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.

中文翻译：

---

涉及Liouville-Weyl分数阶导数的Kirchhoff分数阶微分方程的解的多重性

摘要

在本文中，通过使用变分方法和临界点理论，我们研究了以下分数基希霍夫型方程的解的存在性：

$$ \ 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} $ $

其中\（\ alpha \ in（\ frac {1} {2}，1] \），\（{} _ {-\ infty} D_ {t} ^ {\ alpha} \）和\（{} _ {吨} d _ {\ infty} ^ {\阿尔法} \）的左和右刘维-外尔的顺序分数衍生物\（\阿尔法\）对整个轴\（\ mathbb {R} \），分别，\（ u \ in \ mathbb {R} \），\（l：\ mathbb {R} \ to \ mathbb {R} \）是连续的，并且具有正的最小值\（f \ in C（\ mathbb {R} \ times \ mathbb {R}，\ mathbb {R}）\）和\（S：\ mathbb {R} ^ {+} \ to \ mathbb {R} ^ {+} \）是一个连续函数。