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

Abstract

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 \(\varepsilon\to0\).