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Hylomorphic solitons and charged Q-balls: existence and stability

In this paper we give an abstract definition of solitary wave and soliton and we develop an abstract existence theory. This theory provides a powerful tool to study the existence of solitons for the Klein-Gordon equations as well as for gauge theories. Applying this theory, we prove the existence of a continuous family of stable charged Q-balls.


Introduction
Loosely speaking a solitary wave is a solution of a field equation whose energy travels as a localized packet and which preserves this localization in time. A soliton is a solitary wave which exhibits some strong form of stability so that it has a particle-like behavior (see e.g. [35], [38], [37] and the references therein contained). We are interested in a class of solitons which, following [6], [4], [5], [12], we call hylomorphic. Their existence is due to an interplay between energy and charge. These solitons include the Q-balls, which are spherically symmetric solutions of the nonlinear Klein-Gordon equation and which have been studied since the pioneering papers [36] and [20]. Q-balls arise in a theory of bosonic particles (see [29], [30]), when there is an attraction between the particles. Roughly speaking, a Q-ball is a finite-sized "bubble" containing a large number of particles. The Q-ball is stable against fission into smaller Qballs since, due to the attractive interaction, the Q-ball is the lowest-energy configuration of that number of particles. Q-balls also play an important role in the study of the origin of the matter that fills the universe (see [25]).
In this paper we give an abstract definition of solitary wave and soliton and we develope an abstract existence theory. This theory provides a powerful tool to study the existence of solitons for the Klein-Gordon equations as well as for gauge theories (see [7]). Most of the existence results in the present literature can be deduced in the framework of this theory using Th.18 or 19, as it is shown in a forthcoming book [17]. We get a new result applying Th. 19 to the study of charged Q-balls. Let us describe this result.
If the Klein-Gordon equations are coupled with the Maxwell equations (NKGM), then the relative solitary waves are called charged, or gauged Q-balls (see e.g. [20]). The existence of charged Q-balls is stated in [10], [11], [13], [33]. However, in these papers there are not stability results and hence the existence of solitons for NKGM (namely stable charged Q-balls) was an open question.
The problem with the stability of charged Q-balls is that the electric charge tends to brake them since charges of the same sign repel each other. In this respect Coleman, in his celebrated paper [20], says "I have been unable to construct Q-balls when the continuous symmetry is gauged. I think what is happening physically is that the long-range force caused by the gauge field forces the charge inside the Q-ball to migrate to the surface, and this destabilizes the system, but I am not sure of this".
A partial answer to this question is in [15] where the existence of stable charged Q-balls is established provided that the interaction between matter and gauge field is sufficiently small. Theorem 19 allows to extend this result and to prove the existence of a continuous family of stable charged Q-balls. More precisely, we prove that there is a family of Q-balls {u δ } δ∈(0,δ∞) whose energy and charge are decreasing with δ.

Solitary waves and solitons
In this section we construct a functional abstract framework which allows to define solitary waves, solitons and hylomorphic solitons.

