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

TL;DR

This paper proves the existence of multiple solutions for critical nonlocal Kirchhoff problems with variable singular exponents by establishing a new embedding theorem and using bootstrap methods for uniform bounds.

Contribution

It introduces a novel continuous embedding for variable exponent Sobolev spaces and applies it to demonstrate solution multiplicity in critical Kirchhoff problems.

Findings

Established a continuous embedding $W^{s( imes),2}( abla) ightarrow L^{ ext{variable}}( abla)$ near critical exponents.

Proved the existence of multiple solutions for critical nonlocal Kirchhoff problems with variable singular exponents.

Derived uniform $L^{ ext{infinity}}$ bounds for solutions using bootstrap techniques.

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