Existence and concentration results for fractional Schrödinger-Poisson system via penalization method

Electron. J. Differential Equations 2021, Paper No. 14, 31 pp

In this paper, we are concerned with positive solutions for the fractional Schrödinger-Poisson system : \begin{equation*} \varepsilon^{2s}(-\Delta)^su+V(x)u+\phi u=f(u), \end{equation*} \begin{equation*} \varepsilon^{2t}(-\Delta)^t\phi=u^2, \end{equation*} where $\varepsilon>0$ is a small parameter, $(-\Delta)^\alpha$ denotes the fractional Laplacian of order $\alpha=s,t\in(\frac{3}{4},1)$, $V\in C(\mathbb{R}^3,\mathbb{R})$ is the potential function and $f:\mathbb{R}\rightarrow\mathbb{R}$ 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.