Existence results for a class of fourth-order elliptic equations of p(x)-Kirchhoff type

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In this paper, we consider a class of fourth-order elliptic equations of Kirchhoff type with variable exponents { Δ p ( x ) 2 u − M ( ∫ Ω 1 p ( x ) | ∇u | p ( x ) d x ) Δ p ( x ) u + | u | p ( x ) − 2 u = λf ( x , u ) in Ω , u = Δu = 0 on ∂Ω , where \\max \\left \\{1, \\frac {N}{2}\\right \\} $ ]]> p − := inf x ∈ Ω ¯ p ( x ) > max { 1 , N 2 } , λ is a positive parameter, Ω ⊂ R N ( N ≥ 1 ) is a smooth bounded domain, Δ p ( x ) 2 u = Δ ( | Δu | p ( x ) − 2 Δu ) is the operator of fourth order called the p ( x ) -biharmonic operator, Δ p ( x ) u = div ⁡ ( | ∇u | p ( x ) − 2 ∇u ) is the p ( x ) -Laplacian, p : Ω ¯ → R is a log-Hölder continuous function, M : [ 0 , + ∞ ) → R is a continuous function and f : Ω ¯ × R → R is an L 1 -Carathéodory function satisfying some certain conditions. Using variational methods and critical point theory, we prove some existence and multiplicity results for the problem in an appropriate space of functions. Furthermore, we provide two examples to illustrate our main conclusions.