On a critical fourth order Leray–Lions \(p(\cdot )\)-Kirchhoff type problem with no-flux boundary condition

Using variational methods combined with the genus theory, we investigate in this paper, the existence of nontrivial weak solutions for a class of fourth order critical \(p(\cdot )\) $p(·)$ -Kirchhoff type problem involving the Leray–Lions type operators with indefinite weight and no flux boundary condition. Accurately, we show the existence of at least k pairs of nontrivial weak solutions for the given problem.