{\int }_{{\mathbb{R}}^3}|(-△{)}^{\frac{s}{2}}u{|}^{2}dx)(-△{)}^{s}u=\lambda u+|u{|}^{q-2}u+\mu |u{|}^{p-2}u,x\in {\mathbb{R}}^3,$$

with prescribed $L^2$-norm mass

$${\int }_{{\mathbb{R}}^3}{u}^{2}dx=c^2$$

where $s\in (\frac{3}{4},1),a,b,c>0,\frac{6+8s}{3}<q<2_s^{\ast },p\ge 2_s^{\ast }(2_s^{\ast }=\frac{6}{3-2s}),\mu >0$ and $\lambda \in \mathbb{R}$ as a Langrange multiplier. By combining an appropriate truncation argument with Moser iteration method, we prove that the existence of normalized solutions for the above equation when the parameter $\mu$ is sufficiently small.