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

Liu, Zhisu; Zhang, Jianjun

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

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 $V$.

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