AIMS Mathematics, 2020, 5(5): 4494-4511. doi: 10.3934/math.2020289

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

Department of Applied Mathematics, Lanzhou University of Technology, Lanzhou, Gansu, 730050, P. R. China

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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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