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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 forovercoming the lack of compactness of the critical Sobolev imbedding. Usingthis result we obtain several existence and multiplicity results for a class ofKirchhoff type problems involving $p(\cdot)$-biharmonic operator and criticalgrowth.

On a $p(\cdot)$-biharmonic problem of Kirchhoff type involving critical growth