Multiple solutions for a fractional p-Kirchhoff equation with critical growth and low order perturbations

In this article, we deal with the following fractional $p$-Kirchhoff type equation

$$\{\begin{array}{ll}M\left({\int }_{{\mathbb{R}}^N}{\int }_{{\mathbb{R}}^N}\frac{|u(x)-u(y){|}^p}{|x-y{|}^{N+ps}}dxdy\right)(-\mathrm{\Delta }{)}_p^{s}u=\frac{|u{|}^{p_{\alpha }^{\ast }-2}u}{|x{|}^{\alpha }}+\frac{\lambda }{|x{|}^{\beta }},& \mathrm{i}\mathrm{n}\mathrm{\Omega },\\ u>0,& \mathrm{i}\mathrm{n}\mathrm{\Omega },\\ u=0,& \mathrm{i}\mathrm{n}{\mathbb{R}}^{\mathrm{N}}\mathrm{\setminus }\mathrm{\Omega },\end{array}$$

where $\mathrm{\Omega }\subset {\mathbb{R}}^N$ is a smooth bounded domain containing $0$, $(-\mathrm{\Delta }{)}_p^s$ denotes the fractional $p$-Laplacian, $M(t)=a+b{t}^{k-1}$ for $t\ge 0$ and $k>1$, $a,b>0$, $\lambda >0$ is a parameter, $0<s<1$, $0\le \alpha <ps<N$, $\frac{N(p-2)+ps}{p-1}<\beta <\frac{N(p_{\alpha }^{\ast }-1)+\alpha }{p_{\alpha }^{\ast }}$, $1<p<pk<p_{\alpha }^{\ast }=\frac{p(N-\alpha )}{N-ps}$ is the fractional critical Hardy-Sobolev exponent. With aid of the variational method and the concentration compactness principle, we prove the existence of two distinct positive solutions.