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.