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

Abstract

<abstract><p>This paper was concerned with the following Kirchhoff type equation involving the fractional Laplace operator $ (-\Delta)^{s} $</p> <p><disp-formula> <label/> <tex-math id="FE1"> \begin{document}$ \begin{cases} \left(1+\alpha\int_{\mathbb{R}^{3}}|(-\Delta)^{\frac{s}{2}}u|^{2}dx\right)(-\Delta)^{s} u+\mu K(x)u = g(x)|u|^{p-2}u, &amp;{\rm in}\ \mathbb{R}^{3}, \\ u\in H^{s}(\mathbb{R}^{3}), \ \end{cases} $\end{document} </tex-math></disp-formula></p> <p>where $ \alpha, \ \mu &gt; 0 $, $ s\in [\frac{3}{4}, 1) $, $ 2 &lt; p &lt; 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 $.</p></abstract>