On a p⋅-biharmonic problem of Kirchhoff type involving critical growth

ABSTRACT

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.

Keywords:

• Variable exponent Lebesgue–Sobolev spaces
• concentration-compactness principle
• $p(\cdot )$-biharmonic operator
• genus theory
• variational methods

2010 Mathematics Subject Classifications:

• 35B33
• 35J20
• 35J60
• 35G30
• 46E35
• 49J35

Acknowledgments

The second author was supported by the University of Economics Ho Chi Minh City, Vietnam.

Disclosure statement

No potential conflict of interest was reported by the author(s).