In this paper, we investigate the following fractional Kirchhoff equation: $(a+b{\int }_{{\mathbb{R}}^3}|(-\mathrm{\Delta }{)}^{\frac{s}{2}}u|\mathrm{d}x)(-\mathrm{\Delta }{)}^{s}u+V(x)u=f(u),x\in {\mathbb{R}}^3,$where $a>0,b\ge 0$, $(-\mathrm{\Delta }{)}^s$ denotes the fractional Laplacian operator with order $s\in (\frac{3}{4},1)$, V is a positive continuous potential and f is supercubic but subcritical functional at infinity with some valid conditions. We prove that there exist multiple sign-changing solutions for the above problem via the method of invariant sets of descending flow. In particular, the nonlinear term includes the power-type nonlinearity $f(u)=|u{|}^{p-2}u$ for the less studied case $p\in (3,4)$. Even for s = 1, our result is new and extends the existing results in the literature.

在本文中，我们研究了以下分数基尔霍夫方程：$(\mathrm{一个}+b{\int }_{{\mathbb{R}}^3}|(-\mathrm{\Delta }{)}^{\frac{s}{2}}你|\mathrm{d}X)(-\mathrm{\Delta }{)}^s你+五(X)你=F(你),X\in {\mathbb{R}}^3,$在哪里$\mathrm{一个}>0,b\ge 0$,$(-\mathrm{\Delta }{)}^s$表示带阶的分数拉普拉斯算子$s\in (\frac{3}{4},1)$, V是一个正连续势，f是超立方但在无穷远处具有一些有效条件的亚临界泛函。我们通过降流不变集的方法证明了上述问题存在多个符号变化解。特别是，非线性项包括幂型非线性$F(你)=|你{|}^{p-2}你$对于研究较少的案例$p\in (3,4)$. 即使对于s  = 1，我们的结果也是新的，并且扩展了文献中的现有结果。