Existence and Concentration of Multiple Positive Solutions for a Logarithmic Fractional Schrödinger--Poisson System

We study a logarithmic fractional Schrödinger–Poisson system in ${\mathbb{R}}^3$: \begin{equation*} \begin{cases} \varepsilon^{2α}(-Δ)^αu+V(x)u+φu=u\log u^{2}+|u|^{p-2}u, & $\text{in }$\R^{3},\ \varepsilon^{2α}(-Δ)^αφ=u^{2}, & $\text{in }$\R^{3}. \end{cases} \end{equation*} Here $\alpha \in (\frac{3}{4},1)$, $4<p<2_{\alpha }^{\ast }=\frac{6}{3-2\alpha }$, 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 $\epsilon >0$, the system admits at least ${\mathrm{cat}}_{M_{\delta }}(M)$ distinct positive solutions. Moreover, the maximum points of these solutions concentrate near the global minimum set of $V$ as $\epsilon \to 0$.