Definitions of solitons
Solitary waves and solitons are particular states of a dynamical system described by one or more partial differential equations. Thus, we assume that the states of this system are described by one or more fields which mathematically are represented by functions u : where V is a vector space with norm | · | V and which is called the internal parameters space. We assume the system to be deterministic; this means that it can be described as a dynamical system (X, γ) where X is the set of the states and γ : R × X → X is the time evolution map. If u 0 (x) ∈ X, the evolution of the system will be described by the function u (t, x) := γ t u 0 (x).
We assume that the states of X have "finite energy" so that they decay at ∞ sufficiently fast and that X ⊂ L 1 loc R N , V .
Thus we are lead to give the following definition: Definition 1 A dynamical system (X, γ) is called of FT type (field-theory-type) if X is a Hilbert space of fuctions of type (2).
For every τ ∈ R N , and u ∈ X, we set Clearly, the group is a unitary representation of the group of translations.
Definition 2 A set Γ ⊂ X is called compact up to space tanslations or Tcompact if for any sequence u n (x) ∈ Γ there is a subsequence u n k and a sequence τ k ∈ R N such that u n k (x − τ k ) is convergent.
Now, we want to give a very abstract definition of solitary wave. As we told in the introduction, a solitary wave is a field whose energy travels as a localized packet and which preserves this localization in time. For example, consider a solution of a field equation having the following form: for every x 0 , v ∈ R N , ω ∈ R, u (t, x) is a solitary wave. The evolution of a solitary wave is a translation plus a mild change of the internal parameters (in this case the phase). This situation can be formalized by the following definition: Definition 3 If u 0 ∈ X, we define the closure of the orbit of u 0 as follows: A state u 0 ∈ X is called solitary wave if Clearly, (5) describes a solitary wave according to the definition above. The standing waves, namely objects of the form probably are the "simplest" solitary waves. In this case the orbit O (u 0 ) itself is compact. Take X = L 2 (R N ) and u ∈ X; if γ t u = u x e t , u is not a solitary wave, since (i) of the above definition is violated; if γ t u = 1 e t u x e t , u is not a solitary wave since (ii) of Def. 3 does not hold. Also, according to our definition, a "couple" of solitary waves is not a solitary wave: for example is not a solitary wave since (ii) is violated.
The solitons are solitary waves characterized by some form of stability. To define them at this level of abstractness, we need to recall some well known notions in the theory of dynamical systems.
Definition 5 Let (X, d) be a metric space and let (X, γ) be a dynamical system.
Now we are ready to give the definition of soliton: The above definition needs some explanation. First of all notice that every u ∈ Γ is a soliton and that every soliton is a solitary wave. Now for simplicity, we assume that Γ is a manifold (actually, in many concrete models, this is the generic case). Then (ii) implies that Γ is finite dimensional. Since Γ is invariant, u 0 ∈ Γ ⇒ γ t u 0 ∈ Γ for every time. Thus, since Γ is finite dimensional, the evolution of u 0 is described by a finite number of parameters. The dynamical system (Γ, γ) behaves as a point in a finite dimensional phase space. By the stability of Γ, a small perturbation of u 0 remains close to Γ. However, in this case, its evolution depends on an infinite number of parameters. Thus, this system appears as a finite dimensional system with a small perturbation.
Example. We will illustrate the definition 6 with an example. Consider the solitary wave (5) and the set (5) is a soliton provided that Γ v is stable; in fact the following conditions are satisfied: • The dynamics on Γ v is given by the following equation: This dynamics implies that Γ v is invariant and that (iii) holds.
• we have assumed that Γ v is stable; in this case any perturbation of our soliton has the following structure: where x 0 (t), θ(t) are suitable functions and w(t, x) is a perturbation small in H 1 R N , C .
• Γ v is T -compact; actually it is isomorphic to R N × S 1 .

Definition of hylomorphic solitons
We now assume that the dynamical system (X, γ) has two constants of motion: the energy E and the charge C. At the level of abstractness of this section (and the next one), the name energy and charge are conventional, but in our applications, E and C will be the energy and the charge as defined in section 5.2.
Definition 7 A solitary wave u 0 ∈ X is called standing hylomorphic soliton if it is a soliton according to Def. 6 and if Γ has the following structure where Notice that, by (8), we have that a hylomorphic soliton u 0 satisfies the following nonlinear eigenvalue problem: In general, a minimizer u 0 of E on M c 0 = {u ∈ X |C(u)| = c 0 } is not a soliton; in fact, according to Def. 6, it is necessary to prove the following facts: • (i) The set Γ (e 0 , c 0 ) is stable.
In concrete cases, the point (i) is the most delicate point to prove. If (i) does not hold, according to our definitions, u 0 is a solitary wave but not a soliton. Now let us see the general definition of hylomorphic soliton.
Definition 8 Let (X, γ) be a dynamical system of type FT and invariant for the action of a Lie group G, namely, for any u ∈ X, ∀g ∈ G, gγ t u =γ t gu.
u is called hylomorphic soliton if u =gu 0 where u 0 is a standing hylomorphic soliton and g is a suitable element of G.
In the application G will be a representation of the Galileo or of the Lorentz group. Now let us illustrate with an example Def. 7 and Def. 8.
Example. Let us consider the example (6). The standing wave u(x)e −iωt is a hylomorphic soliton if satisfies the request in Definition 6 and if Γ 0 = Γ (e 0 , c 0 ) (see (7)) for a suitable c 0 .

