Nonlinear Schrödinger equation in the Bopp–Podolsky electrodynamics: Solutions in the electrostatic case

Abstract

We study the following nonlinear Schrödinger–Bopp–Podolsky system{−Δu+ωu+q2ϕu=|u|p−2u−Δϕ+a2Δ2ϕ=4πu2 inR3 with a,ω>0. We prove existence and nonexistence results depending on the parameters q,p. Moreover we also show that, in the radial case, the solutions we find tend to solutions of the classical Schrödinger–Poisson system as a→0.