The purpose of the present paper is to study the existence of solutions for the following nonhomogeneous singular Kirchhoff problem involving the p(x)\documentclass[12pt]{minimal}
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\begin{document}$p(x)$\end{document}-biharmonic operator:
{M(t)(Δp(x)2u+a(x)|u|p(x)−2u)=g(x)u−γ(x)∓λf(x,u),in Ω,Δu=u=0,on ∂Ω,\documentclass[12pt]{minimal}
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\begin{document} $$\begin{aligned} \left \{ \textstyle\begin{array}{l} M(t)\Big(\Delta ^{2}_{p(x)}u+a(x)|u|^{p(x)-2}u\Big) =g(x)u^{-\gamma (x)} \mp \lambda f(x,u),\quad \mbox{in }\Omega , \\ \Delta u=u=0, \quad \mbox{on }\partial \Omega , \end{array}\displaystyle \right . \end{aligned}$$ \end{document} where Ω⊂RN\documentclass[12pt]{minimal}
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\begin{document}$\Omega \subset {\mathbb{R}}^{N}$\end{document}, (N≥3)\documentclass[12pt]{minimal}
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\begin{document}$(N\geq 3)$\end{document} be a bounded domain with C2\documentclass[12pt]{minimal}
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\begin{document}$C^{2}$\end{document} boundary, λ\documentclass[12pt]{minimal}
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\begin{document}$\lambda $\end{document} is a positive parameter, γ:Ω‾⟶(0,1)\documentclass[12pt]{minimal}
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\begin{document}$\gamma : \overline{\Omega }\longrightarrow (0,1)$\end{document} be a continuous function, p∈C(Ω‾)\documentclass[12pt]{minimal}
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\begin{document}$p\in C(\overline{\Omega })$\end{document} with 1<p−:=infx∈Ωp(x)≤p+:=supx∈Ωp(x)<N2\documentclass[12pt]{minimal}
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\begin{document}$\displaystyle 1< p^{-}:=\inf _{x\in \Omega }p(x)\leq p^{+}:=\sup _{x \in \Omega }p(x)<\frac{N}{2}$\end{document}, as usual, p∗(x)=Np(x)N−2p(x)\documentclass[12pt]{minimal}
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\begin{document}$p^{*}(x)=\displaystyle \frac{N p(x)}{N-2p(x)}$\end{document}, g∈Lp∗(x)p∗(x)+γ(x)−1(Ω)\documentclass[12pt]{minimal}
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\begin{document}$g \in L^{\frac{p^{*}(x)}{p^{*}(x)+\gamma (x)-1}}(\Omega )$\end{document}. We assume that M(t)\documentclass[12pt]{minimal}
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\begin{document}$M(t)$\end{document} is a continuous function with
t:=∫Ω1p(x)(|Δu|p(x)+a(x)|u|p(x))dx,\documentclass[12pt]{minimal}
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\begin{document}$$ t:=\int _{\Omega }\frac{1}{p(x)}(|\Delta u|^{p(x)}+a(x)|u|^{p(x)})dx, $$\end{document} and assumed to verify assertions (M1)-(M3) in Sect. 3, moreover f(x,u)\documentclass[12pt]{minimal}
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\begin{document}$f(x,u)$\end{document} are assumed to satisfy assumptions (f1)-(f6). In the proofs of our results we use variational techniques and monotonicity arguments combined with the theory of the generalized Lebesgue Sobolev spaces.
