Groundstates for Kirchhoff-Type Equations with Hartree-Type Nonlinearities

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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 $$N\ge 3$$ , $$\max \{0,N-4\}<\alpha <N$$ , $$2<p<\frac{N+\alpha }{N-2}$$ , $$a>0,b\ge 0$$ 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.