Positive Solutions for the Kirchhoff-Type Equation with Hartree Nonlinearities

Abstract

In this paper, we consider the following Kirchhoff-type equation:

$$\begin{aligned} \left\{ \begin{array}{ll} \displaystyle -\left( a+b\int _{\mathbb {R}^{N}}|\nabla u|^{2}\mathrm {d}x\right) \Delta u&{}\\ \qquad + \displaystyle \lambda V(x)u+\mu \phi |u|^{p-2}u=|u|^{q-2}u, &{} \text{ in } \mathbb {R}^{N},\\ (-\Delta )^{\frac{\alpha }{2}} \phi =|u|^{p}, &{} \text{ in } \mathbb {R}^{N},\\ \end{array} \right. \end{aligned}$$

where $a,b>0$, $0<\alpha <N$, $N\ge 3$, $2\le p<\frac{N+\alpha }{N-2}$, $2<q<\min \{4, 2^{*}\}$, $2^{*}=2N/(N-2)$ and $\lambda ,~\mu >0$ are parameters. Using the truncation technique and the parameter-dependent compactness lemma, we prove the existence of positive solutions for the above problem when $b,\mu$ are sufficiently small, and $\lambda$ is large enough. Furthermore, the decay rate and the asymptotic behavior of the positive solutions are also explored.