Multiple positive solutions to the fractional Kirchhoff-type problems involving sign-changing weight functions

This paper was concerned with the following Kirchhoff type equation involving the fractional Laplace operator $(-\mathrm{\Delta }{)}^s$

$$\{\begin{array}{ll}\left(1+\alpha {\int }_{{\mathbb{R}}^3}|(-\mathrm{\Delta }{)}^{\frac{s}{2}}u{|}^{2}dx\right)(-\mathrm{\Delta }{)}^{s}u+\mu K(x)u=g(x)|u{|}^{p-2}u,& \mathrm{i}\mathrm{n}{\mathbb{R}}^3,\\ u\in H^s({\mathbb{R}}^3),\end{array}$$

where $\alpha ,\mu >0$, $s\in [\frac{3}{4},1)$, $2<p<4$. By filtration of the Nehari manifold and variational techniques, we obtained the existence of one and two positive solutions under some conditions imposed on $K$ and $g$.