Existence results of hylomorphic solitons
In the previous section, we have seen that the existence of hylomorphic soliton is related to the existence of minimizers of the energy. In this section we will investigate the following minimization problem and under which conditions the set of the minimizers is stable.

The abstract framework
The following definitions could be given in a more abstract framework. Nevertheless, for the sake of definitess, in the following we shall assume that (X, γ) is a dynamical system of FT-type (see Def 1) and that G is a subgroup of T (see(4)).
Definition 10 A sequence u n in X is called G-compact if there is a subsequence u n k and a sequence g k ∈ G such that g k u n k is convergent. A subset Γ ⊂ X is called G-compact if every sequence in Γ is G-compact.
Observe that the above definition reduces to Definition 2 if G = T . If G = {Id} or, more in general, it is a compact group, G-compactness implies compactness. If G is not compact such as the tranlation group T , Gcompactness is a weaker notion than compactness.
Clearly a G-compact functional has a G-compact set of minimizers.

Definition 12
We say that a functional F on X has the splitting property if given a sequence u n = u + w n ∈ X such that w n converges weakly to 0, we have that Remark 13 Every quadratic form, which is continuous and symmetric, satisfies the splitting property; in fact, in this case, we have that F (u) := Lu, u for some continuous selfajoint operator L; then, given a sequence u n = u + w n with w n ⇀ 0 weakly, we have that Definition 14 A sequence u n ∈ X is called G-vanishing sequence if it is bounded and if for any subsequence u n k and for any sequence g k ⊂ G the sequence g k u n k converges weakly to 0.
So, if u n → 0 strongly, u n is a G-vanishing sequence. However, if u n ⇀ 0 weakly, it might happen that it is not a G-vanishing sequence; namely it might exist a subsequence u n k and a sequence g k ⊂ G such that g k u n k is weakly convergent to someū = 0. Let see an example; if u 0 ∈ X is a solitary wave and t n → +∞, then the sequence γ tn u 0 is not a T -vanishing sequence.
In the following E and C will denote two constants of the motion for the dynamical system (in the applications they will be the energy and the charge). We will assume that E and C are C 1 and bounded functionals on X.
We set Since E and C are constants of motion, also Λ is a constant of motion; it will be called hylenic ratio (see the definition of charge, sec. 5.2) and, as we will see it will play a central role in this theory.
The notions of G-vanishing sequence and of hylenic ratio allow to introduce the following (important) definition:

Definition 15
We say that the hylomorphy condition holds if where Moreover, we say that u 0 ∈ X satisfies the hylomorphy condition if, By this definition, using the above notation, we have the following: In order to apply the existence theorems of sect. 3.2, it is necessary to estimate Λ 0 ; the following propositons may help to do this.

Proposition 16
Assume that there exists a seminorm ♯ on X such that Then lim inf Proof. It follows directly from the definition (12) of Λ 0 and (14).

Proposition 17
If E and C are twice differentiable in 0 and Proof. By the above proposition, Now, finally, we can give some abstract theorems relative to the existence of hylomorphic solitons.

Statement of the abstract existence theorems
We formulate the assumptions on E and C: • (EC-2)(Splitting property) E and C satisfy the splitting property.
Theorem 19 Let the assumptions of theorem 18 hold. Moreover assume that Then for every δ ∈ (0, δ ∞ ) , δ ∞ > 0, there exists a hylomorphic soliton u δ . Moreover, if δ 1 < δ 2 , the corresponding solitons u δ 1 , u δ 2 are distinct, namely we have that The proofs of the above results are in the remaining part of this section. In subsection 3.3 we prove the existence of minimizers, namely that Γ(e, c) = ∅ (see (7)) and in subsection 3.4, we prove the stability of Γ(e, c), namely that the minimizers are hylomorphic solitons.

A minimization result
We start with a technical lemma.
Lemma 20 Let u n = u + w n ∈ X be a sequence such that w n converges weakly to 0. Then, up to a subsequence, we have and the equality holds if and only if Λ (u) = lim Λ (w n ) .

