In this paper, we consider the following fractional Kirchhoff problem with strong singularity: $$ \textstyle\begin{cases} (1+ b\int _{\mathbb{R}^{3}}\int _{\mathbb{R}^{3}} \frac{ \vert u(x)-u(y) \vert ^{2}}{ \vert x-y \vert ^{3+2s}}\,\mathrm{d}x \,\mathrm{d}y )(-\Delta )^{s} u+V(x)u = f(x)u^{-\gamma }, & x \in \mathbb{R}^{3}, \\ u>0,& x\in \mathbb{R}^{3}, \end{cases} $$ where $(-\Delta )^{s}$ is the fractional Laplacian with $0< s<1$ , $b>0$ is a constant, and $\gamma >1$ . Since $\gamma >1$ , the energy functional is not well defined on the work space, which is quite different with the situation of $0<\gamma <1$ and can lead to some new difficulties. Under certain assumptions on V and f, we show the existence and uniqueness of a positive solution $u_{b}$ by using variational methods and the Nehari manifold method. We also give a convergence property of $u_{b}$ as $b\rightarrow 0$ , where b is regarded as a positive parameter.

中文翻译：

---

具有奇异性的分数阶Kirchhoff问题的唯一性和集中性

在本文中，我们考虑以下具有奇异性的分数阶Kirchhoff问题：$$ \ textstyle \ begin {cases（（1+ b \ int _ {\ mathbb {R} ^ {3}} \ int _ {\ mathbb {R } ^ {3}} \ frac {\ vert u（x）-u（y）\ vert ^ {2}} {\ vert xy \ vert ^ {3 + 2s}} \，\ mathrm {d} x \， \ mathrm {d} y）（-\ Delta）^ {s} u + V（x）u = f（x）u ^ {-\ gamma}，＆x \ in \ mathbb {R} ^ {3}， \\ u> 0，＆x \ in \ mathbb {R} ^ {3}，\ end {cases} $$，其中$（-\ Delta）^ {s} $是小数拉普拉斯算子，其中$ 0 <s <1 $ ，$ b> 0 $是常数，而$ \ gamma> 1 $。由于$ \ gamma> 1 $，因此在工作空间上的能量函数定义不明确，这与$ 0 <\ gamma <1 $的情况大不相同，并且可能导致一些新的困难。在关于V和f的某些假设下，我们使用变分方法和Nehari流形方法证明了正解$ u_ {b} $的存在和唯一性。