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This paper discusses the existence and multiplicity of solutions for a class of $p (x)$ -Kirchhoff type problems with Dirichlet boundary data of the following form $\}\begin{matrix} - (a + b \int_{Ω} \frac{1}{p (x)} {| \nabla u |}^{p (x)} d x) {div (| \nabla u |}^{p (x) - 2} \nabla u) = f (x, u), & i n Ω \\ u = 0 & o n \partial Ω, \end{matrix}$ where $Ω$ is a smooth open subset of $ℝ^{N}$ and $p \in C (\bar{Ω})$ with $N < p^{-} = {inf}_{x \in Ω} p (x) \leq p^{+} = {sup}_{x \in Ω} p (x) < + \infty$ , $a$ , $b$ are positive constants and $f : \bar{Ω} \times ℝ \to ℝ$ is a continuous function. The proof is based on critical point theory and variable exponent Sobolev space theory.

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