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

Abstract

We establish a continuous embedding W s(•),2 (Ω) ֒→ L α(•) (Ω), where the variable exponent α(x) can be close to the critical exponent 2, with s(x) = s(x, x) for all x ∈ Ω. 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 ∞ -estimate of these infinite solutions by a bootstrap argument.