Magnetostatic solutions for a semilinear perturbation of the Maxwell equations

Abstract

In this paper we consider a model introduced in [3] which describes the interaction between the matter and the electromagnetic field from a unitarian standpoint. This model is based on a semilinear perturbation of the Maxwell equations and, in the magnetostatic case, reduces to the following nonlinear elliptic degenerate equation: $$\nabla \times (\nabla \times {\bf A})=W'(|{\bf A}|^2){\bf A},$$ where "$\nabla\times$" is the curl operator, $W:\mathbb R\to\mathbb R$ is a suitable nonlinear term, and ${\bf A}:\mathbb R^3\to\mathbb R^3$ is the gauge potential associated with the magnetic field ${\bf H}$. We prove the existence of a nontrivial finite energy solution with a kind of cylindrical symmetry. The proof is carried out by using a suitable variational framework based on the Hodge decomposition, which is crucial in order to handle the strong degeneracy of the equation. Moreover, the use of a natural constraint and a concentration-compactness argument are also required.