Proof. Given four real numbers
In fact, suppose that A B ≥ a b ; then

Notice that the equality holds if and only if
Using the splitting property and the above inequality, up to a subsequence, we have that For any δ > 0, set By the hylomorphy condition (11) we have Theorem 21 Assume that E and C satisfy (EC-0),...,(EC-3) and the hylomorphy condition (11). Then, for every δ ∈ (0, δ ∞ ) , J δ is G-compact and it has a minimizer u δ = 0. Moreover, u δ ∈ Γ (e δ , c δ ) (see (7)) where e δ = E(u δ ), Proof. Let u n be a minimizing sequence of J δ (δ ∈ (0, δ ∞ )). This sequence u n is bounded in X. In fact, arguing by contradiction, assume that, up to a subsequence, u n → ∞. Then by (EC-3) (ii), E(u n ) → ∞ and hence J δ (u n ) → ∞ which contradicts the fact that u n is a minimizing sequence of J δ .
We now set Since δ ∈ (0, δ ∞ ), where δ ∞ is defined in (18), we have that Moreover, since E ≥ 0,we have Then, up to a subsequence, Λ (u n ) → λ < Λ 0 . So, by definition (12) of Λ 0 , u n is not a G−vanishing sequence. Hence, by Def. 14, we can extract a subsequence u n k and we can take a sequence g k ⊂ G such that u ′ k := g k u n k is weakly convergent to some We can write u ′ n = u δ + w n with w n ⇀ 0 weakly. We want to prove that w n → 0 strongly. By lemma 20 and by the splitting property of E, we have, up to a subsequence, that Now there are two possibilities (up to subsequences): . We will show that the possibility 1 cannot occur. In fact, if it holds, we have that and hence, we get that E (u δ ) ≤ 0, contradicting (20). Then possibility 2 holds and we have that Then, lim E (w n ) = 0 and, by (EC-3)(iii), w n → 0 strongly.
To prove the second part of the theorem, we set:.
In the following u δ will denote a minimizer of J δ .
(b) We argue indirectly and assume that By (a), we have that Also, we have that (26)) and so and by (27) we get a contradiction.
(c) Since the conclusion follows from (a) and (b).

The stability result
In order to prove Theorem 18 it is sufficient to show that the minimizers in Th. 21 provide solitons, so we have to prove that the set Γ (e, c) is stable. To do this, we need the (well known) Liapunov theorem in following form: Theorem 24 Let Γ be an invariant set and assume that there exists a differentiable function V (called Liapunov function) defined on a neighborhood of Γ such that Then Γ is stable.
Proof. For completeness, we give a proof of this well known result. Arguing by contradiction, assume that Γ, satisfying the assumptions of Th. 24, is not stable. Then there exists ε > 0 and sequences u n ∈ X and t n > 0 such that d(u n , Γ) → 0 and d(γ tn (u n ) , Γ) > ε.
Then we have where the first and the third implications are consequence of property (c). The second implication follows from property (b). Clearly, this fact contradicts (28). Proof : The fact that Γ is G-compact, is a trivial consequence of the fact that Γ is the set of minimizers of a G-compact functional V . Now we prove (c). First we show the implication ⇒ . Let u n be a sequence such that V (u n ) → 0. By contradiction, we assume that d(u n , Γ) 0, namely that there is a subsequence u ′ n such that Since V (u n ) → 0 also V (u ′ n ) → 0, and, since V is G compact, there exists a sequence g n in G such that, for a subsequence u ′′ n , we have g n u ′′ n → u 0 . Then and this contradicts (29). Now we prove the other implication ⇐ . Let u n be a sequence such that d(u n , Γ) → 0, then there exists v n ∈ Γ s.t.
Since V is G-compact, also Γ is G-compact; so, for a suitable sequence g n , we have g n v n →w ∈ Γ. We get the conclusion if we show that V (u n ) → 0. We have by (30), that d(u n , v n ) → 0 and hence d(g n u n , g n v n ) → 0 and so, since g n v n →w, we have g n u n →w ∈ Γ. Therefore, by the continuity of V and sincē w ∈ Γ, we have V (g n u n ) → V (w) = 0 and we can conclude that V (u n ) → 0.
Proof of Th. 18. By Theorem 21 for every δ ∈ (0, δ ∞ ) J δ is G−compact and it has a minimizer u δ = 0 with So, in order to show that u δ is an hylomorphic soliton, we need to show that is G-compact and stable (see Definitions 6 and 7). We set V is G compact. In fact: Let w n be a minimizing sequence for V, then V (w n ) → 0 and consequently E (w n ) → e δ and C (w n ) → c δ . Now, since we have that w n is a minimizing sequence also for J δ . Then, since by Theorem 21 J δ is G-compact, we get w n is G-compact.
So we conclude that V is G-compact. Then, by Lemma 25, we deduce that V −1 (0) = Γ (e δ , c δ ) is G compact and that V satisfies the point (c) in Theorem 24. Moreover V satisfies also the points (a) and (b) in Theorem 24. So we conclude that Γ (e δ , c δ ) is stable.

