Existence of solutions for Kirchhoff-type systems with critical Sobolev exponents in ${\mathbb{R}}^3$

In this paper, we study the following Kirchhoff-type system:

(0.1) $$\{\begin{array}{l}-(a_1+b_1{\int }_{{\mathbb{R}}^3}|\mathrm{\nabla }u{|}^{2}dx)\mathrm{\Delta }u=\frac{2\alpha }{\alpha +\beta }|u{|}^{\alpha -2}u|v{|}^{\beta }+\epsilon f(x),\\ -(a_2+b_2{\int }_{{\mathbb{R}}^3}|\mathrm{\nabla }v{|}^{2}dx)\mathrm{\Delta }v=\frac{2\beta }{\alpha +\beta }|u{|}^{\alpha }|v{|}^{\beta -2}v+\epsilon g(x),\\ (u,v)\in D^{1,2}({\mathbb{R}}^3)\times D^{1,2}({\mathbb{R}}^3),\end{array}$$

where $a_1,a_2\ge 0,b_1,b_2>0,\alpha ,\beta >1,\alpha +\beta =6$ and $f(x),g(x)\ge 0,f(x),g(x)\in L^{\frac{6}{5}}({\mathbb{R}}^3).$ The aim of this paper is to demonstrate the existence of at least two solutions for system (0.1), utilizing the variational method. To achieve this, we construct an energy functional and analyze its critical points by applying the Ekeland variational principle, the mountain pass lemma and the concentration compactness principle.