Ground states to a Kirchhoff equation with fractional Laplacian

Dengfeng Lu; Shuwei Dai; Dengfeng Lu; Shuwei Dai

doi:10.3934/math.20231248

AIMS Mathematics

2023, Volume 8, Issue 10: 24473-24483. doi: 10.3934/math.20231248

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Research article

Ground states to a Kirchhoff equation with fractional Laplacian

Dengfeng Lu ¹,
Shuwei Dai ^{2
,
,}

1.
School of Mathematics and Statistics, Hubei Engineering University, Xiaogan 432000, China
2.
College of Technology, Hubei Engineering University, Xiaogan 432000, China

Received: 28 March 2023 Revised: 16 May 2023 Accepted: 29 May 2023 Published: 18 August 2023
MSC : 35J60, 35B40, 35R11

The aim of this paper is to deal with the Kirchhoff type equation involving fractional Laplacian operator

$ \left(\alpha+\beta \int_{\mathbb{R}^{3}}|(-\Delta)^{\frac{s}{2}}\psi|^{2}\,\mathrm{d} x\right)(-\Delta)^{s}\psi+\kappa \psi = |\psi|^{p-2}\psi \ \ \ \mbox{in} \ \mathbb{R}^{3}, $

where $ \alpha, \beta, \kappa > 0 $ are constants. By constructing a Palais-Smale-Pohozaev sequence at the minimax value $ c_{mp} $, the existence of ground state solutions to this equation for all $ p\in(2, 2_{s}^{*}) $ is established by variational arguments. Furthermore, the decay property of the ground state solution is also investigated.
- fractional Kirchhoff equation,
- ground state solution,
- (PSP)-sequence,
- variational methods
Citation: Dengfeng Lu, Shuwei Dai. Ground states to a Kirchhoff equation with fractional Laplacian[J]. AIMS Mathematics, 2023, 8(10): 24473-24483. doi: 10.3934/math.20231248

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Abstract

The aim of this paper is to deal with the Kirchhoff type equation involving fractional Laplacian operator

$ \left(\alpha+\beta \int_{\mathbb{R}^{3}}|(-\Delta)^{\frac{s}{2}}\psi|^{2}\,\mathrm{d} x\right)(-\Delta)^{s}\psi+\kappa \psi = |\psi|^{p-2}\psi \ \ \ \mbox{in} \ \mathbb{R}^{3}, $

where $ \alpha, \beta, \kappa > 0 $ are constants. By constructing a Palais-Smale-Pohozaev sequence at the minimax value $ c_{mp} $, the existence of ground state solutions to this equation for all $ p\in(2, 2_{s}^{*}) $ is established by variational arguments. Furthermore, the decay property of the ground state solution is also investigated.

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