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October 2020 On critical fractional systems with Hardy–Littlewood–Sobolev nonlinearities
Qianyu Hong, Yang Yang
Rocky Mountain J. Math. 50(5): 1661-1683 (October 2020). DOI: 10.1216/rmj.2020.50.1661

Abstract

In this paper, we study the critical fractional Laplacian system with Choquard type nonlinearities

( Δ ) s u = k 1 ( N | u ( y ) | 2 μ | x y | μ d y ) | u | 2 μ 2 u + α γ 2 μ ( N | v ( y ) | β | x y | μ d y ) | u | α 2 u  in  N , ( Δ ) s v = k 2 ( N | v ( y ) | 2 μ | x y | μ d y ) | v | 2 μ 2 v + β γ 2 μ ( N | u ( y ) | α | x y | μ d y ) | v | β 2 v  in  N , u , v D s ( N ) ,

where (Δ)s is the well known fractional Laplacian operator, s(0,1), N>2s, k1,k2>0, γ0, μ(0,N), α,β>1 and 2μ=(2Nμ)(N2s) is the fractional upper critical exponent in the Hardy–Littlewood–Sobolev inequality, Ds(N) is the Hilbert space as the completion of Cc(N). We show the existence and nonexistence results for the critical nonlocal system. These extend results in the literature for the case s=1 and α=β=2μ, where 2μ=(2Nμ)(N2) is the critical exponent in the sense of the Hardy–Littlewood–Sobolev inequality. Particularly, we establish new existence and multiplicity of positive solutions for the system when α+β=22μ, whose key is providing an accurate upper bound of the least energy.

Citation

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Qianyu Hong. Yang Yang. "On critical fractional systems with Hardy–Littlewood–Sobolev nonlinearities." Rocky Mountain J. Math. 50 (5) 1661 - 1683, October 2020. https://doi.org/10.1216/rmj.2020.50.1661

Information

Received: 7 August 2019; Revised: 31 January 2020; Accepted: 3 February 2020; Published: October 2020
First available in Project Euclid: 5 November 2020

zbMATH: 07274827
MathSciNet: MR4170679
Digital Object Identifier: 10.1216/rmj.2020.50.1661

Subjects:
Primary: 35A15, 35R09, 35R11

Rights: Copyright © 2020 Rocky Mountain Mathematics Consortium

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Vol.50 • No. 5 • October 2020
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