On Fractional Kirchhoff Problems with Liouville–Weyl Fractional Derivatives

Abstract

In this paper, we study the following fractional Kirchhoff-type problem with Liouville–Weyl fractional derivatives:

$$\begin{cases}\left[a+b\left(\int\limits_{\mathbb{R}}(|u|^{2}+|{{}_{-\infty}}D_{x}^{\beta}u|^{2})dx\right)^{\varrho-1}\right]({{}_{x}}D_{\infty}^{\beta}({{}_{-\infty}}D_{x}^{\beta}u)+u)=|u|^{2^{*}_{\beta}-2}u,in~\mathbb{R},\\ u\in\mathbb{I}_{-}^{\beta}(\mathbb{R}),\end{cases}$$

where \(\beta\in(0,\frac{1}{2})\), \(\varrho>1\), \({{}_{-\infty}}D_{x}^{\beta}u(\cdot)\), and \({{}_{x}}D_{\infty}^{\beta}u(\cdot)\) denote the left and right Liouville–Weyl fractional derivatives, \(2_{\beta}^{*}=\frac{2}{1-2\beta}\) is fractional critical Sobolev exponent \(a\geq 0\) and \(b>0\). Under suitable values of the parameters \(\varrho\), \(a\) and \(b\), we obtain a nonexistence result of nontrivial solutions of infinitely many nontrivial solutions for the above problem.