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

Abstract

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 $2_s^{\ast }(x)=\frac{2N}{N-2\bar{s}(x)}$, 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.