Multiplicity of Solutions for Kirchhoff Fractional Differential Equations Involving the Liouville-Weyl Fractional Derivatives

Abstract

In this paper, by using variational methods and critical point theory we investigate the existence of solutions for the following fractional Kirchhoff-type equation:

$$\begin{cases}S\left(\int\limits_{\mathbb{R}}{|{}_{-\infty}D_{t}^{\alpha}u(t)|}^{2}dt\right){}_{t}D_{\infty}^{\alpha}({}_{-\infty}D_{t}^{\alpha}u(t))+l(t)u(t)=f(t,u(t)),\quad t\in\mathbb{R},\\ u\in H^{\alpha}(\mathbb{R}),\end{cases}$$

where $\alpha\in(\frac{1}{2},1]$, ${}_{-\infty}D_{t}^{\alpha}$ and ${}_{t}D_{\infty}^{\alpha}$ are the left and right Liouville-Weyl fractional derivatives of order $\alpha$ on the whole axis $\mathbb{R}$, respectively, $u\in\mathbb{R}$, $l:\mathbb{R}\to\mathbb{R}$ is continuous and has a positive minimum, $f\in C(\mathbb{R}\times\mathbb{R},\mathbb{R})$, and $S:\mathbb{R}^{+}\to\mathbb{R}^{+}$ is a continuous function.