Magnetostatic solutions for a semilinear perturbation of the Maxwell equations

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: ∇×(∇×A)=W′(|A|^2)A, where "∇×" is the curl operator, W:R→R is a suitable nonlinear term, and A:R^3→R^3 is the gauge potential associated with the magnetic field 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.