Existence of static solutions of the semilinear Maxwell equations

Abstract

In this paper we study a model which describes the relation of the matter and the electromagnetic field from a unitarian standpoint in the spirit of the ideas of Born and Infeld. This model, introduced in [1], is based on a semilinear perturbation of the Maxwell equation (SME). The particles are described by the finite energy solitary waves of SME whose existence is due to the presence of the nonlinearity. In the magnetostatic case (i.e. when the electric field ${\bf E}=0$ and the magnetic field ${\bf H}$ does not depend on time) the semilinear Maxwell equations reduce to semilinear equation where “ $\nabla\times $ ” is the curl operator, f′ is the gradient of a smooth function $f:{\mathbb{R}}^3\to{\mathbb{R}}$ and ${\bf A}:{\mathbb{R}}^3\to{\mathbb{R}}^3$ is the gauge potential related to the magnetic field ${\bf H}$ ( ${\bf H}=\nabla\times {\bf A}$ ). The presence of the curl operator causes (1) to be a strongly degenerate elliptic equation. The existence of a nontrivial finite energy solution of (1) having a kind of cylindrical symmetry is proved. The proof is carried out by using a variational approach based on two main ingredients: the Principle of symmetric criticality of Palais, which allows to avoid the difficulties due to the curl operator, and the concentration-compactness argument combined with a suitable minimization argument. Keywords: Maxwell equations, Natural constraint, Minimizing sequence Mathematics Subject Classification (2000): 35B40, 35B45, 92C15