In this paper, by using the concentration-compactness principle of Lions for variable exponents found in [Bonder JF, Silva A. Concentration-compactness principal for variable exponent space and applications. Electron J Differ Equ. 2010;141:1–18.] and the Mountain Pass Theorem without the Palais–Smale condition given in [Rabinowitz PH. Minimax Methods in Critical Point Theory with Applications to Differential Equations, CBMS Reg. Conf. Ser. Math., Vol. 65, Amer. Math. Soc., Providence, RI, 1986.], we obtain the existence and multiplicity solutions $u=(u_1,u_2,\dots .u_n)$, for a class of Kirchhoff-Type Potential Systems with critical exponent, namely $\begin{array}{r}\{\begin{array}{ll}-M_i({\mathcal{A}}_i(u_i))\mathrm{div}({\mathcal{B}}_i(\mathrm{\nabla }{u}_i))=|u_i{|}^{s_i(x)-2}{u}_i+\lambda F_{u_i}(x,u)& \mathrm{in}\mathrm{\Omega },\\ u=0& \mathrm{on}\mathrm{\partial }\mathrm{\Omega },\end{array}\end{array}$ where Ω is a bounded smooth domain in ${\mathbb{R}}^N(N\ge 2)$, and ${\mathcal{B}}_i(\mathrm{\nabla }{u}_i)=a_i(|\mathrm{\nabla }{u}_i{|}^{p_i(x)})|\mathrm{\nabla }{u}_i{|}^{p_i(x)-2}\mathrm{\nabla }{u}_i.$ The functions $M_i$, ${\mathcal{A}}_i$, $a_i$ and $a_i$ ($1\le i\le n$) are given functions, whose properties will be introduced hereafter, λ is the positive parameter, and the real function F belongs to $C^1(\mathrm{\Omega }\times {\mathbb{R}}^n)$, $F_{u_i}$ denotes the partial derivative of F with respect to $u_i$. Our results extend, complement and complete in several ways some of many works in particular [Chems Eddine N. Existence of solutions for a critical (p1(x), . . . , pn(x))-Kirchhoff-type potential systems. Appl Anal. 2020.]. We want to emphasize that a difference of some previous research is that the conditions on $a_i(.)$ are general enough to incorporate some differential operators of great interest. In particular, we can cover a general class of nonlocal operators for $p_i(x)>1$ for all $x\in \overline{\mathrm{\Omega }}$.

在本文中，通过使用在 [Bonder JF, Silva A. Concentration-compactness principal for variable exponent space and applications. 电子 J 不同方程。2010;141:1–18.] 和没有 Palais-Smale 条件的山口定理 [Rabinowitz PH. 临界点理论中的极小极大方法及其在微分方程中的应用，CBMS Reg。会议。系列 数学，卷。65 岁，阿梅尔。数学。Soc., Providence, RI, 1986.]，我们获得了存在解和多重解$你=({你}_{\mathrm{1个}},{你}_{\mathrm{2个}},\dots \dots .{你}_n)$，对于一类具有临界指数的基尔霍夫型势系统，即$\begin{array}{r}\{\begin{array}{ll}-{米}_{我}({\mathcal{A}}_{我}({你}_{我}))\mathrm{分区}({\mathcal{乙}}_{我}(\mathrm{\nabla }{你}_{我}))=|{你}_{我}{|}^{{秒}_{我}(X)-\mathrm{2个}}{你}_{我}+\lambda F_{{你}_{我}}(X,你)& \mathrm{在}\mathrm{欧姆},\\ 你=0& \mathrm{在}\mathrm{\partial }\mathrm{欧姆},\end{array}\end{array}$其中 Ω 是一个有界平滑域${\mathbb{R}}^{否}(否\ge \mathrm{2个})$， 和${\mathcal{乙}}_{我}(\mathrm{\nabla }{你}_{我})=A_{我}(|\mathrm{\nabla }{你}_{我}{|}^{p_{我}(X)})|\mathrm{\nabla }{你}_{我}{|}^{p_{我}(X)-\mathrm{2个}}\mathrm{\nabla }{你}_{我}.$功能${米}_{我}$,${\mathcal{A}}_{我}$,$A_{我}$和$A_{我}$($\mathrm{1个}\le 我\le n$)为给定函数，其性质在后面介绍，λ为正参数，实函数F属于$C^{\mathrm{1个}}(\mathrm{欧姆}\times {\mathbb{R}}^n)$,$F_{{你}_{我}}$表示F关于的偏导数${你}_{我}$. 我们的结果以多种方式扩展、补充和完善了许多工作中的一些，特别是 [Chems Eddine N. Existence of solutions for a critical (p1(x), . . . . , pn(x))-Kirchhoff-type potential systems。应用肛门。2020.]。我们想强调的是，与之前的一些研究不同的是，条件$A_{我}(.)$足够通用，可以合并一些非常有趣的微分算子。特别是，我们可以涵盖一类通用的非局部运算符$p_{我}(X)>\mathrm{1个}$对全部$X\in \overline{\mathrm{欧姆}}$.