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

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

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