Ricerche di Matematica manuscript No.
(will be inserted by the editor)
Antonio Azzollini · Vieri Benci ·
Teresa D’Aprile · Donato Fortunato
Existence of Static Solutions of the
Semilinear Maxwell Equations
Received: date / Accepted: date
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 E = 0 and the magnetic field H does not depend on time)
the semilinear Maxwell equations reduce to the following semilinear equation
∇ × (∇ × A) = f
0
(A) (1)
where “∇×” is the curl operator, f
0
is the gradient of a smo oth function
f : R
3
→ R and A : R
3
→ R
3
is the gauge potential related to the magnetic
field H (H = ∇ × 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
A. Azzollini
Dipartimento di Matematica, Universit`a degli Studi di Bari
via E. Orabona 4, 70125 Bari (Italy)
E-mail: azzollini@dm.uniba.it
V. Benci
Dipartimento di Matematica Applicata “U. Dini”, Universit`a degli Studi di Pisa
via Bonanno 25/b, 56126 Pisa (Italy)
E-mail: b enci@dma.unipi.it
T. D’Aprile
Dipartimento di Matematica, Universit`a degli Studi di Roma “Tor Vergata”
via della Ricerca Scientifica 1, 00133 Roma (Italy)
E-mail: daprile@mat.uniroma2.it
D. Fortunato
Dipartimento di Matematica, Universit`a degli Studi di Bari
via E. Orabona 4, 70125 Bari, (Italy) and INFN Sezione di Bari
E-mail: fortunat@dm.uniba.it
2 Antonio Azzollini et al.
proof is carried out by using a variational approach based on two main ingre-
dients: 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
1 Introduction
The study of the relation of matter and the electromagnetic field is a classi-
cal, intriguing problem both from physical and mathematical point of view.
In the framework of a classical relativistic theory, particles must be con-
sidered pointwise. However charged pointwise particles have infinite energy
and therefore infinite inertial mass. This fact gives rise to well known diffi-
culties (see for example [4,5,9]). The use of nonlinear equations in classical
electrodynamics permits in some situations to avoid these difficulties. In a
pioneering paper ([3]) Born and Infeld intro duced a nonlinear formulation of
the Maxwell equations. This theory avoids the divergences, however it is not
unitarian, i.e. the nonlinearity they introduce does not allow the existence
of a self-induced electromagnetic field and an external source is needed (see
chapter 12 in [10]).
Following these lines of thought, in [1] a unitarian field theory has been
introduced. This theory is based on a semilinear perturbation of the Maxwell
equations. More precisely the usual Maxwell action for the gauge potentials
A : R
4
→ R
3
, ϕ : R
4
→ R
S
M
(A, ϕ) =
1
2
Z Z
µ
¯
¯
¯
¯
∂A
∂t
+∇ϕ
¯
¯
¯
¯
2
− |∇ × A|
2
¶
dxdt
is modified as follows:
S (A, ϕ) =
1
2
Z Z
µ
¯
¯
¯
¯
∂A
∂t
+∇ϕ
¯
¯
¯
¯
2
− |∇ × A|
2
+ W
¡
|A|
2
− ϕ
2
¢
¶
dxdt (2)
where W : R → R and “∇×” denotes the curl operator.
The argument of W is |A|
2
− |ϕ|
2
in order to make the action invariant
for the Poincar´e group and the equations consistent with Special Relativity.
Making the variation of S with respect to δA, δϕ respectively, we get the
equations
∂
∂t
µ
∂A
∂t
+ ∇ϕ
¶
+ ∇ × (∇ × A) = W
0
¡
|A|
2
− ϕ
2
¢
A, (3)
−∇ ·
µ
∂A
∂t
+∇ϕ
¶
= W
0
¡
|A|
2
− ϕ
2
¢
ϕ. (4)
Existence of Static Solutions of the Semilinear Maxwell Equations 3
If we set
ρ = ρ (A, ϕ) = W
0
¡
|A|
2
− ϕ
2
¢
ϕ, (5)
J = J (A, ϕ) = W
0
¡
|A|
2
− ϕ
2
¢
A, (6)
equations (3) and (4) are formally the Maxwell equations in the presence
of matter if we interpret ρ (A, ϕ) as charge density and J (A, ϕ) as current
density. Notice that ρ and J are not assigned functions representing external
sources: they depend on the gauge potentials, so that we are in the presence
of an unitarian theory. We make the following assumptions on W :
(W1) W ∈ C
1
(R, R); W (0) = 0;
(W2) there exists ξ > 0 such that W (ξ) > 0;
(W3) there exist positive constants c, p, q with 2 < p < 6 < q such that
|W
0
(s)| ≤ c|s|
p/2−1
for |s| ≥ 1,
|W
0
(s)| ≤ c|s|
q /2− 1
for |s| ≤ 1.
The set
Ω
t
=
n
x ∈ R
3
: 1 ≤
¯
¯
|A(x, t)|
2
− |ϕ(x, t)|
2
¯
¯
o
is interpreted as the region of the space filled with matter at time t (see
section 2 of [1]). Observe that the above assumptions allow to take W (s) = 0
for |s| ≤ 1 − ε (ε > 0), so that ρ and J vanish outside a neighbourhood of Ω
t
and in this region equations (3) and (4) reduce to the Maxwell equations in
the empty space.
Equations (3) and (4) have been extensively studied in [1] where, among
other things, the existence of a finite energy (magnetostatic) solution (A , 0),
with A depending only on the space variable x, has b een stated. However
the proof contains a gap, which will be overcome by Theorem 1 below.
