In this paper, we investigate the following fractional p-Kirchhoff type problem a+b[u]s,pp(θ-1)(-Δ)psu=(Iμ∗|u|q)|u|q-2u+|u|pα∗-2u|x|α,u>0,inΩ,u=0,inRN\Ω,\documentclass[12pt]{minimal}
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\begin{document}$$\begin{aligned} \left\{ \begin{array}{ll} \left( a+b[u]_{s,p}^{p(\theta -1)}\right) (-\Delta )^s_pu = \Big ({\mathcal {I}}_\mu *|u|^q\Big )|u|^{q-2}u+\frac{|u|^{p_{\alpha }^*-2}u}{|x|^\alpha },\ u>0, &{}\text{ in }\ \Omega ,\\ u=0, \ &{} \mathrm{in}\ {\mathbb {R}}^N\backslash \Omega , \end{array} \right. \end{aligned}$$\end{document}where [u]s,pp=∬R2N|u(x)-u(y)|p|x-y|N+psdxdy\documentclass[12pt]{minimal}
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\begin{document}$$[u]_{s,p}^{p}=\displaystyle \iint _{{\mathbb {R}}^{2N}} \frac{|u(x) - u(y)|^{p}}{|x - y|^{N+ps}}\, dxdy$$\end{document}, Ω\documentclass[12pt]{minimal}
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\begin{document}$$\Omega $$\end{document} is a bounded smooth domain of RN\documentclass[12pt]{minimal}
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\begin{document}$${\mathbb {R}}^N$$\end{document} containing 0 with Lipschitz boundary, (-Δ)ps\documentclass[12pt]{minimal}
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\begin{document}$$(-\Delta )_{p}^{s}$$\end{document} denotes the fractional p-Laplacian, 0≤α<ps<N\documentclass[12pt]{minimal}
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\begin{document}$$0\le \alpha<ps<N$$\end{document} with s∈(0,1)\documentclass[12pt]{minimal}
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\begin{document}$$s\in (0,1)$$\end{document}, p>1\documentclass[12pt]{minimal}
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\begin{document}$$p>1$$\end{document}, a≥0\documentclass[12pt]{minimal}
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\begin{document}$$a\ge 0$$\end{document}, b>0\documentclass[12pt]{minimal}
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\begin{document}$$b>0$$\end{document}, 1<θ≤pα∗/p\documentclass[12pt]{minimal}
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\begin{document}$$1<\theta \le p_\alpha ^*/ p$$\end{document}, pα∗=(N-α)pN-ps\documentclass[12pt]{minimal}
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\begin{document}$$p_\alpha ^*=\frac{(N-\alpha )p}{N-ps}$$\end{document} is the fractional critical Hardy-Sobolev exponent, Iμ(x)=|x|-μ\documentclass[12pt]{minimal}
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\begin{document}$${\mathcal {I}}_\mu (x)=|x|^{-\mu }$$\end{document} is the Riesz potential of order μ∈(0,min{N,2ps})\documentclass[12pt]{minimal}
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\begin{document}$$\mu \in (0,\min \{N,2ps\})$$\end{document}, q∈(1,Np/(N-ps))\documentclass[12pt]{minimal}
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\begin{document}$$q\in (1, Np/(N-ps))$$\end{document} 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.
