The concentration behavior of ground state solutions for a critical fractional Schrödinger-Poisson system

Math. Nachr. 292, no. 8, 1837–1868

In this paper, we study the following critical fractional Schrödinger-Poisson system \begin{equation*} \varepsilon^{2s}(-\Delta)^s u +V(x)u+\phi u=P(x)f(u)+Q(x)|u|^{2_s^*-2}u, \end{equation*} \begin{equation*} \varepsilon^{2t}(-\Delta)^t \phi=u^2, \end{equation*} where $\varepsilon>0$ is a small parameter, $s\in (\frac{3}{4},1),t\in(0,1)$ and $2s+2t>3$, $2_s^*:=\frac{6}{3-2s}$ is the fractional critical exponent for 3-dimension, $V(x)\in \mathcal{C}(\mathbb{R}^3)$ has a positive global minimum, and $P(x)$, $Q(x)\in \mathcal{C}(\mathbb{R}^3)$ are two positive and have global maximum. We prove the existence of a positive ground state solution by using variational methods, and we determine a concrete set related to the potentials $V,P$ and $Q$ as the concentration position of these ground state solutions as $\varepsilon\to0$. Moreover, we consider some properties of these ground state solutions, such as convergence and decay estimate.