Groundstates for Kirchhoff-Type Equations with Hartree-Type Nonlinearities

Abstract

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.