Magneto-Static Vortices in Two Dimensional Abelian Gauge Theories

Abstract

We study the existence of vortices of the Klein-Gordon-Maxwell equations in the two dimensional case. In particular we find sufficient conditions for the existence of vortices in the magneto-static case, i.e. when the electric potential φ = 0. This result, due to the lack of suitable embedding theorems for the vector potential A is achieved with the help of a penalization method. Mathematics Subject Classification (2000). Primary 35J50; Secondary 35Q60.

Existence of hylomorphic solitary waves in Klein–Gordon and in Klein–Gordon–Maxwell equations

“…From mathematical viewpoint, the existence of vortices for (NKG) and for (NKGM) has been studied in some recent papers ( [16], [4], [5], [13], [14], [15], [8], [9]).…”

Vortex ground states for Klein-Gordon-Maxwell-Proca type systems

“…As in [6, Section 3.2] we can prove the following 3 and its critical points that satisfy the additional condition div A = 0 are solutions of (S µ ) in the sense of distributions, i.e.…”

“…where φ n = φ un . Since {u n } is bounded inĤ 1 , by Lemma 2.3, Lemma 2.6 and (3.2), also {φ n } and {A n } are bounded respectively in H µ and (H µ ) 3 . Thus there exist φ 0 ∈ H µ and A 0 ∈ (H µ ) 3 such that, up to a subsequence,…”

“…In this paper, instead, we are interested in the electromagnetostatic case (nontrivial φ = φ(x) and A = A(x)) which has been studied in a smaller number of articles (see [3,5,6]), only for the Klein-Gordon-Maxwell system (µ = 0). In particular, from the physical point of view, it is important to find ground state solutions which are minimizers of the energy functional among a set of nontrivial solutions and, in this work, we find symmetric ground states for any µ ∈ R. Our results are also new when µ = 0.…”

A minimization method and applications to the study of solitons

“…the more recent ones [24], [29], [30], [28] with their references). From mathematical viewpoint, the existence of vortices for the nonlinear Klein-Gordon equations (NKG), for nonlinear Schroedinger equations (NSE) and for nonlinear Klein-Gordon-Maxwell equations (NKGM) has been studied in some recent papers ( [18], [2], [3], [10], [16], [7], [8], [15]).…”