In this paper, by using the concentration–compactness principle of Lions for variable exponents found in \cite{Bonder} and the Mountain Pass Theorem without the Palais-Smale condition given in \cite{Rabinowitz}, we obtain the existence and multiplicity solutions $ u=(u_1,u_2,….u_n)$, for a class of Kirchhoff-Type Potential Systems with critical exponent, namely \begin{eqnarray*} \begin{cases} -M_i\Big(\mathcal{A}_i(u_i)\Big) \textrm{div}\,\Big( \mathcal{B}_i(\nabla u_i)\Big)=|u_i|^{s_i (x)-2}u_i +\lambda F_{u_i}(x,u) & \text{in }\Omega, \\ u=0 & \text{on }\partial\Omega; \end{cases} \end{eqnarray*} \\ where $\Omega$ is a bounded smooth domain in $\mathbb{R}^N (N\geq 2)$, and $$\mathcal{B}_i(\nabla u_i)=a_i\Big(|\nabla u_i|^{p_i(x)}\Big) |\nabla u_i|^{p_i(x)-2} \nabla u_i . $$ The functions $M_i$, $ \mathcal{A}_i$, $a_i$ and $a_i$, ($1\leq i \leq n$), are given functions, whose properties will be introduced hereafter, $\lambda$ is positive parameter, and the real function $F$ belongs to $C^1(\Omega \times \mathbb{R}^{n})$, $F_{u_i}$ denotes the partial derivative of $F$ with respect to $u_i$. Our result extend, complement and complete in several ways some of many works in particular \cite{chems1}. 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 $x\in \overline{\Omega}$.