The structure of hylomorphic solitons 4.1 The meaning of the hylenic ratio
Let (X, γ) be a dynamical system of type FT. If u ∈ X is a finite energy field, usually it disperses as time goes on, namely V is the internal parameter space (cf. pag. 3) and B 1 (x) = y ∈ R N : |x − y| < 1 . However, if the hylomorphy condition (11) is satisfied, this dispersion in general does not occur. In fact we have the following result: where u(t) = γ t u and γ 0 u = u 0 .
Proof : Let t n → ∞ be a sequence of times such that Since Λ is a constant of motion Λ (u (t n )) = Λ (u 0 ) < Λ 0 then, by the definition of Λ 0 , may be taking a subsequence, there is a sequence of translations T xn such that whereū = 0 and w n ⇀ 0 in X. Without loss of generality, we may assume that u = 0 in B 1 (0). Since X is compactly embedded into L 1 loc R N , V , we have that By (33), we have that Then, using (35) Finally, by (32) and (36), we get Thus the hylomorphy condition prevents the dispersion. As we have seen in the preceding section, (11) is also a fundamental assumption in proving the existence of hylomorphic solitons. Now, we assume E and C to be local quantities, namely, given u ∈ X, there exist the density functions ρ E,u (x) and ρ C,u (x) ∈ L 1 (R N ) such that Energy and hylenic densities ρ E,u , ρ C,u allow to define the density of binding energy as follows: − denotes the negative part of f. If u satisfies the hylomorphy condition, we have that E (u) < Λ 0 |C (u)| and hence he have that β u (t, x) = 0 for some x ∈ R N .
The support of the binding energy density is called bound matter region; more precisely we have the following definition Definition 27 Given any configuration u, we define the bound matter region as follows If u 0 is a soliton, the set Σ (u 0 ) is called support of the soliton at time t.
In the situation considered in this article, we will see that the solitons satisfy the hylomorphy condition. Thus we may think that a soliton u 0 consists of bound matter localized in a precise region of the space, namely Σ (u 0 ). This fact gives the name to this type of soliton from the Greek words "hyle"="matter " and "morphe"="form".

The swarm interpretation
Clearly the physical interpretation of hylomorphic solitons depends on the model which we are considering. However we can always assume a conventional interpretations which we will call swarm interpretation since the soliton is regarded as a swarm of particles bound together. This iterpretation is consitent with the model of the Q-ball.
We assume that u is a field which describes a fluid consisting of particles; the particles density is given by the function ρ C (t, x) = ρ C,u (t, x) which, of course satisfies a continuity equation where J C is the flow of particles. Hence C is the total number of particles. Here the particles are not intended to be as in "particle theory" but rather as in fluid dynamics, so that C does not need to be an integer number. Alternatively, if you like, you may think that C is not the number of particles but it is proportional to it. Also, in some equations as for example in NKG, C can be negative; in this case, the existence of antiparticles is assumed. Thus, the hylomorphy ratio represents the average energy of each particle (or antiparticle). The number Λ 0 defined in (12) is interpreted as the rest energy of each particle when they do not interact with each other. If Λ (u) > Λ 0 , then the average energy of each particle is bigger than the rest energy; if Λ (u) < Λ 0 , the opposite occurs and this fact means that particles act with each other with an attractive force. If the particles were at rest and they were not acting on each other, their energy density would be Λ 0 · |ρ C (t, x)| .
If ρ E (t, x) denotes the energy density and if then, in the point x at time t, the particles attract each other with a force which is stronger than the repulsive forces; this explains the name density of binding energy given to β(t, x) in (37). Thus a soliton relative to the state u can be considered as a "rigid" object occupying the region of space Σ (u) (cf. Def. 27); it consists of particles which stick to each other; the energy to destroy the soliton is given by

