  We establish a concentration-compactness principle for the Sobolev space
$W^{2,p(\cdot)}(\Omega)\cap W_0^{1,p(\cdot)}(\Omega)$ that is a tool for
overcoming the lack of compactness of the critical Sobolev imbedding. Using
this result we obtain several existence and multiplicity results for a class of
Kirchhoff type problems involving $p(\cdot)$-biharmonic operator and critical
growth.
