Mengyun Zhou, Yongyi Lan, Existence of radial sign-changing solutions for fractional Kirchhoff-type problems in $R^3$

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DOI: 10.23952/jnva.7.2023.2.04

Volume 7, Issue 2, 1 April 2023, Pages 223-234

Abstract. In this paper, the following fractional Kirchhoff-type problem
$\Bigg(a + b\int_{\mathbb{R}^{3}}|(-\Delta)^{\frac{s}{2}} u|^{2} \mbox{d}x\Bigg)(-\Delta)^{s}u + V(x) u = f(x,u),\,\,\,\,x\in \mathbb{R}^{3},$
where $a,b > 0$ are constants, $s \in (\frac{3}{4},1)$ , $2_{s}^{\ast} = \frac{6}{3-2s}$ , $V : \mathbb{R}^{3} \rightarrow \mathbb{R}$ is a continuous function, and $f: \mathbb{R}^{3}\times \mathbb{R} \rightarrow \mathbb{R}$ is a continuous function, is considered. It is demonstrated that the fractional Kirchhoff-type equation has a radial sign-changing solution $u_{b}$ and a radial solution $\overline{u}_{b}$ when $f$ does not satisfy the subcritical growth condition and the usual Nehari-type monotonicity condition. The main tools are the constraint variational method and some analysis techniques.

How to Cite this Article:
M. Zhou, Y. Lan, Existence of radial sign-changing solutions for fractional Kirchhoff-type problems in $\mathbb{R}^{3}$ , J. Nonlinear Var. Anal. 7 (2023), 223-234.