Basic features
The nonlinear Klein-Gordon equation for a complex valued field ψ, defined on the space-time R 4 , can be written as follows: and, with some abuse of notation, for some smooth function W : [0, ∞) → R. Hereafter x = (x 1 , x 2 , x 3 ) and t will denote the space and time variables. The field ψ : R 4 → C will be called matter field. If W ′ (s) is linear, W ′ (s) = m 2 s, m = 0, equation (39) reduces to the Klein-Gordon equation.
Consider the Abelian gauge theory in R 4 equipped with the Minkowski metric and described by the Lagrangian density (see e.g. [9], [38], [37]) where Here q denotes a positive parameter, ∇× and ∇ denote respectively the curl and the gradient operators with respect to the x variable, are the covariant derivatives and finally ϕ ∈ R and A= (A 1 , A 2 , A 3 ) ∈ R 3 are the gauge potentials. Now consider the total action Making the variation of S with respect to ψ, ϕ and A we get the following system of equations The abelian gauge theory, namely equations (43, 44, 45), provides a very elegant way to couple the Maxwell equation with matter if we interpret ψ as a matter field.
In order to give a more meaningful form to these equations, we will write ψ in polar form ψ(x, t) = u(x, t) e iS(x,t) , u ≥ 0, S ∈ R/2πZ So (42) takes the following form and the equations (43, 44, 45) take the form: In order to show the relation of the above equations with the Maxwell equations, we make the following change of variables: So (48) and (49) are the second couple of the Maxwell equations with respect to a matter distribution whose charge and current density are respectively ρ and j: (50) and (51) give rise to the first couple of the Maxwell equations Using the variables j and ρ, equation (46) can be written as follows and finally Equation (47) is the charge continuity equation Notice that equation (54) is a consequence of (gauss) and (ampere). In conclusion, an Abelian gauge theory, via equations (gauss,..,matter), provides a model of interaction of the matter field ψ with the electromagnetic field (E, H). In fact that equations (gauss,..,matter) are equivalent to (46,..,49).

Energy and charge
Let examine the invariants of NKGM which are relevant for us, namely the energy and the charge. In this subsection we compute these invariants using the gauge invariant variables u, ρ, j, E, H.
Energy. Energy, by definition, is the quantity which is preserved by the time invariance of the Lagrangian. Using the gauge invariant variables, the energy E calculated along the solutions of equation (gauss) takes the following form where Proof. By the Noether's theorem (see e.g. [28] or [6]), we have that, given the Lagrangian the density of energy takes the following form: Now we will compute each term. We have: Multiplying by ϕ equation (gauss) and integrating, we get − E·∇ϕ = ρϕ Thus, replacing this expression in the above formula, we get Moreover, using the notation (50, 51, 52, 53), we have that Then, by (56,...,59) and the above expression for L we get Charge. Using (54), we see that the electric charge has the following expression In order to be consistent with the previous literature ( [6], [4], [5], [12], [16], [15], [13], [17]), we will call charge the following quantity: C(u) is a dimensionless quantity which, in some interpretation of NKGM, represents the number of particles (see [20], [6], [17]). In some of the quoted papers, C(u) is called hylenic charge and hence the ratio (10) is called hylenic ratio.

