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

We establish a continuous embedding $W^{s(\cdot),2}(Ω)\hookrightarrow L^{α(\cdot)}(Ω)$, where the variable exponent $α(x)$ 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 $x\in\barΩ$. 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.

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