The aim of this work is to study the existence of weak solutions for a nonhomogeneous singular $p(x)$-Kirchhoff problem of the following form \begin{equation*} (\textbf{P}_{\pm \lambda}) \quad \left\{ \begin{aligned} M(t)\Delta(|\Delta u|^{p(x)-2}\Delta u) &=a(x) u^{-\gamma (x)}\pm \lambda u^{q(x)-2}u, &\ &\mbox{in }\Omega, \\ \Delta u&=u=0, & \ &\mbox{on }\partial\Omega, \end{aligned} \right. \end{equation*} by using variational techniques and monotonicity arguments combined with the theory of the generalized Lebesgue Sobolev spaces.

中文翻译：

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包含奇异非线性和Navier边界条件的Kirchhoff $ p（x）$-双调和问题

这项工作的目的是研究以下形式\ begin {equation *}（\ textbf {P} _ {\ pm \ lambda}）的非齐奇奇异$ p（x）$-Kirchhoff问题的弱解的存在\ quad \ left \ {\ begin {aligned} M（t）\ Delta（| \ Delta u | ^ {p（x）-2} \ Delta u）＆= a（x）u ^ {-\ gamma（x ）} \ pm \ lambda u ^ {q（x）-2} u，＆\＆\ mbox {in} \ Omega，\\ \ Delta u＆= u = 0，＆\＆\ mbox {on} \ partial \欧米茄，\ end {aligned} \ right。\ end {equation *}通过使用变分技术和单调论证与广义Lebesgue Sobolev空间的理论相结合。