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

Jin-Long Zhang Da-Bin Wang

*Corresponding author: Da-Bin Wang wangdb96@163.com

Math2020,5,4494doi:10.3934/math.2020289

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 $a, \mu>0$, $b, l\geq0$, $\alpha\in(0,3)$, $p\in[2,3)$ 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.

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