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Here and in the following we assume c (light velocity) = 1. In the empty space ρ = 0, J = 0. The first three equations (3), (4) and (5) are respectively the Amp`ere, Gauss and Faraday laws. Observe that from (3) we get ∂∇ · E + ∇·J =0. ∂t Then, using (4), we get that ρ and J are related by the continuity equation ∂ρ + ∇·J = 0. ∂t Now introduce the gauge potentials A, ϕ which permit to write (3), (4) as second order equations and to satisfy identically (5) and (6). A, ϕ are related to E and H by H=∇×A E=− ∂A − ∇ϕ.

56) There are positive constants c1 , c2 , p, q with 2 < p < 6 < q such that p c1 |ξ| ≤ f (ξ) for |ξ| ≥ 1 (57) 48 V. Bienci and D. Fortunato q c1 |ξ| ≤ f (ξ) for |ξ| ≤ 1 (58) |f (ξ)| ≤ c2 |ξ|p−1 for |ξ| ≥ 1 (59) q−1 |f (ξ)| ≤ c2 |ξ| for |ξ| ≤ 1. (60) We are interested in finding nontrivial, finite energy, weak solutions A : R → R3 of the equation (61) ∇ × ∇ × A = f (A) 3 where f denotes the gradient of f. A weak solution of (61) means that both A and f (A) are in L1loc and that for all ϕ ∈ C0∞ (R3 , R3 ) (A |∇ × (∇ × ϕ))dx = (f (A) | ϕ) dx where (· | ·) denotes the Euclidean inner product in R3 .

By the presence of the nonlinear term the equations of this theory are not gauge invariant. We point out that the gauge invariance is destroyed only in 34 V. Bienci and D. e. in the region where the nonlinear term is negligible so that we have essentially the Maxwell equations in the empty space. 1). • Any matter particle (charged or not) carries an intrinsic magnetic moment; this is a kind of classical analogue of the spin (see proposition 3 and remark 4). The main attention is devoted to static solutions of (SME).