Existence of charged Q-balls
We shall make the following assumptions on W : then (W-0) implies that the energy E in (55) is positive; if this condition does not hold, it is possible to have solitary waves, but not hylomorphic solitons.
(W-ii) is the crucial assumption which characterizes the nonlinearity which might produce hylomorphic solitons.
The hylomorphy condition (W-ii) can also be written as follows: (W-iii) if W and hence N is of class C 3 , then (66) reduces to |N ′ (s)| ≤ cs q−1 for q < 6, and this is the usual subcritical growth condition. Now we introduce the phase space X. First observe that the term ρ 2 + j 2 /u 2 in (55) is singular, so we introduce new gauge invariant variables which eliminate this singularity: Using these new variables the energy takes the form: The generic point in the phase space X is given by u = (u,û, θ, Θ, E, H) whereû = ∂ t u is considered as independent variable; the phase space is given by where H is the Hilbert space of the functions equipped with the norm defined by the quadratic part of the energy, namely: In these new variables the energy and the hylenic charge become two continuous functionals on X having the form Our equations (matter, gauss, ampere, faraday, nomonopole) become Remark 28 In the following we shall assume that the Cauchy problem for (NKGM) is well posed in X. Actually, in the literature there are few results relative to this problem (we know only [26], [32], [34]) and we do not know which are the assumptions that W should satisfy. Also we refer to [14] for a discussion and some partial results on this issue.
We have the following existence results.
The next two lemmas, whose proofs follow standard arguments, state that E satisfies the coercivity assumption (EC-3) and that both E and C satisfy the splitting property (EC-2).
Lemma 31 Let the assumptions of Theorem 29 be satisfied, then E defined by (70) satisfies (EC-3), namely for any sequence u n = (u n ,û n , θ n , Θ n , E n , H n ) in H such that E(u n ) → 0 (respectively E(u n ) bounded), we have u n → 0 (respectively u n bounded), where · is defined in (69).

Lemma 32
Let the assumptions of Theorem 29 be satisfied, then E and C satisfy the splitting property (EC-2).
It remains to prove that the hylomorphy condition (11) holds.

Analysis of the hylenic ratio
First of all we set: where r, q are introduced in (66). With some abuse of notation we shall write max ( u L r , u L q ) = u ♯ .
where T is defined in (4).
Proof. Let u n be a bounded sequence in H 1 R 3 such that, up to a subsequence, . We need to show that u n is not T − vanishing. May be taking a subsequence, we have that at least one of the following holds: Now suppose that (i) holds (If (ii) holds, we will argue in the same way replacing r with q).
We set for j ∈ Z 3 where Q is now the cube defined as follows Clearly Now let c be the constant for the Sobolev embedding H 1 (Q j ) ⊂ L t (Q j ) . We have 0 < a r ≤ |u n | r = j Qj where M and a are the constants respectively in (76) and (77). Then Then, for any n, there exists j n ∈ Z 3 such that Then, since (T jn u n ) (x) = u n (x + j n ) (see ( 3)), we have Since u n is bounded, also T jn u n is bounded in H 1 (R 3 ). Then we have, up to a subsequence, that T jn u n ⇀ u 0 weakly in H 1 (R 3 ) and hence strongly in L r (Q). By (79), u 0 = 0.
So we conclude that u n is not T − vanishing.
Now, as usual, we set By the definition of Λ 0 and Λ ♯ and lemma 33, we have that The following lemma holds: Lemma 34 Let W satisfy assumption (66), then the following inequality holds Proof. First of all observe that by (66) we have So, if we take u ♯ = 1 and ε > 0,we get By the Sobolev embeddings, there is k 3 > 0 such that Now, choose 2 < s < min(r, q).

Lemma 35
Assume that W satisfies (W-0),...,(W-iii) and (66) then Proof. We shall prove that So (87) will follow from (81) and (82) and (88). Let R > 0; set where R > 1. By the hylomorphy assumption (63) there exist α ∈ (0, m) such that Now let ϕ R ∈ D 1,2 denote the solution of the following equation We have Λ * = inf where the last inequality is a consequence of (90). In order to estimate the term containing ϕ R in (92), we remember that ϕ R is the solution of (91). Observe that u 2 R has radial symmetry and that the electric field outside any spherically symmetric charge distribution is the same as if all of the charge were concentrated into a point. So |∇ϕ R (r)| corresponds to the strength of an electrostatic field at distance r, created by an electric charge given by By (92), we get Λ * ≤ α + c 1 αR + c 6 q 2 αs 2 0 R 2 .
With these choices of R and q, a direct calculation shows that α + c 1 αR + c 6 q 2 αs 2 0 R 2 < m.
Then, by (94) and (95), we get that there exists a positive constant c such that, .
So all the assumptions of the Theorem 19 are satisfied and the conclusion follows.
From the above equations we clearly get (73). (74) is given by the constraint (97).