On critical variable-order Kirchhoff type problems with variable singular exponent

We establish a continuous embedding $W^{s(\cdot),2}(\Omega)\hookrightarrow L^{\alpha(\cdot)}(\Omega)$, where the variable exponent $\alpha(x)$ can be close to critical $2_{s}^*(x)=\frac{2N}{N-2\bar{s}(x)}$, with $\bar{s}(x)=s(x,x)$ for all $x\in\bar{\Omega}$. Subsequently, this is used prove multiplicity of solutions nonlocal degenerate Kirchhoff problems singular exponent. Moreover, we also obtain uniform $L^{\infty}$-estimate these infinite by bootstrap argument.