On the Schrödinger–Born–Infeld System

Abstract

In this paper we study a system which we propose as a model to describe the interaction between matter and electromagnetic field from a dualistic point of view. This system arises from a suitable coupling of the Schrödinger and the Born-Infeld lagrangians, this latter replacing the role that, classically, is played by the Maxwell lagrangian. We use a variational approach to find an electrostatic radial ground state solution by means of suitable estimates on the functional of the action.2010 Mathematics Subject Classification. 35J50, 35J93, 35Q60.

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

“…Moreover from (i) and the Hölder inequality we easily get (ii). [4,6). In this case any (PS) sequence for J q is bounded.…”

“…See e.g. [3,24] for the Maxwell theory, [4,9,23,26] for the Born-Infeld one, [19] for the Chern-Simon one, and references therein. However, to the best of our knowledge, this is the first time that a system like (1.1), which involves the Bopp-Podolsky electromagnetic theory, appears in the mathematical literature.…”

“…In this case, that is for p ∈ (3, 6), the Mountain Pass Geometry holds for every q. However the boundedness of the (PS) sequences is obtained in two different way: it is standard if p ∈ [4,6) while it is obtained by means of the monotonicity trick (see [29,37]) if p ∈ (3,4). Even now the compactness (and then the existence of a solution) can be recovered by using the Splitting Lemma; nevertheless, just to use a different (and simpler) argument, we prefer to give the proof in the radial setting.…”

Spacelike graphs with prescribed mean curvature on exterior domains in the Minkowski spacetime

“…Here ρ can be considered as an assigned charges source. We also refer to [6], where the Born-Infeld equation is coupled with the nonlinear Schrödinger one. Therefore, Theorem 1.7 can also be seen as an existence result for the Born-Infeld problem on an exterior domain with assigned boundary conditions.…”

A perturbation approach for the Schrödinger-Born-Infeld system: Solutions in the subcritical and critical case

“…We use a new perturbation approach to prove the existence and multiplicity of nontrivial solutions of the above system in the subcritical and critical case. We emphasise that our results cover the case f (u) = |u| p−1 u for p ∈ (2, 5/2] and µ = 0 which was left in [2] as an open problem.…”