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