In this paper, we investigate the following fractional p-Kirchhoff type problem

where \([u]_{s,p}^{p}=\displaystyle \iint _{{\mathbb {R}}^{2N}} \frac{|u(x) - u(y)|^{p}}{|x - y|^{N+ps}}\, dxdy\), \(\Omega \) is a bounded smooth domain of \({\mathbb {R}}^N\) containing 0 with Lipschitz boundary, \((-\Delta )_{p}^{s}\) denotes the fractional p-Laplacian, \(0\le \alpha<ps<N\) with \(s\in (0,1)\), \(p>1\), \(a\ge 0\), \(b>0\), \(1<\theta \le p_\alpha ^*/ p\), \(p_\alpha ^*=\frac{(N-\alpha )p}{N-ps}\) is the fractional critical Hardy-Sobolev exponent, \({\mathcal {I}}_\mu (x)=|x|^{-\mu }\) is the Riesz potential of order \(\mu \in (0,\min \{N,2ps\})\), \(q\in (1, Np/(N-ps))\) satisfies some restrictions. By the concentration-compactness principle and mountain pass theorem, we obtain the existence of a positive weak solution for the above problem as q satisfies suitable ranges.

在本文中，我们研究以下分数p -Kirchhoff 类型问题

其中\([u]_{s,p}^{p}=\displaystyle \iint _{{\mathbb {R}}^{2N}} \frac{|u(x) - u(y)|^ {p}}{|x - y|^{N+ps}}\, dxdy\) , \(\Omega \)是包含 0的\({\mathbb {R}}^N\)的有界光滑域与 Lipschitz 边界，\((-\Delta )_{p}^{s}\)表示分数p -Laplacian，\(0\le \alpha<ps<N\)与\(s\in (0, 1)\) , \(p>1\) , \(a\ge 0\) , \(b>0\) , \(1<\theta \le p_\alpha ^*/ p\) , \( p_\alpha ^*=\frac{(N-\alpha )p}{N-ps}\)是分数临界 Hardy-Sobolev 指数，\({\mathcal {I}}_\mu (x)=| x|^{-\mu }\)是\(\mu \in (0,\min \{N,2ps\})\)阶的 Riesz 势，\(q\in (1, Np/(N-ps))\)满足一些限制。根据集中紧性原理和山口定理，当q满足合适的范围时，我们得到上述问题的正弱解的存在性。