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

Abstract

This paper was concerned with the following Kirchhoff type equation involving the fractional Laplace operator $ (-\Delta)^{s} $ $ \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, &{\rm in}\ \mathbb{R}^{3}, \\ u\in H^{s}(\mathbb{R}^{3}), \ \end{cases} $ 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 $.

Keywords

• fractional laplace operator
• kirchhoff type problems
• sign-changing
• nehari manifold
• variational methods