|Publication ahead of print|
|Published online||08 August 2017|
Multiplicity and concentration of positive solutions for the fractional Schrödinger–Poisson systems with critical growth
1 School of Mathematics and Physics, University of South China, Hengyang, Hunan 421001, P.R. China.
2 College of Mathematics and Statistics, Chongqing Jiaotong University, Chongqing 400074, P.R. China.
3 Chern Institute of Mathematics, Nankai University, Tianjin 300071, P.R. China.
Received: 5 February 2016
Revised: 2 August 2016
Accepted: 13 September 2016
In this paper, we study the multiplicity and concentration of solutions for the following critical fractional Schrödinger–Poisson system: where ϵ> 0 is a small parameter, (− △ )α denotes the fractional Laplacian of order α = s,t ∈ (0,1), where 2α is the fractional critical exponent in Dimension 3; V ∈ C1(ℝ3,ℝ+) and f is subcritical. We first prove that for ϵ> 0 sufficiently small, the system has a positive ground state solution. With minimax theorems and Ljusternik–Schnirelmann theory, we investigate the relation between the number of positive solutions and the topology of the set where V attains its minimum for small ϵ. Moreover, each positive solution uϵ converges to the least energy solution of the associated limit problem and concentrates around a global minimum point of V.
Mathematics Subject Classification: 35B25 / 35B38 / 35J65
Key words: Fractional Schrödinger–Poisson system / positive solution / critical growth / variational method
© EDP Sciences, SMAI 2017
Current usage metrics show cumulative count of Article Views (full-text article views including HTML views, PDF and ePub downloads, according to the available data) and Abstracts Views on Vision4Press platform.
Data correspond to usage on the plateform after 2015. The current usage metrics is available 48-96 hours after online publication and is updated daily on week days.
Initial download of the metrics may take a while.