This article is devoted to the following Kirchhoff-type equation with the convolution nonlinearity: \(\begin{aligned} -\left( a+b\int _{{\mathbb R}^2}|\nabla u|^2\textrm{d}x\right) \Delta u+V(\varepsilon x)u=\mu (I_\alpha *|u|^q)|u|^{q-2}u+f(u)\ \ \text{ in }\ {\mathbb R}^2, \end{aligned}\) where \(\varepsilon , \mu >0\) are parameters, \(a>0, b\geqslant 0\) , \(q>(6+\alpha )/2\) , \(I_\alpha \) is the Riesz potential, \(0<\alpha <2\) , \(V\in \mathcal {C}({\mathbb R}^2,{\mathbb R})\) and \(f\in \mathcal {C}({\mathbb R}, {\mathbb R})\) . By applying variational methods, we obtain the existence of positive ground state solutions for the above equation, which concentrates at a global minimum of V in the semi-classical limit as \(\varepsilon \rightarrow 0\) . Additionally, we demonstrate that this solution fulfills the property of exponential decay. At last, we use Ljusternik-Schnirelmann theory and Morse theory to investigate the multiplicity of solutions for the above problem.
