On the logarithmic fractional Schrödinger–Poisson system with saddle-like potential

Abstract

In this paper, we use variational methods to prove the existence of a positive solution for the following class of logarithmic fractional Schrödinger–Poisson system: { ϵ 2 s ( − Δ ) s u + V ( x ) u − ϕ ( x ) u = u log ⁡ u 2  in  R 3 , ϵ 2 t ( − Δ ) t ϕ = | u | 2  in  R 3 , where ϵ > 0 , s , t ∈ ( 0 , 1 ) , ( − Δ ) α is the fractional Laplacian and V is a saddle-like potential.