Existence and asymptotical behavior of solutions for a fractional logarithmic Schrödinger-Poisson system

We study the following fractional Schrödinger-Poisson system \begin{equation}\label{eq:0.1} \left\{\begin{array}{ll} (-\Delta)^s u + V(x) u + \lambda\phi (x) u =\mu u+u\log{u^{2}},\ x\in \mathbb{R}^3, \\ (-\Delta)^s\phi = u^2, \displaystyle \ \lim_{|x|\to +\infty}\phi (x)=0, \end{array}\right. \end{equation} where $s\in (\frac{1}{2},1)$, and $\mu,\lambda$ are real parameters. Assuming that $V\in C(\mathbb{R}^3,\mathbb{R}^+)$ and $\lim_{|x|\to +\infty}V(x)=\infty$, we show that problem (0.1) has a nonzero solution under suitable assumptions on the parameters $\lambda$ and $\mu$. Moreover, the asymptotical behavior of solutions as $\lambda\to 0$ has also been discussed.