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

Jin-Long Zhang; Da-Bin Wang; Jin-Long Zhang; Da-Bin Wang

doi:10.3934/math.2020289

AIMS Mathematics

2020, Volume 5, Issue 5: 4494-4511. doi: 10.3934/math.2020289

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Research article

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

Jin-Long Zhang ,
Da-Bin Wang ^,

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

Received: 31 March 2020 Accepted: 10 May 2020 Published: 20 May 2020
MSC : 35J60, 35J20

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.
- nonlocal term,
- variation methods,
- nodal solutions
Citation: Jin-Long Zhang, Da-Bin Wang. Existence of least energy nodal solution for Kirchhoff-type system with Hartree-type nonlinearity[J]. AIMS Mathematics, 2020, 5(5): 4494-4511. doi: 10.3934/math.2020289

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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.

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