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

We establish a concentration-compactness principle for the Sobolev space $W^{2,p(\cdot )}(\mathrm{\Omega })\cap W_0^{1,p(\cdot )}(\mathrm{\Omega })$ 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.