Multiplicity and concentration of positive solutions for the fractional Schrödinger–Poisson systems with critical growth

Liu, Zhisu¹ ; Zhang, Jianjun^2, 3

¹School of Mathematics and Physics, University of South China, Hengyang, Hunan 421001, P.R. China.
²College of Mathematics and Statistics, Chongqing Jiaotong University, Chongqing 400074, P.R. China.
³Chern Institute of Mathematics, Nankai University, Tianjin 300071, P.R. China.

ESAIM: Control, Optimisation and Calculus of Variations, Tome 23 (2017) no. 4, pp. 1515-1542

Résumé

In this paper, we study the multiplicity and concentration of solutions for the following critical fractional Schrödinger–Poisson system:

$$\begin{array}{c}\hfill \left\{\begin{array}{cc}{ϵ}^{2s}{(-▵)}^{s}u+V\left(x\right)u+\varphi u=f\left(u\right)+{\left|u\right|}^{2_s^{*}-2}u\hfill & \text{in}{ℝ}^3,\hfill \\ {ϵ}^{2t}{(-▵)}^t\varphi =u^2\hfill & \text{in}{ℝ}^3,\hfill \end{array}\right.\end{array}$$

where $ϵ>0$ is a small parameter, ${(-△)}^{\alpha }$ denotes the fractional Laplacian of order $\alpha =s,t\in (0,1)$, where 2${}_{\alpha }^{*}$6/3−2α is the fractional critical exponent in Dimension $3$; $V\in C^1({ℝ}^3,{ℝ}^{+})$ 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 $V$.

Détail

DOI : 10.1051/cocv/2016063

Classification : 35B25, 35B38, 35J65
Keywords: Fractional Schrödinger–Poisson system, positive solution, critical growth, variational method