Electronic Journal of Differential Equations, 
Vol. 2021 (2021), No. 14, pp. 1-31.

Title: Existence and concentration results for fractional Schrodinger-Poisson system
       via penalization method
       
Authors: Zhipeng Yang (Yunnan Normal Univ., Kunming, China)
         Wei Zhang (Yunnan Univ. of Finance and Economics, Kunming, China)
         Fukun Zhao (Yunnan Normal Univ., Kunming, China)

Abstract:
 This article concerns the positive solutions for the fractional Schrodinger-Poisson system
 $$\displaylines{ 
 \varepsilon^{2s}(-\Delta)^su+V(x)u+\phi u=f(u)\quad \text{in }\mathbb{R}^3,\cr
 \varepsilon^{2t}(-\Delta)^t\phi=u^2\quad \text{in }\mathbb{R}^3,
 }$$
 where $\varepsilon>0$ is a small parameter, $(-\Delta)^\alpha$ denotes the fractional
 Laplacian of orders $\alpha=s,t\in(3/4,1)$, $V\in C(\mathbb{R}^3,\mathbb{R})$ is
 the potential function and $f:\mathbb{R}\to\mathbb{R}$ is continuous and
 subcritical. Under a local condition imposed on the potential function, we relate the
 number of positive solutions with the topology of the set where the potential attains
 its minimum values. Moreover, we considered some properties of these positive solutions,
 such as concentration behavior and decay estimate. In the proofs we apply variational methods,
 penalization techniques and Ljusternik-Schnirelmann theory.

Submitted October 5, 2018. Published March 16, 2021.
Math Subject Classifications: 49J35, 58E05.
Key Words: Penalization method; fractional Schrodinger-Poisson;
           Lusternik-Schnirelmann theory.