Existence and concentration phenomenon of multiple solutions for the fractional logarithmic Schr\"{o}dinger-Poisson system via penalization method

Abstract

This paper concerns the existence of multiple solutions for the fractional logarithmic Schr\"odinger-Possion system of the form \begin{equation*} \begin{cases} {\varepsilon}^{2\alpha} (-\Delta )^{\alpha}u+V(x) u+\phi u=u \log u^{2}+u^{q-1}, & \text{in}\quad \mathbb{R}^{3}, {\varepsilon}^{2\alpha} (-\Delta )^{\alpha}\phi=u^2, & \text{in}\quad \mathbb{R}^{3}. \end{cases} \end{equation*} where $\varepsilon>0$ is a small parameter, $q \in (4, 2_\alpha^*)$ with $\alpha\in(\frac{3}{4},1)$, $V: \mathbb{R}^{3} \rightarrow \mathbb{R}$ 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 $\varepsilon>0$ small enough, the system has a positive ground state solution and each positive solution concentrates around a local minimum point of $V$.