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

TL;DR

This paper proves the existence of multiple solutions for a fractional logarithmic Schr"odinger-Poisson system, showing solutions concentrate around local minima of the potential as a small parameter tends to zero.

Contribution

It introduces a new Banach space making the energy functional differentiable, enabling the use of Lusternik-Schnirelmann theory to establish multiple solutions.

Findings

Existence of positive ground state solutions for small epsilon.

Solutions concentrate near local minima of the potential.

Multiple solutions are obtained via topological methods.

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 $V$.