Volume 23, Number 4, October-December 2017
|Page(s)||1515 - 1542|
|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∗6/3−2α 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
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