Existence and concentration phenomenon of multiple solutions for the fractional logarithmic Schrödinger-Poisson system via penalization method

This paper concerns the existence of multiple solutions for the fractional logarithmic Schrödinger-Possion system of the form \begin{equation*} \begin{cases} {\varepsilon}^{2α} (-Δ)^αu+V(x) u+φu=u \log u^{2}+u^{q-1}, & $\text{in}$\quad $\mathbb{R}$^{3}, {\varepsilon}^{2α} (-Δ)^αφ=u^2, & $\text{in}$\quad $\mathbb{R}$^{3}. \end{cases} \end{equation*} where $\epsilon >0$ is a small parameter, $q\in (4,2_{\alpha }^{\ast })$ with $\alpha \in (\frac{3}{4},1)$, $V:{\mathbb{R}}^3\to \mathbb{R}$ is a continuous function that satisfies some local potential hypothesis. By introducing a new Banach space, the energy functional become $C^1$, which create the conditions for studying the multiplicity of solutions involving Lusternik-Schnirelmann category. We prove that for $\epsilon >0$ small enough, the system has a positive ground state solution and each positive solution concentrates around a local minimum point of $V$.