Existence of least energy nodal solution for Kirchhoff-type system with Hartree-type nonlinearity

Abstract

This paper deals with following Kirchhoff-type system with critical growth $ \begin{cases} -(a+ b\int _{\mathbb{R}^3}|\nabla u|^{2}dx)\Delta u+ V(x)u+\phi|u|^{p-2}u = |u|^{4}u+\mu f(u), ~ x\in\mathbb{R}^3, (-\Delta)^{\alpha/2}\phi = l|u|^p, ~ x\in \mathbb{R}^3, \end{cases} $ where $a, \mu \gt 0$, $b, l\geq0$, $\alpha\in(0, 3)$, $p\in[2, 3)$ and $\phi|u|^{p-2}u$ is a Hartree-type nonlinearity. By the minimization argument on the nodal Nehari manifold and the quantitative deformation lemma, we prove that the above system has a least energy nodal solution. Our result improve and generalize some interesting results which were obtained in subcritical case.