Existence and Concentration of Multiple Positive Solutions for a Logarithmic Fractional Schr\"odinger--Poisson System

We study a logarithmic fractional Schr\"odinger--Poisson system in $\R^{3}$: \begin{equation*} \begin{cases} \varepsilon^{2\alpha}(-\Delta)^{\alpha}u+V(x)u+\phi u=u\log u^{2}+|u|^{p-2}u, & \text{in }\R^{3},\\ \varepsilon^{2\alpha}(-\Delta)^{\alpha}\phi=u^{2}, & \text{in }\R^{3}. \end{cases} \end{equation*} Here $\alpha\in\bigl(\frac34,1\bigr)$, $4<p<2_{\alpha}^{*}=\frac{6}{3-2\alpha}$, and $V$ satisfies a global potential condition. Using a suitable Orlicz-type Banach space, we establish a $C^{1}$ variational framework for the problem and combine the Nehari manifold method with Lusternik--Schnirelmann category theory. We then prove that, for every fixed $\delta>0$ and all sufficiently small $\varepsilon>0$, the system admits at least $\operatorname{cat}_{M_{\delta}}(M)$ distinct positive solutions. Moreover, the maximum points of these solutions concentrate near the global minimum set of $V$ as $\varepsilon\to0$.