In the magnetostatic case, equations (3) and (4) reduce to
∇ × (∇ × A) = W
0
¡
|A|
2
¢
A. (7)
In this paper we study equation (7) and we prove the existence of a
nontrivial, finite energy solution A = (A
1
, A
2
, A
3
) whose components are
related to each others by some kind of cylindrical symmetry. More precisely
the following theorem holds:
Theorem 1 Assume that hypotheses (W1), (W2), (W3) hold. Then equation
(7) has a nontrivial, weak solution A having the following form:
A(x) = A
³
q
x
2
1
+ x
2
2
, x
3
´
(−x
2
, x
1
, 0)
where A : (0, +∞) × R → R. Moreover A has finite energy, i.e.
Z
R
3
|∇A|
2
dx < +∞.
4 Antonio Azzollini et al.
The main difficulty in dealing with the equation (7) lies in the fact that
the energy functional related to it
E[A] =
1
2
Z
R
3
|∇ × A|
2
dx −
Z
R
3
W (|A|
2
)dx (8)
is, in general, strongly indefinite in the sense that it is not bounded from
below or from above and any possible critical point has infinite Morse in-
dex; namely the second variation of (8) is negative definite (if W is strongly
convex) on the infinite dimensional space
©
A = ∇ϕ : ϕ ∈ C
∞
0
(R
3
, R)
ª
.
To overcome this difficulty, in section 2 we introduce a suitable space D
1
F
whose elements are divergence free, so that for A ∈ D
1
F
we have
Z
R
3
|∇ × A|
2
dx =
Z
R
3
|∇A|
2
dx.
It can be shown that D
1
F
is a natural constraint for (8), so that we are reduced
to look for critical points of E|
D
1
F
. Furthermore the maps in D
1
F
have a sort
of cylindrical symmetry and the functional E|
D
1
F
has a lack of compactness
due to its invariance under the translations along the x
3
axis. The proof of
the existence of critical points for E|
D
1
F
is carried out in section 4 combining
a concentration-compactness type result, proved in section 3, with a suitable
minimization argument.
2 The Variational Setting
In this section we collect some preliminary results concerning the variational
structure of the system (7). Let C
∞
0
(R
3
, R
3
) be the set of the C
∞
vector
fields A : R
3
→ R
3
having compact support. Then let D
1
(R
3
, R
3
) denote the
completion of C
∞
0
(R
3
, R
3
) with respect to the norm
kAk
2
D
1
=
Z
R
3
|∇A|
2
dx, A ∈ D
1
(R
3
, R
3
).
D
1
(R
3
, R
3
) is a Hilbert space with the scalar product
Z
R
3
(∇A|∇B)dx, A, B ∈ D
1
(R
3
, R
3
),
where A = (A
1
, A
2
, A
3
), B = (B
1
, B
2
, B
3
) and (∇A|∇B) =
P
3
i=1
∇A
i
·∇B
i
,
being “ · ” the scalar product in R
3
. By the Sobolev inequalities, D
1
(R
3
, R
3
)
is continuously embedded into L
6
(R
3
, R
3
):
D
1
(R
3
, R
3
) → L
6
(R
3
, R
3
). (9)
Consequently, for every Ω ⊂ R
3
open and bounded we have
D
1
(R
3
, R
3
) → H
1
(Ω, R
3
) (10)
Existence of Static Solutions of the Semilinear Maxwell Equations 5
with continuous embedding.
The functional associated to (7) is
E[A] =
1
2
Z
R
3
|∇ × A|
2
dx −
1
2
Z
R
3
W (|A|
2
)dx.
Observe that, according to assumption (W3), we have
|W (s
2
)| ≤
c
3
|s|
6
, |W
0
(s
2
)| ≤ c|s|
4
. (11)
Then, by (9), it is easy to prove that E is well defined on D
1
(R
3
, R
3
) and
belongs to the class C
1
(D
1
(R
3
, R
3
), R). Hence critical p oints of E correspond
to solutions of (7).
The main difficulty in dealing with the functional E lies in its strongly
indefinite nature: indeed, if W is p ositive, it is negatively definite on the
infinite-dimensional subspace
{A = ∇φ | φ ∈ C
∞
0
(R
3
, R)}.
Hence it does not exhibit a mountain pass geometry. In order to remove this
indefiniteness, we are going to restrict our functional to a suitable subspace
of D
1
(R
3
, R
3
). More precisely consider the following space
F =
(
A : R
3
→ R
3
¯
¯
¯
¯
¯
∃ A : (0, +∞) ×R → R s.t.
A(x)=A(r, x
3
)(−x
2
, x
1
, 0) a.e. in R
3
)
where
r = r
x
= |(x
1
, x
2
)| =
q
x
2
1
+ x
2
2
, (12)
and set
D
1
F
= D
1
(R
3
, R
3
) ∩ F.
It is obvious that D
1
F
is a closed subspace of D
1
(R
3
, R
3
). Furthermore, for
all A ∈ D
1
F
we have that div A = 0, by which
∇ × (∇ × A) = −∆A ∀A ∈ D
1
F
; (13)
hence the restricted functional E
|D
1
F
has the following form
E
|D
1
F
[A] =
1
2
Z
R
3
|∇A|
2
dx −
1
2
Z
R
3
W (|A|
2
)dx. (14)
The first object is to prove that D
1
F
is a natural constraint for E, i.e. a
suitable subspace where to find solutions of (7). To this aim we recall the
following Principle of symmetric criticality of Palais ([8]):
Principle of symmetric criticality. Assume that there exists a topological group
of transformations G which acts isometrically on a Hilbert space X and define
Fix G := {A ∈ X | GA = A ∀G ∈ G}. (15)