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 the critical exponent $2_s^ast = frac{2N}{N-2bar{s}(x)}$, 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