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.
liuzhisu183@sina.com
² College of Mathematics and Statistics, Chongqing Jiaotong University, Chongqing 400074, P.R. China.
zhangjianjun09@tsinghua.org.cn
³ Chern Institute of Mathematics, Nankai University, Tianjin 300071, P.R. China.

Received: 5 February 2016
Revised: 2 August 2016
Accepted: 13 September 2016

Abstract

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 2_α^∗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 u_ϵ converges to the least energy solution of the associated limit problem and concentrates around a global minimum point of V.