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This paper deals with a class of p(x)-Kirchhoff Dirichlet problems possessing a vari- ational structure which corresponds to the variational functional E defined on W 1,p(x) 0 (Ω) .W e prove a Brezis-Nirenberg type theorem which asserts that every local minimizer of E in the C 1 (Ω) topology is also a local minimizer of E in the W 1,p(x) 0 (Ω) topology. Some applications of this theorem are given.

A Brezis-Nirenberg type theorem on local minimizers for p(x)-Kirchhoff Dirichlet problems and applications