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April 2020 Ground state solutions for the periodic fractional Schrödinger–Poisson systems with critical Sobolev exponent
Mingzhu Yu, Haibo Chen, Weihong Xie
Rocky Mountain J. Math. 50(2): 719-732 (April 2020). DOI: 10.1216/rmj.2020.50.719

Abstract

We study the fractional Schrödinger–Poisson system with critical Sobolev exponent

( Δ ) s u + V ( x ) u + ϕ u = f ( x , u ) + K ( x ) | u | 2 s 2 u  in  3 , ( Δ ) t ϕ = u 2  in  3 ,

where ( Δ ) α denotes the fractional Laplacian of order α = s , t ( 0 , 1 ) ; V ( x ) , f ( x , u ) and K ( x ) are 1 -periodic in the x -variables; 2 s = 6 ( 3 2 s ) is the fractional critical Sobolev exponent in dimension 3 . Under some weaker conditions on f , we prove the existence of ground state solutions for such a system via the mountain pass theorem in combination with the concentration-compactness principle. Our results are new even for s = t = 1 .

Citation

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Mingzhu Yu. Haibo Chen. Weihong Xie. "Ground state solutions for the periodic fractional Schrödinger–Poisson systems with critical Sobolev exponent." Rocky Mountain J. Math. 50 (2) 719 - 732, April 2020. https://doi.org/10.1216/rmj.2020.50.719

Information

Received: 26 February 2019; Revised: 8 October 2019; Accepted: 31 October 2019; Published: April 2020
First available in Project Euclid: 29 May 2020

zbMATH: 07210992
MathSciNet: MR4104407
Digital Object Identifier: 10.1216/rmj.2020.50.719

Subjects:
Primary: 35B33 , 35J60

Keywords: critical Sobolev exponent , fractional Schrödinger–Poisson systems , ground state solutions , Mountain pass theorem , periodic

Rights: Copyright © 2020 Rocky Mountain Mathematics Consortium

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Vol.50 • No. 2 • April 2020
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