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

TL;DR

This paper develops a concentration-compactness principle for variable exponent Sobolev spaces and applies it to prove existence and multiplicity of solutions for Kirchhoff type problems with critical growth involving a p()-biharmonic operator.

Contribution

It introduces a new concentration-compactness principle for variable exponent Sobolev spaces and applies it to Kirchhoff problems with critical growth, addressing compactness issues.

Findings

Established a concentration-compactness principle for variable exponent Sobolev spaces.

Proved existence of solutions for Kirchhoff type problems with critical growth.

Demonstrated multiplicity results for the same class of problems.

Abstract

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.