Ground states for Kirchhoff equation with Hartree-type nonlinearities

In this paper, we consider the following nonlinear problem of Kirchhoff-type with Hartree-type nonlinearities: $$\begin{aligned} \left\{ \begin{array}{ll} -\left( a+b\int _{\mathbb {R}^N}|Du|^2\right) \Delta u+V(x)u=(I_{\alpha }*|u|^{p})|u|^{p-2}u,&{}\quad x\in \mathbb {R}^N,\\ \\ u\in H^1(\mathbb {R}^N),\quad u>0,&{}\quad x\in \mathbb {R}^N, \end{array}\right. \end{aligned}$$ where \(N\ge 3\), \(\max \{0,N-4\} 0,b\ge 0\) are constants, \(I_{\alpha }\) is the Riesz potential and \(V{:}\,\mathbb {R}^N\rightarrow \mathbb {R}\) is a potential function. Under certain assumptions on V, we prove that the problem has a positive ground state solution by using global compactness lemma, monotonicity technique and some new tricks recently given in the literature.