Existence and concentration of ground state solutions for critical Kirchhoff-type equation involving Hartree-type nonlinearities

Abstract

We consider the following Kirchhoff-type equation of the form

$$\begin{aligned} -\left( a+b\int \limits _{\,\,\,\mathbb {R}^3}|\nabla u|^2\right) \Delta u+(1+\mu g(x))u=\lambda \left( \frac{1}{|x|^{\alpha }}*|u|^p\right) |u|^{p-2}u+|u|^4u,\quad x\in \mathbb {R}^3\end{aligned}$$

where $a>0, b\ge 0$ are constants, $\lambda , \mu$ are positive parameters, $\alpha \in (0,3), p\in \left( 2, 6-\alpha \right)$ and $g\in C(\mathbb {R}^3)$ satisfies some conditions. By the mountain pass theorem, we establish the existence of ground state solutions. Besides, the concentration of ground state solutions is also described as $\mu \rightarrow \infty$.