Infinitely many sign-changing solutions for nonlinear fractional Kirchhoff equations

Abstract

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.

Keywords:

• Fractional Kirchhoff equation
• sign-changing solutions
• variational methods

2010 Mathematics Subject Classifications:

• 35R11
• 35A15
• 47G20

Acknowledgments

The authors would like to thank the anonymous referee for his or her careful readings of the paper and many helpful comments. The research was done when G. Gu visited Department of Mathematics, University of Texas at San Antonio under the support of China Scholarship Council, and the first author thanks professor Changfeng Gui for his invitation and Department of Mathematics, University of Texas at San Antonio for their support and kind hospitality.

Disclosure statement

No potential conflict of interest was reported by the author(s).

Additional information

Funding

This work was supported by National Natural Science Foundation of China (11771385).