On a Class of p(z)-Biharmonic Kirchhoff Type Problems with Indefinite Weight and No-Flow Boundary Condition

In this paper we study the existence of weak solutions for a fourth order variable exponent Kirchhoff type problem involving p(z)-biharmonic operator with indefinite weight and no flux boundary condition. The proof of the existence result relies on employing the concept of a Fredholm-type results for a pair of nonlinear operators \((\mathfrak {O},\mathfrak {S})\) $(\mathfrak{O},\mathfrak{S})$ , in conjunction with the theory of variable exponent Sobolev spaces.