Existence of static solutions of the semilinear Maxwell equations

Abstract

In this paper we study a model which describes the relation of the matter\ud and the electromagnetic field from a unitarian standpoint in the spirit of the\ud ideas of Born and Infeld. This model, introduced in [1], is based on a semilinear\ud perturbation of the Maxwell equation (SME). The particles are described by the\ud finite energy solitary waves of SME whose existence is due to the presence of\ud the nonlinearity. In the magnetostatic case (i.e. when the electric field E = 0 and\ud the magnetic field H does not depend on time) the semilinear Maxwell equations\ud reduce to the following semilinear equation\ud ∇×(∇×A) = f \ud (A) (1)\ud where “∇×” is the curl operator, f is the gradient of a smooth function f :R3→R\ud and A : R3 → R3 is the gauge potential related to the magnetic field H (H =\ud ∇×A). The presence of the curl operator causes (1) to be a strongly degenerateelliptic equation. The existence of a nontrivial finite energy solution of (1) having\ud a kind of cylindrical symmetry is proved. The proof is carried out by using a\ud variational approach based on two main ingredients: the Principle of symmetric\ud criticality of Palais, which allows to avoid the difficulties due to the curl operator,\ud and the concentration-compactness argument combined with a suitable minimization\ud argument

On the Schrödinger–Born–Infeld System

“…According to this unitarian point of view, the system giving a complete description of the dynamics arose variationally starting from a nonlinear version of the Maxwell lagrangian. This unitarian approach was also taken up by Benci and Fortunato in [4] (see also [1] and [12]). On the other hand there is the dualistic point of view, based on the idea that the dynamics can be described coupling equations related with particles and equations related with the electromagnetic field through a suitable combination of the lagrangians.…”

Magneto-Static Vortices in Two Dimensional Abelian Gauge Theories

“…To overcome this difficulty, in [8] the authors first introduced the manifold of free divergence vector fields on which the "bad" term reduces to R 2 |∇A| 2 dx, second they proved that this manifold is a natural constraint for the functional associated to (1.11)- (1.12). This idea has also been used successfully in [3] and [6].…”

On a “Zero Mass” Nonlinear Schrödinger Equation

“…Moreover, the presence of the potential V does not permit us to use any symmetry to recover compactness in a suitable natural constraint of D 1,2 (R N ). In order to overcome this difficulty, we are going to use a concentration-compactness argument as in [5] (see also [19,20]). The following lemma provides the boundedness and the concentration of the (PS)-sequences (actually we consider a more general situation).…”