In this paper, we use variational approaches to establish the existence of weak solutions for a class of Kirchhoff-type equations with fractional \(p_{1}(x)\) & \(p_{2}(x)\)-Laplacian operator, for \(1\le p_{1}(x,y)<p_{2}(x,y)\), \(sp_{2}(x,y)<N\) for all \((x,y)\in {\overline{\varOmega }}\times {\overline{\varOmega }}\), and a Carathéodory reaction term which does not satisfy the Ambrosetti–Rabinowitz type growth condition. By mountain pass theorem with Cerami condition and the theory of the fractional variable exponent Sobolev space, we prove the existence of nontrivial solution for the problems in an appropriate space of functions. Furthermore, a multiplicity result of the problem is proved for odd nonlinearity.

在本文中，我们使用变分方法来建立一类带有分数\（p_ {1}（x）\）＆\（p_ {2}（x）\）- Laplacian的Kirchhoff型方程的弱解的存在性运算符，\（1 \文件P_ {1}（X，Y）<P_ {2}（X，Y）\） ，\（以sp_ {2}（X，Y）<N \）对所有\（（ x，y）\ in {\ overline {\ varOmega}} \ times {\ overline {\ varOmega}} \\），以及不满足Ambrosetti-Rabinowitz型生长条件的Carathéodory反应项。通过具有Cerami条件的山口定理和分数变量指数Sobolev空间的理论，我们证明了在适当的函数空间中问题的非平凡解的存在。此外，对于奇数非线性证明了该问题的多重结果。