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

Abstract

We study the following nonlinear Schrödinger-Bopp-Podolsky systemwith 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.

The existence of nontrivial solution of a class of Schrödinger–Bopp–Podolsky system with critical growth

“…To the best of our knowledge, there are very few papers related to the existence of solutions to problem (1.1). In [1], d' Avenia and Siciliano studied the following Schrödinger-Bopp-Podolsky equation:…”

“…It is easy to show that D is a Hilbert space continuously embedded into D 1,2 (R 3 ) and consequently in L ∞ (R 3 ), see [1].…”

Electro-magneto-static study of the nonlinear Schrödinger equation coupled with Bopp-Podolsky electrodynamics in the Proca setting

“…with an external massive vector field (ϕ, A) which is governed by the Bopp-Podolsky-Proca Lagrangian L BP P . A very nice presentation of this (without the Proca contribution) is in D'Avenia and Siciliano [10]. More on the Proca contribution can be found in Goldhaber and Nieto [23,24], Luo, Gillies and Tu [36], and Ruegg and Ruiz-Altaba [39].…”

“…Such type of solutions were introduced in the paper [3] by Benci and Fortunato for the Klein-Gordon-Maxwell equations in R 3 (see also D'Avenia, Mederski and Pomponio [9]). We learned from the Bopp-Podolsky setting in the paper [10] by D'Avenia and Siciliano where the existence of electrostatic solutions is established for the Schrödinger equation in the Bopp-Podolsky electrodynamics in the case of R 3 .…”

Normalized solutions for a Schrödinger-Bopp-Podolsky system

“…In this paper we treat a modification of the problem dealt with by Pisani and Siciliano consisting in the addition of a biharmonic term in the equation of the electrostatic potential and imposing appropriate boundary conditions. The problem studied can be interpreted as a coupling of the Schrödinger equation with the Bopp-Podolsky electrodynamics (for more details on this matter, see [3] and the references therein). However here we focus on the mathematical aspects of the problem.…”